Datasets:

akjadhav
/

leandojo-lean4-formal-informal-strings-split

Dataset card Data Studio Files Files and versions Community

Split (3)

train · 47.4k rows

formal stringlengths 41 427k	informal stringclasses 1 value
Metric.isBounded_range_of_tendsto_cofinite_uniformity α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) ⊢ IsBounded (range f) rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with ⟨s, hsf, hs1⟩ case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} ⊢ IsBounded (range f) rw [← image_union_image_compl_eq_range] case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} ⊢ IsBounded (f '' ?m.498800 ∪ f '' ?m.498800ᶜ) α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} ⊢ Set β refine (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩) case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} x : β hx : x ∈ sᶜ y : β hy : y ∈ sᶜ ⊢ dist (f x) (f y) ≤ 1 exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩) ** Qed
CauchySeq.isBounded_range α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s t : Set α r : ℝ f : ℕ → α hf : CauchySeq f ⊢ Cauchy (map f Filter.cofinite) rwa [Nat.cofinite_eq_atTop] ** Qed
Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) have : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) := by rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left] exact fun x hx ↦ disjoint_left_comm.2 <\| tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx) α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x this : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) rcases ((((hasBasis_nhdsSet _).inf_principal _)).disjoint_iff ((basis_sets _).comap _)).1 this with ⟨U, ⟨hUo, hkU⟩, t, ht, hd⟩ case intro.intro.intro.intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t✝ : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x this : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) U : Set β hUo : IsOpen U hkU : k ⊆ U t : Set α ht : t ∈ cobounded α hd : Disjoint (U ∩ s) (f ⁻¹' id t) ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) refine ⟨U, hkU, hUo, (isBounded_compl_iff.2 ht).subset ?_⟩ case intro.intro.intro.intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t✝ : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x this : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) U : Set β hUo : IsOpen U hkU : k ⊆ U t : Set α ht : t ∈ cobounded α hd : Disjoint (U ∩ s) (f ⁻¹' id t) ⊢ f '' (U ∩ s) ⊆ tᶜ rwa [image_subset_iff, preimage_compl, subset_compl_iff_disjoint_right] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x ⊢ Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x ⊢ ∀ (x : β), x ∈ k → Disjoint (𝓝 x) (comap f (cobounded α) ⊓ 𝓟 s) exact fun x hx ↦ disjoint_left_comm.2 <\| tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx) ** Qed
Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s t : Set α r : ℝ inst✝ : TopologicalSpace β k : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousAt f x ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) simp_rw [← continuousWithinAt_univ] at hf α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s t : Set α r : ℝ inst✝ : TopologicalSpace β k : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f univ x ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) simpa only [inter_univ] using exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf ** Qed
Metric.isCompact_of_isClosed_isBounded α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s t : Set α r : ℝ inst✝ : ProperSpace α hc : IsClosed s hb : IsBounded s ⊢ IsCompact s rcases eq_empty_or_nonempty s with (rfl \| ⟨x, -⟩) case inl α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α t : Set α r : ℝ inst✝ : ProperSpace α hc : IsClosed ∅ hb : IsBounded ∅ ⊢ IsCompact ∅ exact isCompact_empty case inr.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x✝ : α s t : Set α r : ℝ inst✝ : ProperSpace α hc : IsClosed s hb : IsBounded s x : α ⊢ IsCompact s rcases hb.subset_closedBall x with ⟨r, hr⟩ case inr.intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x✝ : α s t : Set α r✝ : ℝ inst✝ : ProperSpace α hc : IsClosed s hb : IsBounded s x : α r : ℝ hr : s ⊆ closedBall x r ⊢ IsCompact s exact (isCompact_closedBall x r).of_isClosed_subset hc hr ** Qed
Metric.diam_subsingleton α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α hs : Set.Subsingleton s ⊢ diam s = 0 simp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal] ** Qed
Metric.diam_pair α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α ⊢ diam {x, y} = dist x y simp only [diam, EMetric.diam_pair, dist_edist] ** Qed
Metric.diam_triple α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α ⊢ diam {x, y, z} = max (max (dist x y) (dist x z)) (dist y z) simp only [Metric.diam, EMetric.diam_triple, dist_edist] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α ⊢ ENNReal.toReal (max (max (edist x y) (edist x z)) (edist y z)) = max (max (ENNReal.toReal (edist x y)) (ENNReal.toReal (edist x z))) (ENNReal.toReal (edist y z)) rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt] ** Qed
Metric.dist_le_diam_of_mem' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : EMetric.diam s ≠ ⊤ hx : x ∈ s hy : y ∈ s ⊢ dist x y ≤ diam s rw [diam, dist_edist] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : EMetric.diam s ≠ ⊤ hx : x ∈ s hy : y ∈ s ⊢ ENNReal.toReal (edist x y) ≤ ENNReal.toReal (EMetric.diam s) rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : EMetric.diam s ≠ ⊤ hx : x ∈ s hy : y ∈ s ⊢ edist x y ≤ EMetric.diam s exact EMetric.edist_le_diam_of_mem hx hy ** Qed
Metric.ediam_univ_eq_top_iff_noncompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α s : Set α x y z : α inst✝ : ProperSpace α ⊢ EMetric.diam univ = ⊤ ↔ NoncompactSpace α rw [← not_compactSpace_iff, compactSpace_iff_isBounded_univ, isBounded_iff_ediam_ne_top, Classical.not_not] ** Qed
Metric.diam_univ_of_noncompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α s : Set α x y z : α inst✝¹ : ProperSpace α inst✝ : NoncompactSpace α ⊢ diam univ = 0 simp [diam] ** Qed
Metric.diam_eq_zero_of_unbounded α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : ¬IsBounded s ⊢ diam s = 0 rw [diam, ediam_of_unbounded h, ENNReal.top_toReal] ** Qed
Metric.diam_union α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ diam (s ∪ t) ≤ diam s + dist x y + diam t simp only [diam, dist_edist] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ ENNReal.toReal (EMetric.diam (s ∪ t)) ≤ ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (edist x y) + ENNReal.toReal (EMetric.diam t) refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans (add_le_add_right ENNReal.toReal_add_le _) case refine_1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ EMetric.diam s + edist x y = ⊤ → EMetric.diam (s ∪ t) = ⊤ simp only [ENNReal.add_eq_top, edist_ne_top, or_false] case refine_1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ EMetric.diam s = ⊤ → EMetric.diam (s ∪ t) = ⊤ exact fun h ↦ top_unique <\| h ▸ EMetric.diam_mono (subset_union_left _ _) case refine_2 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ EMetric.diam t = ⊤ → EMetric.diam (s ∪ t) = ⊤ exact fun h ↦ top_unique <\| h ▸ EMetric.diam_mono (subset_union_right _ _) ** Qed
Metric.diam_union' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α h : Set.Nonempty (s ∩ t) ⊢ diam (s ∪ t) ≤ diam s + diam t rcases h with ⟨x, ⟨xs, xt⟩⟩ case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x✝ y z : α t : Set α x : α xs : x ∈ s xt : x ∈ t ⊢ diam (s ∪ t) ≤ diam s + diam t simpa using diam_union xs xt ** Qed
Metric.diam_le_of_subset_closedBall ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α r : ℝ hr : 0 ≤ r h : s ⊆ closedBall x r a : α ha : a ∈ s b : α hb : b ∈ s ⊢ r + r = 2 * r simp [mul_two, mul_comm] Qed
IsComplete.nonempty_iInter_of_nonempty_biInter α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) ⊢ Set.Nonempty (⋂ n, s n) let u N := (h N).some α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) ⊢ Set.Nonempty (⋂ n, s n) have I : ∀ n N, n ≤ N → u N ∈ s n := by intro n N hn apply mem_of_subset_of_mem _ (h N).choose_spec intro x hx simp only [mem_iInter] at hx exact hx n hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n ⊢ Set.Nonempty (⋂ n, s n) have : CauchySeq u := by apply cauchySeq_of_le_tendsto_0 _ _ h' intro m n N hm hn exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn) α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u ⊢ Set.Nonempty (⋂ n, s n) obtain ⟨x, -, xlim⟩ : ∃ x ∈ s 0, Tendsto (fun n : ℕ => u n) atTop (𝓝 x) := cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) ⊢ Set.Nonempty (⋂ n, s n) refine' ⟨x, mem_iInter.2 fun n => _⟩ case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) n : ℕ ⊢ x ∈ s n apply (hs n).mem_of_tendsto xlim case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) n : ℕ ⊢ ∀ᶠ (x : ℕ) in atTop, u x ∈ s n filter_upwards [Ici_mem_atTop n] with p hp case h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) n p : ℕ hp : p ∈ Ici n ⊢ u p ∈ s n exact I n p hp α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) ⊢ ∀ (n N : ℕ), n ≤ N → u N ∈ s n intro n N hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N ⊢ u N ∈ s n apply mem_of_subset_of_mem _ (h N).choose_spec α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N ⊢ ⋂ n, ⋂ (_ : n ≤ N), s n ⊆ s n intro x hx α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N x : α hx : x ∈ ⋂ n, ⋂ (_ : n ≤ N), s n ⊢ x ∈ s n simp only [mem_iInter] at hx α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N x : α hx : ∀ (i : ℕ), i ≤ N → x ∈ s i ⊢ x ∈ s n exact hx n hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n ⊢ CauchySeq u apply cauchySeq_of_le_tendsto_0 _ _ h' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n ⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) ≤ diam (s N) intro m n N hm hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n m n N : ℕ hm : N ≤ m hn : N ≤ n ⊢ dist (u m) (u n) ≤ diam (s N) exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn) ** Qed
Metric.exists_isLocalMin_mem_ball α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝⁴ : PseudoMetricSpace α inst✝³ : ProperSpace α inst✝² : TopologicalSpace β inst✝¹ : ConditionallyCompleteLinearOrder β inst✝ : OrderTopology β f : α → β a z : α r : ℝ hf : ContinuousOn f (closedBall a r) hz : z ∈ closedBall a r hf1 : ∀ (z' : α), z' ∈ sphere a r → f z < f z' ⊢ ∃ z, z ∈ ball a r ∧ IsLocalMin f z simp_rw [← closedBall_diff_ball] at hf1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝⁴ : PseudoMetricSpace α inst✝³ : ProperSpace α inst✝² : TopologicalSpace β inst✝¹ : ConditionallyCompleteLinearOrder β inst✝ : OrderTopology β f : α → β a z : α r : ℝ hf : ContinuousOn f (closedBall a r) hz : z ∈ closedBall a r hf1 : ∀ (z' : α), z' ∈ closedBall a r \ ball a r → f z < f z' ⊢ ∃ z, z ∈ ball a r ∧ IsLocalMin f z exact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1 isOpen_ball ** Qed
Metric.cobounded_eq_cocompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α ⊢ cobounded α = cocompact α nontriviality α α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α ✝ : Nontrivial α ⊢ cobounded α = cocompact α inhabit α α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α ✝ : Nontrivial α inhabited_h : Inhabited α ⊢ cobounded α = cocompact α exact cobounded_le_cocompact.antisymm <\| (hasBasis_cobounded_compl_closedBall default).ge_iff.2 fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact ** Qed
comap_dist_left_atTop_eq_cocompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α x : α ⊢ comap (dist x) atTop = cocompact α simp [cobounded_eq_cocompact] ** Qed
MetricSpace.ext α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 m m' : MetricSpace α h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ m = m' cases m case mk α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 m' : MetricSpace α toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ mk eq_of_dist_eq_zero✝ = m' cases m' case mk.mk α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 toPseudoMetricSpace✝¹ : PseudoMetricSpace α eq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ mk eq_of_dist_eq_zero✝¹ = mk eq_of_dist_eq_zero✝ congr case mk.mk.e_toPseudoMetricSpace α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 toPseudoMetricSpace✝¹ : PseudoMetricSpace α eq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ toPseudoMetricSpace✝¹ = toPseudoMetricSpace✝ ext1 case mk.mk.e_toPseudoMetricSpace.h α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 toPseudoMetricSpace✝¹ : PseudoMetricSpace α eq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ PseudoMetricSpace.toDist = PseudoMetricSpace.toDist assumption ** Qed
zero_eq_dist α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ 0 = dist x y ↔ x = y rw [eq_comm, dist_eq_zero] ** Qed
dist_ne_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ dist x y ≠ 0 ↔ x ≠ y simpa only [not_iff_not] using dist_eq_zero ** Qed
dist_le_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ dist x y ≤ 0 ↔ x = y simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y ** Qed
dist_pos α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ 0 < dist x y ↔ x ≠ y simpa only [not_le] using not_congr dist_le_zero ** Qed
eq_of_nndist_eq_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ nndist x y = 0 → x = y simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero] ** Qed
nndist_eq_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ nndist x y = 0 ↔ x = y simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero] ** Qed
zero_eq_nndist α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ 0 = nndist x y ↔ x = y simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist] ** Qed
Metric.subsingleton_closedBall α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ r : ℝ hr : r ≤ 0 ⊢ Set.Subsingleton (closedBall x r) rcases hr.lt_or_eq with (hr \| rfl) case inl α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ r : ℝ hr✝ : r ≤ 0 hr : r < 0 ⊢ Set.Subsingleton (closedBall x r) rw [closedBall_eq_empty.2 hr] case inl α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ r : ℝ hr✝ : r ≤ 0 hr : r < 0 ⊢ Set.Subsingleton ∅ exact subsingleton_empty case inr α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ hr : 0 ≤ 0 ⊢ Set.Subsingleton (closedBall x 0) rw [closedBall_zero] case inr α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ hr : 0 ≤ 0 ⊢ Set.Subsingleton {x} exact subsingleton_singleton ** Qed
Metric.uniformEmbedding_iff' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α γ : Type w inst✝¹ : MetricSpace γ x : γ s : Set γ inst✝ : MetricSpace β f : γ → β ⊢ UniformEmbedding f ↔ (∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧ ∀ (δ : ℝ), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff] ** Qed
Metric.isClosed_of_pairwise_le_dist α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x : γ s✝ s : Set γ ε : ℝ hε : 0 < ε hs : Set.Pairwise s fun x y => ε ≤ dist x y ⊢ Set.Pairwise s fun x y => ¬(x, y) ∈ {p \| dist p.1 p.2 < ε} simpa using hs ** Qed
Metric.closedEmbedding_of_pairwise_le_dist α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α✝ γ : Type w inst✝² : MetricSpace γ x : γ s : Set γ α : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : DiscreteTopology α ε : ℝ hε : 0 < ε f : α → γ hf : Pairwise fun x y => ε ≤ dist (f x) (f y) ⊢ Pairwise fun x y => ¬(f x, f y) ∈ {p \| dist p.1 p.2 < ε} simpa using hf ** Qed
Metric.uniformEmbedding_bot_of_pairwise_le_dist α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x : γ s : Set γ β : Type u_3 ε : ℝ hε : 0 < ε f : β → α hf : Pairwise fun x y => ε ≤ dist (f x) (f y) ⊢ UniformSpace α infer_instance α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x : γ s : Set γ β : Type u_3 ε : ℝ hε : 0 < ε f : β → α hf : Pairwise fun x y => ε ≤ dist (f x) (f y) ⊢ Pairwise fun x y => ¬(f x, f y) ∈ {p \| dist p.1 p.2 < ε} simpa using hf ** Qed
Metric.finite_isBounded_inter_isClosed α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α γ : Type w inst✝² : MetricSpace γ x : γ s✝ : Set γ inst✝¹ : ProperSpace α K s : Set α inst✝ : DiscreteTopology ↑s hK : IsBounded K hs : IsClosed s ⊢ Set.Finite (K ∩ s) refine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure) case refine_1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α γ : Type w inst✝² : MetricSpace γ x : γ s✝ : Set γ inst✝¹ : ProperSpace α K s : Set α inst✝ : DiscreteTopology ↑s hK : IsBounded K hs : IsClosed s ⊢ IsCompact (closure K ∩ s) exact hK.isCompact_closure.inter_right hs case refine_2 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α γ : Type w inst✝² : MetricSpace γ x : γ s✝ : Set γ inst✝¹ : ProperSpace α K s : Set α inst✝ : DiscreteTopology ↑s hK : IsBounded K hs : IsClosed s ⊢ DiscreteTopology ↑(closure K ∩ s) exact DiscreteTopology.of_subset inferInstance (Set.inter_subset_right _ s) ** Qed
MetricSpace.replaceUniformity_eq α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : UniformSpace γ m : MetricSpace γ H : 𝓤 γ = 𝓤 γ ⊢ replaceUniformity m H = m ext case h.dist.h.h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : UniformSpace γ m : MetricSpace γ H : 𝓤 γ = 𝓤 γ x✝¹ x✝ : γ ⊢ dist x✝¹ x✝ = dist x✝¹ x✝ rfl ** Qed
MetricSpace.replaceTopology_eq α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : TopologicalSpace γ m : MetricSpace γ H : U = UniformSpace.toTopologicalSpace ⊢ replaceTopology m H = m ext case h.dist.h.h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : TopologicalSpace γ m : MetricSpace γ H : U = UniformSpace.toTopologicalSpace x✝¹ x✝ : γ ⊢ dist x✝¹ x✝ = dist x✝¹ x✝ rfl ** Qed
MetricSpace.replaceBornology_eq α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α✝ γ : Type w inst✝ : MetricSpace γ α : Type u_3 m : MetricSpace α B : Bornology α H : ∀ (s : Set α), IsBounded s ↔ IsBounded s ⊢ replaceBornology m H = m ext case h.dist.h.h α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α✝ γ : Type w inst✝ : MetricSpace γ α : Type u_3 m : MetricSpace α B : Bornology α H : ∀ (s : Set α), IsBounded s ↔ IsBounded s x✝¹ x✝ : α ⊢ dist x✝¹ x✝ = dist x✝¹ x✝ rfl ** Qed
Metric.secondCountable_of_countable_discretization α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ⊢ SecondCountableTopology α refine secondCountable_of_almost_dense_set fun ε ε0 => ?_ α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 ⊢ ∃ s, Set.Countable s ∧ ∀ (x : α), ∃ y, y ∈ s ∧ dist x y ≤ ε rcases H ε ε0 with ⟨β, fβ, F, hF⟩ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε ⊢ ∃ s, Set.Countable s ∧ ∀ (x : α), ∃ y, y ∈ s ∧ dist x y ≤ ε let Finv := rangeSplitting F case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F ⊢ ∃ s, Set.Countable s ∧ ∀ (x : α), ∃ y, y ∈ s ∧ dist x y ≤ ε refine ⟨range Finv, ⟨countable_range _, fun x => ?_⟩⟩ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F x : α ⊢ ∃ y, y ∈ range Finv ∧ dist x y ≤ ε let x' := Finv ⟨F x, mem_range_self _⟩ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F x : α x' : α := Finv { val := F x, property := (_ : F x ∈ range F) } ⊢ ∃ y, y ∈ range Finv ∧ dist x y ≤ ε have : F x' = F x := apply_rangeSplitting F _ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F x : α x' : α := Finv { val := F x, property := (_ : F x ∈ range F) } this : F x' = F x ⊢ ∃ y, y ∈ range Finv ∧ dist x y ≤ ε exact ⟨x', mem_range_self _, hF _ _ this.symm⟩ ** Qed
TopologicalSpace.OpenNhds.map_id_obj_unop X Y : TopCat f : X ⟶ Y x : ↑X U : (OpenNhds x)ᵒᵖ ⊢ (map (𝟙 X) x).obj U.unop = U.unop simp ** Qed
TopologicalSpace.OpenNhds.op_map_id_obj X Y : TopCat f : X ⟶ Y x : ↑X U : (OpenNhds x)ᵒᵖ ⊢ (map (𝟙 X) x).op.obj U = U simp ** Qed
TopologicalSpace.Closeds.mem_iInf ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 x : α s : ι → Closeds α ⊢ x ∈ iInf s ↔ ∀ (i : ι), x ∈ s i simp [iInf] ** Qed
TopologicalSpace.Closeds.coe_iInf ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α ⊢ ↑(⨅ i, s i) = ⋂ i, ↑(s i) ext case h ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α x✝ : α ⊢ x✝ ∈ ↑(⨅ i, s i) ↔ x✝ ∈ ⋂ i, ↑(s i) simp ** Qed
TopologicalSpace.Closeds.iInf_def ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α ⊢ ⨅ i, s i = { carrier := ⋂ i, ↑(s i), closed' := (_ : IsClosed (⋂ i, ↑(s i))) } ext1 case h ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α ⊢ ↑(⨅ i, s i) = ↑{ carrier := ⋂ i, ↑(s i), closed' := (_ : IsClosed (⋂ i, ↑(s i))) } simp ** Qed
TopologicalSpace.Closeds.isAtom_iff ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α ⊢ IsAtom s ↔ ∃ x, s = singleton x have : IsAtom (s : Set α) ↔ IsAtom s := by refine' Closeds.gi.isAtom_iff' rfl (fun t ht => _) s obtain ⟨x, rfl⟩ := t.isAtom_iff.mp ht exact closure_singleton ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α this : IsAtom ↑s ↔ IsAtom s ⊢ IsAtom s ↔ ∃ x, s = singleton x simp only [← this, (s : Set α).isAtom_iff, SetLike.ext'_iff, Closeds.singleton_coe] ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α ⊢ IsAtom ↑s ↔ IsAtom s refine' Closeds.gi.isAtom_iff' rfl (fun t ht => _) s ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α t : Set α ht : IsAtom t ⊢ ↑(Closeds.closure t) = t obtain ⟨x, rfl⟩ := t.isAtom_iff.mp ht case intro ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α x : α ht : IsAtom {x} ⊢ ↑(Closeds.closure {x}) = {x} exact closure_singleton ** Qed
TopologicalSpace.Opens.isCoatom_iff ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Opens α ⊢ IsCoatom s ↔ ∃ x, s = Closeds.compl (Closeds.singleton x) rw [← s.compl_compl, ← isAtom_dual_iff_isCoatom] ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Opens α ⊢ IsAtom (↑toDual (Closeds.compl (compl s))) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x) change IsAtom (Closeds.complOrderIso α s.compl) ↔ _ ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Opens α ⊢ IsAtom (↑(Closeds.complOrderIso α) (compl s)) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x) simp only [(Closeds.complOrderIso α).isAtom_iff, Closeds.isAtom_iff, Closeds.compl_bijective.injective.eq_iff] ** Qed
Cube.insertAt_boundary N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I H : (t₀ = 0 ∨ t₀ = 1) ∨ t ∈ boundary { j // j ≠ i } ⊢ ↑(insertAt i) (t₀, t) ∈ boundary N obtain H \| ⟨j, H⟩ := H case inl N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I H : t₀ = 0 ∨ t₀ = 1 ⊢ ↑(insertAt i) (t₀, t) ∈ boundary N use i case h N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I H : t₀ = 0 ∨ t₀ = 1 ⊢ ↑(insertAt i) (t₀, t) i = 0 ∨ ↑(insertAt i) (t₀, t) i = 1 rwa [funSplitAt_symm_apply, dif_pos rfl] case inr.intro N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I j : { j // j ≠ i } H : t j = 0 ∨ t j = 1 ⊢ ↑(insertAt i) (t₀, t) ∈ boundary N use j case h N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I j : { j // j ≠ i } H : t j = 0 ∨ t j = 1 ⊢ ↑(insertAt i) (t₀, t) ↑j = 0 ∨ ↑(insertAt i) (t₀, t) ↑j = 1 rwa [funSplitAt_symm_apply, dif_neg j.prop, Subtype.coe_eta] ** Qed
GenLoop.copy_eq N : Type u_1 X : Type u_2 inst✝ : TopologicalSpace X x : X f : ↑(Ω^ N X x) g : (N → ↑I) → X h : g = ↑f ⊢ copy f g h = f ext x case H N : Type u_1 X : Type u_2 inst✝ : TopologicalSpace X x✝ : X f : ↑(Ω^ N X x✝) g : (N → ↑I) → X h : g = ↑f x : N → ↑I ⊢ ↑(copy f g h) x = ↑f x exact congr_fun h x ** Qed
GenLoop.to_from N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p : Ω (↑(Ω^ { j // j ≠ i } X x)) const ⊢ toLoop i (fromLoop i p) = p simp_rw [toLoop, fromLoop, ContinuousMap.comp_assoc, toContinuousMap_comp_symm, ContinuousMap.comp_id] N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p : Ω (↑(Ω^ { j // j ≠ i } X x)) const ⊢ { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }, source' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 0 = const), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 1 = const) } = p ext case a.h.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p : Ω (↑(Ω^ { j // j ≠ i } X x)) const x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑(↑{ toContinuousMap := ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }, source' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 0 = const), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 1 = const) } x✝) y✝ = ↑(↑p x✝) y✝ rfl ** Qed
GenLoop.homotopicTo N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) ⊢ Homotopic p q → Path.Homotopic (toLoop i p) (toLoop i q) refine' Nonempty.map fun H => ⟨⟨⟨fun t => ⟨homotopyTo i H t, _⟩, _⟩, _, _⟩, _⟩ case refine'_3 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (x_1 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_1) = ↑(toLoop i p).toContinuousMap x_1 case refine'_4 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (x_1 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_1) = ↑(toLoop i q).toContinuousMap x_1 case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 iterate 2 intro; ext; erw [homotopyTo_apply, toLoop_apply]; swap case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 intro t y yH case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} ⊢ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_2) = ↑(toLoop i p).toContinuousMap x_2), map_one_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_2) = ↑(toLoop i q).toContinuousMap x_2) }.toContinuousMap (t, x_1)) y = ↑(toLoop i p).toContinuousMap y ∧ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_2) = ↑(toLoop i p).toContinuousMap x_2), map_one_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_2) = ↑(toLoop i q).toContinuousMap x_2) }.toContinuousMap (t, x_1)) y = ↑(toLoop i q).toContinuousMap y constructor <;> ext <;> erw [homotopyTo_apply] case refine'_5.left.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i p).toContinuousMap y) y✝ case refine'_5.right.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i q).toContinuousMap y) y✝ apply H.eq_fst case refine'_5.left.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N case refine'_5.right.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i q).toContinuousMap y) y✝ on_goal 2 => apply H.eq_snd case refine'_5.left.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N case refine'_5.right.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N all_goals use i; rw [funSplitAt_symm_apply, dif_pos rfl]; exact yH case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I ⊢ ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x rintro y ⟨i, iH⟩ case refine'_1.intro N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(↑(homotopyTo i✝ H) t) y = x rw [homotopyTo_apply, H.eq_fst, p.2] case refine'_1.intro.a N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(Cube.insertAt i✝) (t.2, y) ∈ Cube.boundary N case refine'_1.intro.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(Cube.insertAt i✝) (t.2, y) ∈ Cube.boundary N all_goals apply Cube.insertAt_boundary; right; exact ⟨i, iH⟩ case refine'_1.intro.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(Cube.insertAt i✝) (t.2, y) ∈ Cube.boundary N apply Cube.insertAt_boundary case refine'_1.intro.hx.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ (t.2 = 0 ∨ t.2 = 1) ∨ y ∈ Cube.boundary { j // j ≠ i✝ } right case refine'_1.intro.hx.H.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ y ∈ Cube.boundary { j // j ≠ i✝ } exact ⟨i, iH⟩ case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ Continuous fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) } continuity case refine'_4 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (x_1 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_1) = ↑(toLoop i q).toContinuousMap x_1 case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 intro case refine'_4 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I ⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x✝) = ↑(toLoop i q).toContinuousMap x✝ case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 ext case refine'_4.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑(ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x✝)) y✝ = ↑(↑(toLoop i q).toContinuousMap x✝) y✝ case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 erw [homotopyTo_apply, toLoop_apply] case refine'_4.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((1, x✝).1, ↑(Cube.insertAt i) ((1, x✝).2, y✝)) = ↑q (↑(Cube.insertAt i) (x✝, y✝)) case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 swap case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) apply H.apply_zero case refine'_4.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((1, x✝).1, ↑(Cube.insertAt i) ((1, x✝).2, y✝)) = ↑q (↑(Cube.insertAt i) (x✝, y✝)) apply H.apply_one case refine'_5.right.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i q).toContinuousMap y) y✝ apply H.eq_snd case refine'_5.right.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N use i case h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) i = 0 ∨ ↑(Cube.insertAt i) ((t, y).2, y✝) i = 1 rw [funSplitAt_symm_apply, dif_pos rfl] case h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ((t, y).2, y✝).1 = 0 ∨ ((t, y).2, y✝).1 = 1 exact yH ** Qed
GenLoop.homotopicFrom N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) ⊢ Path.Homotopic (toLoop i p) (toLoop i q) → Homotopic p q refine' Nonempty.map fun H => ⟨⟨homotopyFrom i H, _, _⟩, _⟩ case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = ↑↑p x_1 case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 case refine'_3 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (t : ↑I) (x_1 : N → ↑I), x_1 ∈ Cube.boundary N → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑p x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑q x_1 pick_goal 3 case refine'_3 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (t : ↑I) (x_1 : N → ↑I), x_1 ∈ Cube.boundary N → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑p x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑q x_1 rintro t y ⟨j, jH⟩ case refine'_3.intro N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 ⊢ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_1)) y = ↑↑p y ∧ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_1)) y = ↑↑q y erw [homotopyFrom_apply] case refine'_3.intro N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 ⊢ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑p y ∧ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑q y case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = ↑↑p x_1 case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 obtain rfl \| h := eq_or_ne j i case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = ↑↑p x_1 case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 all_goals intro apply (homotopyFrom_apply _ _ _).trans first \| rw [H.apply_zero] \| rw [H.apply_one] first \| apply congr_arg p \| apply congr_arg q apply (Cube.splitAt i).left_inv case refine'_3.intro.inl N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑p y ∧ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑q y constructor case refine'_3.intro.inl.left N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑p y rw [H.eq_fst] case refine'_3.intro.inl.left N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑(toLoop j p).toContinuousMap ((t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑p y case refine'_3.intro.inl.left.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (t, y).2 j ∈ {0, 1} exacts [congr_arg p ((Cube.splitAt j).left_inv _), jH] case refine'_3.intro.inl.right N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑q y rw [H.eq_snd] case refine'_3.intro.inl.right N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑(toLoop j q).toContinuousMap ((t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑q y case refine'_3.intro.inl.right.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (t, y).2 j ∈ {0, 1} exacts [congr_arg q ((Cube.splitAt j).left_inv _), jH] case refine'_3.intro.inr N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 h : j ≠ i ⊢ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑p y ∧ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑q y rw [p.2 _ ⟨j, jH⟩, q.2 _ ⟨j, jH⟩] case refine'_3.intro.inr.right N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 h : j ≠ i ⊢ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = x apply boundary case refine'_3.intro.inr.right.a N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 h : j ≠ i ⊢ (fun j => (t, y).2 ↑j) ∈ Cube.boundary { j // ¬j = i } exact ⟨⟨j, h⟩, jH⟩ case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 intro case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ ContinuousMap.toFun (homotopyFrom i H) (1, x✝) = ↑↑q x✝ apply (homotopyFrom_apply _ _ _).trans case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑H ((1, x✝).1, (1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ first \| rw [H.apply_zero] \| rw [H.apply_one] case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑(toLoop i q).toContinuousMap ((1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ first \| apply congr_arg p \| apply congr_arg q case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ ↑(Homeomorph.toContinuousMap (Cube.insertAt i)) ((1, x✝).2 i, fun j => (1, x✝).2 ↑j) = x✝ apply (Cube.splitAt i).left_inv case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑H ((0, x✝).1, (0, x✝).2 i)) fun j => (0, x✝).2 ↑j) = ↑↑p x✝ rw [H.apply_zero] case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑H ((1, x✝).1, (1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ rw [H.apply_one] case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑(toLoop i p).toContinuousMap ((0, x✝).2 i)) fun j => (0, x✝).2 ↑j) = ↑↑p x✝ apply congr_arg p case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑(toLoop i q).toContinuousMap ((1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ apply congr_arg q ** Qed
GenLoop.transAt_distrib N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) ⊢ transAt i (transAt j a b) (transAt j c d) = transAt j (transAt i a c) (transAt i b d) ext case H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I ⊢ ↑(transAt i (transAt j a b) (transAt j c d)) y✝ = ↑(transAt j (transAt i a c) (transAt i b d)) y✝ simp_rw [transAt, coe_copy, Function.update_apply, if_neg h, if_neg h.symm] ** case neg N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 ⊢ ↑d (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) = ↑d (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) congr 1 case neg.h.e_6.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 ⊢ Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1)) = Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1)) ext1 case neg.h.e_6.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 x✝ : N ⊢ Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1)) x✝ = Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1)) x✝ simp only [Function.update, eq_rec_constant, dite_eq_ite] case neg.h.e_6.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 x✝ : N ⊢ (if x✝ = j then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1) else if x✝ = i then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1) else y✝ x✝) = if x✝ = i then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1) else if x✝ = j then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1) else y✝ x✝ apply ite_ite_comm case neg.h.e_6.h.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 x✝ : N ⊢ x✝ = j → ¬x✝ = i rintro rfl case neg.h.e_6.h.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 x✝ : N h : i ≠ x✝ h✝ : ¬↑(y✝ x✝) ≤ 1 / 2 ⊢ ¬x✝ = i exact h.symm Qed
HomotopyGroup.auxGroup_indep N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N ⊢ auxGroup i = auxGroup j by_cases h : i = j case neg N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j ⊢ auxGroup i = auxGroup j refine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _) case neg N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j ⊢ ∀ (a b c d : HomotopyGroup N X x), Mul.mul (Mul.mul a b) (Mul.mul c d) = Mul.mul (Mul.mul a c) (Mul.mul b d) rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩ case neg.mk.mk.mk.mk N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j a✝ : HomotopyGroup N X x a : ↑(Ω^ N X x) b✝ : HomotopyGroup N X x b : ↑(Ω^ N X x) c✝ : HomotopyGroup N X x c : ↑(Ω^ N X x) d✝ : HomotopyGroup N X x d : ↑(Ω^ N X x) ⊢ Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r b)) (Mul.mul (Quot.mk Setoid.r c) (Quot.mk Setoid.r d)) = Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d)) change Quotient.mk' _ = _ case neg.mk.mk.mk.mk N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j a✝ : HomotopyGroup N X x a : ↑(Ω^ N X x) b✝ : HomotopyGroup N X x b : ↑(Ω^ N X x) c✝ : HomotopyGroup N X x c : ↑(Ω^ N X x) d✝ : HomotopyGroup N X x d : ↑(Ω^ N X x) ⊢ Quotient.mk' (↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv d) (↑(loopHomeo j).toEquiv c)))) (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv b) (↑(loopHomeo j).toEquiv a)))))) = Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d)) apply congr_arg Quotient.mk' case neg.mk.mk.mk.mk N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j a✝ : HomotopyGroup N X x a : ↑(Ω^ N X x) b✝ : HomotopyGroup N X x b : ↑(Ω^ N X x) c✝ : HomotopyGroup N X x c : ↑(Ω^ N X x) d✝ : HomotopyGroup N X x d : ↑(Ω^ N X x) ⊢ ↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv d) (↑(loopHomeo j).toEquiv c)))) (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv b) (↑(loopHomeo j).toEquiv a))))) = ↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv (↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv d) (↑(loopHomeo i).toEquiv b)))) (↑(loopHomeo j).toEquiv (↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv c) (↑(loopHomeo i).toEquiv a))))) simp only [fromLoop_trans_toLoop, transAt_distrib h, coe_toEquiv, loopHomeo_apply, coe_symm_toEquiv, loopHomeo_symm_apply] case pos N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i = j ⊢ auxGroup i = auxGroup j rw [h] ** Qed
HomotopyGroup.transAt_indep N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f g : ↑(Ω^ N X x) ⊢ Quotient.mk (Homotopic.setoid N x) (transAt i f g) = Quotient.mk (Homotopic.setoid N x) (transAt j f g) simp_rw [← fromLoop_trans_toLoop] ** N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f g : ↑(Ω^ N X x) m : (G : Type ?u.862554) → Group G → G → G → G := fun G x x_1 x_2 => x_1 * x_2 ⊢ Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.trans (toLoop i f) (toLoop i g))) = Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.trans (toLoop j f) (toLoop j g))) exact congr_fun₂ (congr_arg (m <\| HomotopyGroup N X x) <\| auxGroup_indep i j) ⟦g⟧ ⟦f⟧ Qed
HomotopyGroup.symmAt_indep N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f : ↑(Ω^ N X x) ⊢ Quotient.mk (Homotopic.setoid N x) (symmAt i f) = Quotient.mk (Homotopic.setoid N x) (symmAt j f) simp_rw [← fromLoop_symm_toLoop] N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f : ↑(Ω^ N X x) inv : (G : Type ?u.864646) → Group G → G → G := fun G x x_1 => x_1⁻¹ ⊢ Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.symm (toLoop i f))) = Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.symm (toLoop j f))) exact congr_fun (congr_arg (inv <\| HomotopyGroup N X x) <\| auxGroup_indep i j) ⟦f⟧ ** Qed
HomotopyGroup.mul_spec ** N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ (fun x_1 x_2 => x_1 * x_2) (Quotient.mk (Homotopic.setoid N x) p) (Quotient.mk (Homotopic.setoid N x) q) = Quotient.mk (Homotopic.setoid N x) (transAt i q p) rw [transAt_indep _ q, ← fromLoop_trans_toLoop] N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ (fun x_1 x_2 => x_1 * x_2) (Quotient.mk (Homotopic.setoid N x) p) (Quotient.mk (Homotopic.setoid N x) q) = Quotient.mk (Homotopic.setoid N x) (fromLoop ?m.900147 (Path.trans (toLoop ?m.900147 q) (toLoop ?m.900147 p))) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ N apply Quotient.sound case a N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ ↑(loopHomeo (Classical.arbitrary N)).symm (Path.trans (↑(loopHomeo (Classical.arbitrary N)).toEquiv q) (↑(loopHomeo (Classical.arbitrary N)).toEquiv p)) ≈ fromLoop ?m.900147 (Path.trans (toLoop ?m.900147 q) (toLoop ?m.900147 p)) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ N rfl Qed
HomotopyGroup.inv_spec N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ (Quotient.mk (Homotopic.setoid N x) p)⁻¹ = Quotient.mk (Homotopic.setoid N x) (symmAt i p) rw [symmAt_indep _ p, ← fromLoop_symm_toLoop] N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ (Quotient.mk (Homotopic.setoid N x) p)⁻¹ = Quotient.mk (Homotopic.setoid N x) (fromLoop ?m.901306 (Path.symm (toLoop ?m.901306 p))) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ N apply Quotient.sound case a N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ ↑(loopHomeo (Classical.arbitrary N)).symm (Path.symm (↑(loopHomeo (Classical.arbitrary N)).toEquiv p)) ≈ fromLoop ?m.901306 (Path.symm (toLoop ?m.901306 p)) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ N rfl ** Qed
UniformSpaceCat.extension_comp_coe X : UniformSpaceCat Y : CpltSepUniformSpace f : toUniformSpace (CpltSepUniformSpace.of (Completion ↑X)) ⟶ toUniformSpace Y ⊢ extensionHom (completionHom X ≫ f) = f apply Subtype.eq case a X : UniformSpaceCat Y : CpltSepUniformSpace f : toUniformSpace (CpltSepUniformSpace.of (Completion ↑X)) ⟶ toUniformSpace Y ⊢ ↑(extensionHom (completionHom X ≫ f)) = ↑f funext x case a.h X : UniformSpaceCat Y : CpltSepUniformSpace f : toUniformSpace (CpltSepUniformSpace.of (Completion ↑X)) ⟶ toUniformSpace Y x : ↑(toUniformSpace (completionFunctor.obj X)) ⊢ ↑(extensionHom (completionHom X ≫ f)) x = ↑f x exact congr_fun (Completion.extension_comp_coe f.property) x ** Qed
Profinite.exists_clopen_of_cofiltered J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hB := TopCat.isTopologicalBasis_cofiltered_limit.{u, u} (F ⋙ Profinite.toTopCat) (Profinite.toTopCat.mapCone C) (isLimitOfPreserves _ hC) (fun j => {W \| IsClopen W}) ?_ (fun i => isClopen_univ) (fun i U1 U2 hU1 hU2 => hU1.inter hU2) ?_ case refine_3 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U hB : TopologicalSpace.IsTopologicalBasis {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W \| IsClopen W}) j) case refine_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)), V ∈ (fun j => {W \| IsClopen W}) j → ↑((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W \| IsClopen W}) i rotate_left case refine_3 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U hB : TopologicalSpace.IsTopologicalBasis {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V obtain ⟨S, hS, h⟩ := hB.open_eq_sUnion hU.1 case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U hB : TopologicalSpace.IsTopologicalBasis {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V clear hB case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let j : S → J := fun s => (hS s.2).choose case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let V : ∀ s : S, Set (F.obj (j s)) := fun s => (hS s.2).choose_spec.choose case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hV : ∀ s : S, IsClopen (V s) ∧ s.1 = C.π.app (j s) ⁻¹' V s := fun s => (hS s.2).choose_spec.choose_spec case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hUo : ∀ (i : ↑S), IsOpen ((fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hsU : U ⊆ ⋃ (i : ↑S), (fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have := hU.2.isCompact.elim_finite_subcover (fun s : S => C.π.app (j s) ⁻¹' V s) hUo hsU case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i this : ∃ t, U ⊆ ⋃ i ∈ t, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V obtain ⟨G, hG⟩ := this case refine_3.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists (G.image j) case refine_3.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let f : ∀ (s : S) (_ : s ∈ G), j0 ⟶ j s := fun s hs => (hj0 (Finset.mem_image.mpr ⟨s, hs, rfl⟩)).some case refine_3.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let W : S → Set (F.obj j0) := fun s => if hs : s ∈ G then F.map (f s hs) ⁻¹' V s else Set.univ case refine_3.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V refine' ⟨j0, ⋃ (s : S) (_ : s ∈ G), W s, _, _⟩ case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W \| IsClopen W}) j) intro i case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U i : J ⊢ TopologicalSpace.IsTopologicalBasis ((fun j => {W \| IsClopen W}) i) change TopologicalSpace.IsTopologicalBasis {W : Set (F.obj i) \| IsClopen W} case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U i : J ⊢ TopologicalSpace.IsTopologicalBasis {W \| IsClopen W} apply isTopologicalBasis_clopen case refine_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)), V ∈ (fun j => {W \| IsClopen W}) j → ↑((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W \| IsClopen W}) i rintro i j f V (hV : IsClopen _) case refine_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U i j : J f : i ⟶ j V : Set ↑((F ⋙ toTopCat).obj j) hV : IsClopen V ⊢ ↑((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W \| IsClopen W}) i exact ⟨hV.1.preimage ((F ⋙ toTopCat).map f).continuous, hV.2.preimage ((F ⋙ toTopCat).map f).continuous⟩ case hUo J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s ⊢ ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) intro s case hUo J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s s : ↑S ⊢ IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s) exact (hV s).1.1.preimage (C.π.app (j s)).continuous case hsU J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i dsimp only case hsU J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ U ⊆ ⋃ i, (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) rw [h] case hsU J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ ⋃₀ S ⊆ ⋃ i, (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) rintro x ⟨T, hT, hx⟩ case hsU.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) x : ↑C.pt.toCompHaus.toTop T : Set ↑(toTopCat.mapCone C).pt hT : T ∈ S hx : x ∈ T ⊢ x ∈ ⋃ i, (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) refine' ⟨_, ⟨⟨T, hT⟩, rfl⟩, _⟩ case hsU.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) x : ↑C.pt.toCompHaus.toTop T : Set ↑(toTopCat.mapCone C).pt hT : T ∈ S hx : x ∈ T ⊢ x ∈ (fun i => (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V)) { val := T, property := hT } dsimp only [forget_map_eq_coe] case hsU.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) x : ↑C.pt.toCompHaus.toTop T : Set ↑(toTopCat.mapCone C).pt hT : T ∈ S hx : x ∈ T ⊢ x ∈ ↑(C.π.app (Exists.choose (_ : T ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ T = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : T ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) rwa [← (hV ⟨T, hT⟩).2] case refine_3.intro.intro.intro.intro.refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ IsClopen (⋃ s ∈ G, W s) apply isClopen_biUnion_finset case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ ∀ (i : ↑S), i ∈ G → IsClopen (W i) intro s hs case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ s : ↑S hs : s ∈ G ⊢ IsClopen (W s) dsimp case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ s : ↑S hs : s ∈ G ⊢ IsClopen (if hs : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹' Exists.choose (_ : ∃ V, IsClopen V ∧ ↑s = ↑(C.π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ) rw [dif_pos hs] case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ s : ↑S hs : s ∈ G ⊢ IsClopen (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹' Exists.choose (_ : ∃ V, IsClopen V ∧ ↑s = ↑(C.π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V)) exact ⟨(hV s).1.1.preimage (F.map _).continuous, (hV s).1.2.preimage (F.map _).continuous⟩ case refine_3.intro.intro.intro.intro.refine'_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ U = ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s ext x case refine_3.intro.intro.intro.intro.refine'_2.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop ⊢ x ∈ U ↔ x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s constructor case refine_3.intro.intro.intro.intro.refine'_2.h.mp J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop ⊢ x ∈ U → x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s intro hx case refine_3.intro.intro.intro.intro.refine'_2.h.mp J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U ⊢ x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s simp_rw [Set.preimage_iUnion, Set.mem_iUnion] case refine_3.intro.intro.intro.intro.refine'_2.h.mp J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U ⊢ ∃ i i_1, x ∈ ↑(C.π.app j0) ⁻¹' if h : i ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ obtain ⟨_, ⟨s, rfl⟩, _, ⟨hs, rfl⟩, hh⟩ := hG hx case refine_3.intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U s : ↑S hs : s ∈ G hh : x ∈ (fun h => (fun s => ↑(C.π.app (j s)) ⁻¹' V s) s) hs ⊢ ∃ i i_1, x ∈ ↑(C.π.app j0) ⁻¹' if h : i ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ refine' ⟨s, hs, _⟩ case refine_3.intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U s : ↑S hs : s ∈ G hh : x ∈ (fun h => (fun s => ↑(C.π.app (j s)) ⁻¹' V s) s) hs ⊢ x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ rwa [dif_pos hs, ← Set.preimage_comp, ← Profinite.coe_comp, ← Functor.map_comp, C.w] case refine_3.intro.intro.intro.intro.refine'_2.h.mpr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop ⊢ x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s → x ∈ U intro hx case refine_3.intro.intro.intro.intro.refine'_2.h.mpr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s ⊢ x ∈ U simp_rw [Set.preimage_iUnion, Set.mem_iUnion] at hx case refine_3.intro.intro.intro.intro.refine'_2.h.mpr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : ∃ i i_1, x ∈ ↑(C.π.app j0) ⁻¹' if h : i ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ U obtain ⟨s, hs, hx⟩ := hx case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ U rw [h] case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ ⋃₀ S refine' ⟨s.1, s.2, _⟩ case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ ↑s rw [(hV s).2] case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ ↑(C.π.app (j s)) ⁻¹' V s rwa [dif_pos hs, ← Set.preimage_comp, ← Profinite.coe_comp, ← Functor.map_comp, C.w] at hx ** Qed
Profinite.exists_locallyConstant_fin_two J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let U := f ⁻¹' {0} J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g have hU : IsClopen U := f.isLocallyConstant.isClopen_fiber _ J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j, V, hV, h⟩ := exists_clopen_of_cofiltered C hC hU case intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g use j, LocallyConstant.ofClopen hV case h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ f = LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.ofClopen hV) apply LocallyConstant.locallyConstant_eq_of_fiber_zero_eq case h.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ↑f ⁻¹' {0} = ↑(LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.ofClopen hV)) ⁻¹' {0} erw [LocallyConstant.coe_comap _ _ (C.π.app j).continuous] case h.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ↑f ⁻¹' {0} = ↑(LocallyConstant.ofClopen hV) ∘ ↑(C.π.app j) ⁻¹' {0} conv_rhs => rw [Set.preimage_comp] case h.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ↑f ⁻¹' {0} = ↑(C.π.app j) ⁻¹' (↑(LocallyConstant.ofClopen hV) ⁻¹' {0}) erw [LocallyConstant.ofClopen_fiber_zero hV, ← h] ** Qed
Profinite.exists_locallyConstant_finite_aux J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g cases nonempty_fintype α case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let ff := (f.map ι).flip case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g have hff := fun a : α => exists_locallyConstant_fin_two _ hC (ff a) case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) hff : ∀ (a : α), ∃ j g, ff a = LocallyConstant.comap (↑(C.π.app j)) g ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g choose j g h using hff case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let G : Finset J := Finset.univ.image j case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists G case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g have hj : ∀ a, j a ∈ (Finset.univ.image j : Finset J) := by intro a simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply] case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let fs : ∀ a : α, j0 ⟶ j a := fun a => (hj0 (hj a)).some case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let gg : α → LocallyConstant (F.obj j0) (Fin 2) := fun a => (g a).comap (F.map (fs _)) case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let ggg := LocallyConstant.unflip gg case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨j0, ggg, _⟩ case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j0)) ggg have : f.map ι = LocallyConstant.unflip (f.map ι).flip := by simp case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) ⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j0)) ggg rw [this] case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.comap (↑(C.π.app j0)) ggg clear this case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.comap (↑(C.π.app j0)) ggg have : LocallyConstant.comap (C.π.app j0) ggg = LocallyConstant.unflip (LocallyConstant.comap (C.π.app j0) ggg).flip := by simp case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.comap (↑(C.π.app j0)) ggg = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.comap (↑(C.π.app j0)) ggg rw [this] case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.comap (↑(C.π.app j0)) ggg = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) clear this case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) congr 1 case intro.intro.e_f J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.flip (LocallyConstant.map ι f) = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) ext1 a case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ LocallyConstant.flip (LocallyConstant.map ι f) a = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) a change ff a = _ case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ ff a = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) a rw [h] case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ LocallyConstant.comap (↑(C.π.app (j a))) (g a) = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) a dsimp case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ LocallyConstant.comap (↑(C.π.app (j a))) (g a) = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) (LocallyConstant.unflip fun a => LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))) a ext1 x case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(LocallyConstant.comap (↑(C.π.app (j a))) (g a)) x = ↑(LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) (LocallyConstant.unflip fun a => LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))) a) x erw [LocallyConstant.coe_comap _ _ (C.π.app (j a)).continuous] case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ (↑(g a) ∘ ↑(C.π.app (j a))) x = ↑(LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) (LocallyConstant.unflip fun a => LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))) a) x dsimp [LocallyConstant.flip, LocallyConstant.unflip] case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = ↑(LocallyConstant.comap ↑(C.π.app j0) { toFun := fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x, isLocallyConstant := (_ : IsLocallyConstant fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x) }) x a erw [LocallyConstant.coe_comap _ _ (C.π.app j0).continuous] case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = (↑{ toFun := fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x, isLocallyConstant := (_ : IsLocallyConstant fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x) } ∘ ↑(C.π.app j0)) x a dsimp case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) (↑(C.π.app j0) x) rw [LocallyConstant.coe_comap _ _ _] J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) ⊢ ∀ (a : α), j a ∈ Finset.image j Finset.univ intro a J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) a : α ⊢ j a ∈ Finset.image j Finset.univ simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply] J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) simp J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.comap (↑(C.π.app j0)) ggg = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) simp case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = (↑(g a) ∘ ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (↑(C.π.app j0) x) dsimp case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = ↑(g a) (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) (↑(C.π.app j0) x)) congr! 1 case intro.intro.e_f.h.h.h.e'_6.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt e_2✝ : ↑(F.obj (j a)).toCompHaus.toTop = (forget Profinite).obj (F.obj (j a)) ⊢ ↑(C.π.app (j a)) x = ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) (↑(C.π.app j0) x) change _ = (C.π.app j0 ≫ F.map (fs a)) x case intro.intro.e_f.h.h.h.e'_6.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt e_2✝ : ↑(F.obj (j a)).toCompHaus.toTop = (forget Profinite).obj (F.obj (j a)) ⊢ ↑(C.π.app (j a)) x = ↑(C.π.app j0 ≫ F.map (fs a)) x rw [C.w] case intro.intro.e_f.h.h.h.e'_6.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt e_2✝ : ↑(F.obj (j a)).toCompHaus.toTop = (forget Profinite).obj (F.obj (j a)) ⊢ ↑(C.π.app (j a)) x = ↑(C.π.app (j a)) x rfl J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ Continuous ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) exact (F.map _).continuous ** Qed
Profinite.exists_locallyConstant_finite_nonempty J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g inhabit α J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j, gg, h⟩ := exists_locallyConstant_finite_aux _ hC f case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let σ : (α → Fin 2) → α := fun f => if h : ∃ a : α, ι a = f then h.choose else default case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨j, gg.map σ, _⟩ case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default ⊢ f = LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.map σ gg) ext x case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = ↑(LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.map σ gg)) x erw [LocallyConstant.coe_comap _ _ (C.π.app j).continuous] case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = (↑(LocallyConstant.map σ gg) ∘ ↑(C.π.app j)) x dsimp case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg (↑(C.π.app j) x) then Exists.choose h else default have h1 : ι (f x) = gg (C.π.app j x) := by change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _ erw [h, LocallyConstant.coe_comap _ _ (C.π.app j).continuous] rfl case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) ⊢ ↑f x = if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg (↑(C.π.app j) x) then Exists.choose h else default have h2 : ∃ a : α, ι a = gg (C.π.app j x) := ⟨f x, h1⟩ case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ↑f x = if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg (↑(C.π.app j) x) then Exists.choose h else default erw [dif_pos h2] case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ↑f x = Exists.choose h2 apply_fun ι J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ι (↑f x) = ↑gg (↑(C.π.app j) x) change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _ J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑(LocallyConstant.map (fun a b => if a = b then 0 else 1) f) x = ↑gg (↑(C.π.app j) x) erw [h, LocallyConstant.coe_comap _ _ (C.π.app j).continuous] J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ (↑gg ∘ ↑(C.π.app j)) x = ↑gg (↑(C.π.app j) x) rfl case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ι (↑f x) = ι (Exists.choose h2) rw [h2.choose_spec] case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ι (↑f x) = ↑gg (↑(C.π.app j) x) exact h1 case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ Function.Injective ι intro a b hh case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b ⊢ a = b have hhh := congr_fun hh a case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hhh : ι a a = ι b a ⊢ a = b dsimp at hhh case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hhh : (if a = a then 0 else 1) = if b = a then 0 else 1 ⊢ a = b rw [if_pos rfl] at hhh case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hhh : 0 = if b = a then 0 else 1 ⊢ a = b split_ifs at hhh with hh1 case pos J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hh1 : b = a hhh : 0 = 0 ⊢ a = b exact hh1.symm case neg J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hh1 : ¬b = a hhh : 0 = 1 ⊢ a = b exact False.elim (bot_ne_top hhh) ** Qed
Profinite.exists_locallyConstant J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let S := f.discreteQuotient J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let ff : S → α := f.lift J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g cases isEmpty_or_nonempty S case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) ⊢ ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop simp only [← not_nonempty_iff, ← not_forall] case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) ⊢ ¬∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop intro h case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop ⊢ False haveI : ∀ j : J, Nonempty ((F ⋙ Profinite.toTopCat).obj j) := h case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) ⊢ False haveI : ∀ j : J, T2Space ((F ⋙ Profinite.toTopCat).obj j) := fun j => (inferInstance : T2Space (F.obj j)) case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) ⊢ False haveI : ∀ j : J, CompactSpace ((F ⋙ Profinite.toTopCat).obj j) := fun j => (inferInstance : CompactSpace (F.obj j)) case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) ⊢ False have cond := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} (F ⋙ Profinite.toTopCat) case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt ⊢ False suffices : Nonempty C.pt case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝² : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝¹ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt this : Nonempty ↑C.pt.toCompHaus.toTop ⊢ False case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt ⊢ Nonempty ↑C.pt.toCompHaus.toTop exact IsEmpty.false (S.proj this.some) case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt ⊢ Nonempty ↑C.pt.toCompHaus.toTop let D := Profinite.toTopCat.mapCone C case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C ⊢ Nonempty ↑C.pt.toCompHaus.toTop have hD : IsLimit D := isLimitOfPreserves Profinite.toTopCat hC case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C hD : IsLimit D ⊢ Nonempty ↑C.pt.toCompHaus.toTop have CD := (hD.conePointUniqueUpToIso (TopCat.limitConeIsLimit.{u, u} _)).inv case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C hD : IsLimit D CD : (TopCat.limitCone (F ⋙ toTopCat)).pt ⟶ D.pt ⊢ Nonempty ↑C.pt.toCompHaus.toTop exact cond.map CD J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' this.imp fun j hj => _ J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop ⊢ ∃ g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨⟨hj.elim, fun A => _⟩, _⟩ case refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α ⊢ IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A) suffices : (fun a ↦ IsEmpty.elim hj a) ⁻¹' A = ∅ case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α ⊢ (fun a => IsEmpty.elim hj a) ⁻¹' A = ∅ exact @Set.eq_empty_of_isEmpty _ hj _ case refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this✝ : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α this : (fun a => IsEmpty.elim hj a) ⁻¹' A = ∅ ⊢ IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A) rw [this] case refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this✝ : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α this : (fun a => IsEmpty.elim hj a) ⁻¹' A = ∅ ⊢ IsOpen ∅ exact isOpen_empty case refine'_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop ⊢ f = LocallyConstant.comap ↑(C.π.app j) { toFun := fun a => IsEmpty.elim hj a, isLocallyConstant := (_ : ∀ (A : Set α), IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A)) } ext x case refine'_2.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = ↑(LocallyConstant.comap ↑(C.π.app j) { toFun := fun a => IsEmpty.elim hj a, isLocallyConstant := (_ : ∀ (A : Set α), IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A)) }) x exact hj.elim' (C.π.app j x) case inr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let f' : LocallyConstant C.pt S := ⟨S.proj, S.proj_isLocallyConstant⟩ case inr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j, g', hj⟩ := exists_locallyConstant_finite_nonempty _ hC f' case inr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) hj : f' = LocallyConstant.comap (↑(C.π.app j)) g' ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨j, ⟨ff ∘ g', g'.isLocallyConstant.comp _⟩, _⟩ case inr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) hj : f' = LocallyConstant.comap (↑(C.π.app j)) g' ⊢ f = LocallyConstant.comap ↑(C.π.app j) { toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) } ext1 t case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) hj : f' = LocallyConstant.comap (↑(C.π.app j)) g' t : ↑C.pt.toCompHaus.toTop ⊢ ↑f t = ↑(LocallyConstant.comap ↑(C.π.app j) { toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) }) t apply_fun fun e => e t at hj case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : ↑f' t = ↑(LocallyConstant.comap (↑(C.π.app j)) g') t ⊢ ↑f t = ↑(LocallyConstant.comap ↑(C.π.app j) { toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) }) t erw [LocallyConstant.coe_comap _ _ (C.π.app j).continuous] at hj ⊢ case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : ↑f' t = (↑g' ∘ ↑(C.π.app j)) t ⊢ ↑f t = (↑{ toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) } ∘ ↑(C.π.app j)) t dsimp at hj ⊢ case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : DiscreteQuotient.proj (LocallyConstant.discreteQuotient f) t = ↑g' (↑(C.π.app j) t) ⊢ ↑f t = ↑(LocallyConstant.lift f) (↑g' (↑(C.π.app j) t)) rw [← hj] case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : DiscreteQuotient.proj (LocallyConstant.discreteQuotient f) t = ↑g' (↑(C.π.app j) t) ⊢ ↑f t = ↑(LocallyConstant.lift f) (DiscreteQuotient.proj (LocallyConstant.discreteQuotient f) t) rfl ** Qed
ContinuousMap.polynomial_comp_attachBound X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] ⊢ comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f) = ↑(↑(Polynomial.aeval f) g) ext case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] a✝ : X ⊢ ↑(comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f)) a✝ = ↑↑(↑(Polynomial.aeval f) g) a✝ simp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe, Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe, Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply] case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] a✝ : X ⊢ Polynomial.eval (↑(↑(attachBound ↑f) a✝)) g = Polynomial.eval (↑↑f a✝) g erw [ContinuousMap.attachBound_apply_coe] ** Qed
ContinuousMap.polynomial_comp_attachBound_mem X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] ⊢ comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f) ∈ A rw [polynomial_comp_attachBound] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] ⊢ ↑(↑(Polynomial.aeval f) g) ∈ A apply SetLike.coe_mem ** Qed
ContinuousMap.comp_attachBound_mem_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) ⊢ comp p (attachBound ↑f) ∈ Subalgebra.topologicalClosure A have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure := continuousMap_mem_polynomialFunctions_closure _ _ p X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ comp p (attachBound ↑f) ∈ Subalgebra.topologicalClosure A have frequently_mem_polynomials := mem_closure_iff_frequently.mp mem_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ comp p (attachBound ↑f) ∈ Subalgebra.topologicalClosure A apply mem_closure_iff_frequently.mpr X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ ∃ᶠ (x : C(X, ℝ)) in nhds (comp p (attachBound ↑f)), x ∈ ↑A refine' ((compRightContinuousMap ℝ (attachBound (f : C(X, ℝ)))).continuousAt p).tendsto.frequently_map _ _ frequently_mem_polynomials X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ ∀ (x : C(↑(Set.Icc (-‖↑f‖) ‖↑f‖), ℝ)), x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) → ↑(compRightContinuousMap ℝ (attachBound ↑f)) x ∈ ↑A rintro _ ⟨g, ⟨-, rfl⟩⟩ case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) g : ℝ[X] ⊢ ↑(compRightContinuousMap ℝ (attachBound ↑f)) (↑↑(Polynomial.toContinuousMapOnAlgHom (Set.Icc (-‖f‖) ‖f‖)) g) ∈ ↑A simp only [SetLike.mem_coe, AlgHom.coe_toRingHom, compRightContinuousMap_apply, Polynomial.toContinuousMapOnAlgHom_apply] case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) g : ℝ[X] ⊢ comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f) ∈ A apply polynomial_comp_attachBound_mem ** Qed
ContinuousMap.abs_mem_subalgebra_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } ⊢ abs ↑f ∈ Subalgebra.topologicalClosure A let f' := attachBound (f : C(X, ℝ)) X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } f' : C(X, ↑(Set.Icc (-‖↑f‖) ‖↑f‖)) := attachBound ↑f ⊢ abs ↑f ∈ Subalgebra.topologicalClosure A let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => \|(x : ℝ)\| } X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } f' : C(X, ↑(Set.Icc (-‖↑f‖) ‖↑f‖)) := attachBound ↑f abs : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) := mk fun x => \|↑x\| ⊢ ContinuousMap.abs ↑f ∈ Subalgebra.topologicalClosure A change abs.comp f' ∈ A.topologicalClosure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } f' : C(X, ↑(Set.Icc (-‖↑f‖) ‖↑f‖)) := attachBound ↑f abs : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) := mk fun x => \|↑x\| ⊢ comp abs f' ∈ Subalgebra.topologicalClosure A apply comp_attachBound_mem_closure ** Qed
ContinuousMap.inf_mem_subalgebra_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ ↑f ⊓ ↑g ∈ Subalgebra.topologicalClosure A rw [inf_eq_half_smul_add_sub_abs_sub' ℝ] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ 2⁻¹ • (↑f + ↑g - \|↑g - ↑f\|) ∈ Subalgebra.topologicalClosure A refine' A.topologicalClosure.smul_mem (A.topologicalClosure.sub_mem (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property) (A.le_topologicalClosure g.property)) _) _ X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ \|↑g - ↑f\| ∈ Subalgebra.topologicalClosure A exact_mod_cast abs_mem_subalgebra_closure A _ ** Qed
ContinuousMap.inf_mem_closed_subalgebra X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑f ⊓ ↑g ∈ A convert inf_mem_subalgebra_closure A f g case h.e'_5 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ A = Subalgebra.topologicalClosure A apply SetLike.ext' case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑A = ↑(Subalgebra.topologicalClosure A) symm case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑(Subalgebra.topologicalClosure A) = ↑A erw [closure_eq_iff_isClosed] case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ IsClosed ↑A exact h ** Qed
ContinuousMap.sup_mem_subalgebra_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ ↑f ⊔ ↑g ∈ Subalgebra.topologicalClosure A rw [sup_eq_half_smul_add_add_abs_sub' ℝ] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ 2⁻¹ • (↑f + ↑g + \|↑g - ↑f\|) ∈ Subalgebra.topologicalClosure A refine' A.topologicalClosure.smul_mem (A.topologicalClosure.add_mem (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property) (A.le_topologicalClosure g.property)) _) _ X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ \|↑g - ↑f\| ∈ Subalgebra.topologicalClosure A exact_mod_cast abs_mem_subalgebra_closure A _ ** Qed
ContinuousMap.sup_mem_closed_subalgebra X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑f ⊔ ↑g ∈ A convert sup_mem_subalgebra_closure A f g case h.e'_5 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ A = Subalgebra.topologicalClosure A apply SetLike.ext' case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑A = ↑(Subalgebra.topologicalClosure A) symm case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑(Subalgebra.topologicalClosure A) = ↑A erw [closure_eq_iff_isClosed] case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ IsClosed ↑A exact h ** Qed
ContinuousMap.sublattice_closure_eq_top X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L ⊢ closure L = ⊤ apply eq_top_iff.mpr X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L ⊢ ⊤ ≤ closure L rintro f - X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ⊢ f ∈ closure L refine' Filter.Frequently.mem_closure ((Filter.HasBasis.frequently_iff Metric.nhds_basis_ball).mpr fun ε pos => _) X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ x, x ∈ Metric.ball f ε ∧ x ∈ L simp only [exists_prop, Metric.mem_ball] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ x, dist x f < ε ∧ x ∈ L by_cases nX : Nonempty X case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L case neg X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : ¬Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L swap case neg X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : ¬Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L exact ⟨nA.some, (dist_lt_iff pos).mpr fun x => False.elim (nX ⟨x⟩), nA.choose_spec⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L dsimp only [Set.SeparatesPointsStrongly] at sep case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L choose g hg w₁ w₂ using sep f case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y ⊢ ∃ x, dist x f < ε ∧ x ∈ L let U : X → X → Set X := fun x y => {z \| f z - ε < g x y z} case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ⊢ ∃ x, dist x f < ε ∧ x ∈ L let ys : ∀ _, Finset X := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ⊢ ∃ x, dist x f < ε ∧ x ∈ L let ys_w : ∀ x, ⋃ y ∈ ys x, U x y = ⊤ := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose_spec case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ⊢ ∃ x, dist x f < ε ∧ x ∈ L have ys_nonempty : ∀ x, (ys x).Nonempty := fun x => Set.nonempty_of_union_eq_top_of_nonempty _ _ nX (ys_w x) case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) ⊢ ∃ x, dist x f < ε ∧ x ∈ L let h : ∀ _, L := fun x => ⟨(ys x).sup' (ys_nonempty x) fun y => (g x y : C(X, ℝ)), Finset.sup'_mem _ sup_mem _ _ _ fun y _ => hg x y⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } ⊢ ∃ x, dist x f < ε ∧ x ∈ L have lt_h : ∀ x z, f z - ε < (h x : X → ℝ) z := by intro x z obtain ⟨y, ym, zm⟩ := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z dsimp simp only [Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff] exact ⟨y, ym, zm⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z ⊢ ∃ x, dist x f < ε ∧ x ∈ L have h_eq : ∀ x, (h x : X → ℝ) x = f x := by intro x; simp [w₁] case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x ⊢ ∃ x, dist x f < ε ∧ x ∈ L let W : ∀ _, Set X := fun x => {z \| (h x : X → ℝ) z < f z + ε} case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x ⊢ ∃ x, dist x f < ε ∧ x ∈ L let xs : Finset X := (CompactSpace.elim_nhds_subcover W W_nhd).choose case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) ⊢ ∃ x, dist x f < ε ∧ x ∈ L let xs_w : ⋃ x ∈ xs, W x = ⊤ := (CompactSpace.elim_nhds_subcover W W_nhd).choose_spec case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) ⊢ ∃ x, dist x f < ε ∧ x ∈ L have xs_nonempty : xs.Nonempty := Set.nonempty_of_union_eq_top_of_nonempty _ _ nX xs_w case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs ⊢ ∃ x, dist x f < ε ∧ x ∈ L let k : (L : Type _) := ⟨xs.inf' xs_nonempty fun x => (h x : C(X, ℝ)), Finset.inf'_mem _ inf_mem _ _ _ fun x _ => (h x).2⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } ⊢ ∃ x, dist x f < ε ∧ x ∈ L refine' ⟨k.1, _, k.2⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } ⊢ dist (↑k) f < ε rw [dist_lt_iff pos] case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } ⊢ ∀ (x : X), dist (↑↑k x) (↑f x) < ε intro z case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ dist (↑↑k z) (↑f z) < ε rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by intros; simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm]] case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑↑k z < ↑f z + ε ∧ ↑f z - ε < ↑↑k z fconstructor X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} ⊢ ∀ (x y : X), U x y ∈ 𝓝 y intro x y X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ U x y ∈ 𝓝 y refine' IsOpen.mem_nhds _ _ case refine'_1 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ IsOpen (U x y) apply isOpen_lt <;> continuity case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ y ∈ U x y rw [Set.mem_setOf_eq, w₂] case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ ↑f y - ε < ↑f y exact sub_lt_self _ pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } ⊢ ∀ (x z : X), ↑f z - ε < ↑↑(h x) z intro x z X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z : X ⊢ ↑f z - ε < ↑↑(h x) z obtain ⟨y, ym, zm⟩ := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z y : X ym : y ∈ fun i => i ∈ (ys x).val zm : z ∈ U x y ⊢ ↑f z - ε < ↑↑(h x) z dsimp case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z y : X ym : y ∈ fun i => i ∈ (ys x).val zm : z ∈ U x y ⊢ ↑f z - ε < ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z simp only [Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff] case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z y : X ym : y ∈ fun i => i ∈ (ys x).val zm : z ∈ U x y ⊢ ∃ b, b ∈ Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤) ∧ ↑f z - ε < ↑(g x b) z exact ⟨y, ym, zm⟩ X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z ⊢ ∀ (x : X), ↑↑(h x) x = ↑f x intro x X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z x : X ⊢ ↑↑(h x) x = ↑f x simp [w₁] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} ⊢ ∀ (x : X), W x ∈ 𝓝 x intro x X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ W x ∈ 𝓝 x refine' IsOpen.mem_nhds _ _ case refine'_1 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ IsOpen (W x) apply isOpen_lt (continuous_set_coe _ _) case refine'_1 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ Continuous fun b => ↑f b + ε continuity case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ x ∈ W x dsimp only [Set.mem_setOf_eq] case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) x < ↑f x + ε rw [h_eq] case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ ↑f x < ↑f x + ε exact lt_add_of_pos_right _ pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ∀ (a b ε : ℝ), dist a b < ε ↔ a < b + ε ∧ b - ε < a intros X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X a✝ b✝ ε✝ : ℝ ⊢ dist a✝ b✝ < ε✝ ↔ a✝ < b✝ + ε✝ ∧ b✝ - ε✝ < a✝ simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm] case pos.left X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑↑k z < ↑f z + ε dsimp case pos.left X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑(Finset.inf' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤)) xs_nonempty fun x => Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε simp only [Finset.inf'_lt_iff, ContinuousMap.inf'_apply] case pos.left X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ∃ i, i ∈ Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤) ∧ ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑f z - ε < ↑(g i x) z} = ⊤)) (_ : Finset.Nonempty (ys i)) fun y => g i y) z < ↑f z + ε exact Set.exists_set_mem_of_union_eq_top _ _ xs_w z case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑f z - ε < ↑↑k z dsimp case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑f z - ε < ↑(Finset.inf' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤)) xs_nonempty fun x => Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z simp only [Finset.lt_inf'_iff, ContinuousMap.inf'_apply] case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ∀ (i : X), i ∈ Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤) → ↑f z - ε < ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑f z - ε < ↑(g i x) z} = ⊤)) (_ : Finset.Nonempty (ys i)) fun y => g i y) z rintro x - case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z x : X ⊢ ↑f z - ε < ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z apply lt_h ** Qed
ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A ⊢ Subalgebra.topologicalClosure A = ⊤ apply SetLike.ext' case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A ⊢ ↑(Subalgebra.topologicalClosure A) = ↑⊤ let L := A.topologicalClosure case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A L : Subalgebra ℝ C(X, ℝ) := Subalgebra.topologicalClosure A ⊢ ↑(Subalgebra.topologicalClosure A) = ↑⊤ have n : Set.Nonempty (L : Set C(X, ℝ)) := ⟨(1 : C(X, ℝ)), A.le_topologicalClosure A.one_mem⟩ case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A L : Subalgebra ℝ C(X, ℝ) := Subalgebra.topologicalClosure A n : Set.Nonempty ↑L ⊢ ↑(Subalgebra.topologicalClosure A) = ↑⊤ convert sublattice_closure_eq_top (L : Set C(X, ℝ)) n (fun f fm g gm => inf_mem_closed_subalgebra L A.isClosed_topologicalClosure ⟨f, fm⟩ ⟨g, gm⟩) (fun f fm g gm => sup_mem_closed_subalgebra L A.isClosed_topologicalClosure ⟨f, fm⟩ ⟨g, gm⟩) (Subalgebra.SeparatesPoints.strongly (Subalgebra.separatesPoints_monotone A.le_topologicalClosure w)) case h.e'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A L : Subalgebra ℝ C(X, ℝ) := Subalgebra.topologicalClosure A n : Set.Nonempty ↑L ⊢ ↑(Subalgebra.topologicalClosure A) = closure ↑L simp ** Qed
ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : C(X, ℝ) ⊢ f ∈ Subalgebra.topologicalClosure A rw [subalgebra_topologicalClosure_eq_top_of_separatesPoints A w] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : C(X, ℝ) ⊢ f ∈ ⊤ simp ** Qed
ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ g, ‖↑g - f‖ < ε have w := mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f) X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∃ᶠ (x : C(X, ℝ)) in 𝓝 f, x ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε rw [Metric.nhds_basis_ball.frequently_iff] at w X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε obtain ⟨g, H, m⟩ := w ε pos case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A g : C(X, ℝ) H : g ∈ Metric.ball f ε m : g ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε rw [Metric.mem_ball, dist_eq_norm] at H case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A g : C(X, ℝ) H : ‖g - f‖ < ε m : g ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε exact ⟨⟨g, m⟩, H⟩ ** Qed
ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : X → ℝ c : Continuous f ε : ℝ pos : 0 < ε ⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuousMap_of_separatesPoints A w ⟨f, c⟩ ε pos case intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : X → ℝ c : Continuous f ε : ℝ pos : 0 < ε g : { x // x ∈ A } b : ‖↑g - mk f‖ < ε ⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε use g case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : X → ℝ c : Continuous f ε : ℝ pos : 0 < ε g : { x // x ∈ A } b : ‖↑g - mk f‖ < ε ⊢ ∀ (x : X), ‖↑↑g x - f x‖ < ε rwa [norm_lt_iff _ pos] at b ** Qed
Subalgebra.SeparatesPoints.isROrC_to_real 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra ⊢ SeparatesPoints (comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) intro x₁ x₂ hx 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) ∧ f x₁ ≠ f x₂ obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx case intro.intro.intro.intro 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) ∧ f x₁ ≠ f x₂ let F : C(X, 𝕜) := f - const _ (f x₂) case intro.intro.intro.intro 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) ∧ f x₁ ≠ f x₂ refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_normSq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩ 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) ⊢ ↑f x₂ • 1 = const X (↑f x₂) ext1 case h 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) a✝ : X ⊢ ↑(↑f x₂ • 1) a✝ = ↑(const X (↑f x₂)) a✝ simp only [coe_smul, coe_one, smul_apply, one_apply, Algebra.id.smul_eq_mul, mul_one, const_apply] case intro.intro.intro.intro.refine'_1 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ comp (ContinuousMap.mk ↑normSq) F ∈ ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) rw [SetLike.mem_coe, Subalgebra.mem_comap] case intro.intro.intro.intro.refine'_1 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ ↑(AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk ↑normSq) F) ∈ restrictScalars ℝ A.toSubalgebra convert (A.restrictScalars ℝ).mul_mem hFA (star_mem hFA : star F ∈ A) ** case h.e'_4 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ ↑(AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk ↑normSq) F) = F * star F ext1 case h.e'_4.h 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A a✝ : X ⊢ ↑(↑(AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk ↑normSq) F)) a✝ = ↑(F * star F) a✝ exact (IsROrC.mul_conj (K := 𝕜) _).symm case intro.intro.intro.intro.refine'_2 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₁ ≠ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₂ have : f x₁ - f x₂ ≠ 0 := sub_ne_zero.mpr hf case intro.intro.intro.intro.refine'_2 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A this : ↑f x₁ - ↑f x₂ ≠ 0 ⊢ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₁ ≠ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₂ simpa only [comp_apply, coe_sub, coe_const, sub_apply, coe_mk, sub_self, map_zero, Ne.def, normSq_eq_zero, const_apply] using this Qed
ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra ⊢ StarSubalgebra.topologicalClosure A = ⊤ rw [StarSubalgebra.eq_top_iff] 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra ⊢ ∀ (x : C(X, 𝕜)), x ∈ StarSubalgebra.topologicalClosure A let I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ofRealClm.compLeftContinuous ℝ X 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm ⊢ ∀ (x : C(X, 𝕜)), x ∈ StarSubalgebra.topologicalClosure A have key : LinearMap.range I ≤ (A.toSubmodule.restrictScalars ℝ).topologicalClosure := by let A₀ : Submodule ℝ C(X, ℝ) := (A.toSubmodule.restrictScalars ℝ).comap I have SW : A₀.topologicalClosure = ⊤ := haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real congr_arg Subalgebra.toSubmodule this rw [← Submodule.map_top, ← SW] have h₁ := A₀.topologicalClosure_map ((@ofRealClm 𝕜 _).compLeftContinuousCompact X) have h₂ := (A.toSubmodule.restrictScalars ℝ).map_comap_le I exact h₁.trans (Submodule.topologicalClosure_mono h₂) 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) ⊢ ∀ (x : C(X, 𝕜)), x ∈ StarSubalgebra.topologicalClosure A intro f 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) ⊢ f ∈ StarSubalgebra.topologicalClosure A let f_re : C(X, ℝ) := (⟨IsROrC.re, IsROrC.reClm.continuous⟩ : C(𝕜, ℝ)).comp f 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f ⊢ f ∈ StarSubalgebra.topologicalClosure A let f_im : C(X, ℝ) := (⟨IsROrC.im, IsROrC.imClm.continuous⟩ : C(𝕜, ℝ)).comp f 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f ⊢ f ∈ StarSubalgebra.topologicalClosure A have h_f_re : I f_re ∈ A.topologicalClosure := key ⟨f_re, rfl⟩ 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A ⊢ f ∈ StarSubalgebra.topologicalClosure A have h_f_im : I f_im ∈ A.topologicalClosure := key ⟨f_im, rfl⟩ 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A ⊢ f ∈ StarSubalgebra.topologicalClosure A have := A.topologicalClosure.add_mem h_f_re (A.topologicalClosure.smul_mem h_f_im IsROrC.I) 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ (StarSubalgebra.topologicalClosure A).toSubalgebra ⊢ f ∈ StarSubalgebra.topologicalClosure A rw [StarSubalgebra.mem_toSubalgebra] at this 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A ⊢ f ∈ StarSubalgebra.topologicalClosure A convert this case h.e'_4 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A ⊢ f = ↑I f_re + IsROrC.I • ↑I f_im ext case h.e'_4.h 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A a✝ : X ⊢ ↑f a✝ = ↑(↑I f_re + IsROrC.I • ↑I f_im) a✝ apply Eq.symm case h.e'_4.h.h 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A a✝ : X ⊢ ↑(↑I f_re + IsROrC.I • ↑I f_im) a✝ = ↑f a✝ simp [mul_comm IsROrC.I _] 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm ⊢ LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) let A₀ : Submodule ℝ C(X, ℝ) := (A.toSubmodule.restrictScalars ℝ).comap I 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) ⊢ LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) have SW : A₀.topologicalClosure = ⊤ := haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real congr_arg Subalgebra.toSubmodule this 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ ⊢ LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) rw [← Submodule.map_top, ← SW] 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ ⊢ Submodule.map I (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) have h₁ := A₀.topologicalClosure_map ((@ofRealClm 𝕜 _).compLeftContinuousCompact X) 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ h₁ : Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) A₀) ⊢ Submodule.map I (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) have h₂ := (A.toSubmodule.restrictScalars ℝ).map_comap_le I 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ h₁ : Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) A₀) h₂ : Submodule.map I (Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra))) ≤ Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra) ⊢ Submodule.map I (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) exact h₁.trans (Submodule.topologicalClosure_mono h₂) ** Qed
ContinuousMap.algHom_ext_map_X A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ φ = ψ suffices (⊤ : Subalgebra ℝ C(s, ℝ)) ≤ AlgHom.equalizer φ ψ from AlgHom.ext fun x => this (by trivial) A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ ⊤ ≤ AlgHom.equalizer φ ψ rw [← polynomialFunctions.topologicalClosure s] A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ Subalgebra.topologicalClosure (polynomialFunctions s) ≤ AlgHom.equalizer φ ψ exact Subalgebra.topologicalClosure_minimal (polynomialFunctions s) (polynomialFunctions.le_equalizer s φ ψ h) (isClosed_eq hφ hψ) A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) this : ⊤ ≤ AlgHom.equalizer φ ψ x : C(↑s, ℝ) ⊢ x ∈ ⊤ trivial ** Qed
ContinuousMap.starAlgHom_ext_map_X 𝕜 : Type u_1 A : Type u_2 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : Ring A inst✝⁴ : StarRing A inst✝³ : Algebra 𝕜 A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set 𝕜 inst✝ : CompactSpace ↑s φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ φ = ψ suffices (⊤ : StarSubalgebra 𝕜 C(s, 𝕜)) ≤ StarAlgHom.equalizer φ ψ from StarAlgHom.ext fun x => this mem_top 𝕜 : Type u_1 A : Type u_2 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : Ring A inst✝⁴ : StarRing A inst✝³ : Algebra 𝕜 A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set 𝕜 inst✝ : CompactSpace ↑s φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ ⊤ ≤ StarAlgHom.equalizer φ ψ rw [← polynomialFunctions.starClosure_topologicalClosure s] 𝕜 : Type u_1 A : Type u_2 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : Ring A inst✝⁴ : StarRing A inst✝³ : Algebra 𝕜 A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set 𝕜 inst✝ : CompactSpace ↑s φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ topologicalClosure (Subalgebra.starClosure (polynomialFunctions s)) ≤ StarAlgHom.equalizer φ ψ exact StarSubalgebra.topologicalClosure_minimal (polynomialFunctions.starClosure_le_equalizer s φ ψ h) (isClosed_eq hφ hψ) ** Qed
Profinite.effectiveEpiFamily_tfae α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_have 1 → 2 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_have 2 → 3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_have 3 → 1 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b tfae_3_to_1 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_finish case tfae_1_to_2 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B ⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) intro case tfae_1_to_2 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B ✝ : EffectiveEpiFamily X π ⊢ Epi (Sigma.desc π) infer_instance case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b intro e case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Epi (Sigma.desc π) ⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b rw [epi_iff_surjective] at e case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) ⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b let i : ∐ X ≅ finiteCoproduct X := (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _) case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) ⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b intro b case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) b : ↑B.toCompHaus.toTop ⊢ ∃ a x, ↑(π a) x = b obtain ⟨t, rfl⟩ := e b case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t let q := i.hom t case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t refine ⟨q.1, q.2, ?_⟩ case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t have : t = i.inv (i.hom t) case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t rw [this] case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t)) show _ = (i.inv ≫ Sigma.desc π) (i.hom t) case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) suffices : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π rw [Iso.inv_comp_eq] case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π apply colimit.hom_ext case this.w α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ∀ (j : Discrete α), colimit.ι (Discrete.functor X) j ≫ Sigma.desc π = colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π rintro ⟨a⟩ case this.w.mk α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α ⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π = colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app, colimit.comp_coconePointUniqueUpToIso_hom_assoc] case this.w.mk α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α ⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π ext case this.w.mk.w α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α x✝ : (forget Profinite).obj (X a) ⊢ ↑(π a) x✝ = ↑((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝ rfl case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑i.inv (↑i.hom t) show t = (i.hom ≫ i.inv) t case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑(i.hom ≫ i.inv) t simp only [i.hom_inv_id] case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑(𝟙 (∐ X)) t rfl case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this✝ : t = ↑i.inv (↑i.hom t) this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) rw [this] case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this✝ : t = ↑i.inv (↑i.hom t) this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑i.hom t) rfl case tfae_3_to_1 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b ⊢ (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π apply effectiveEpiFamily_of_jointly_surjective ** Qed
Path.Homotopy.eval_zero X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ ⊢ eval F 0 = p₀ ext t case a.h X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ t : ↑I ⊢ ↑(eval F 0) t = ↑p₀ t simp [eval] ** Qed
Path.Homotopy.eval_one X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ ⊢ eval F 1 = p₁ ext t case a.h X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ t : ↑I ⊢ ↑(eval F 1) t = ↑p₁ t simp [eval] ** Qed
Path.Homotopy.hcomp_apply ** X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ x : ↑I × ↑I ⊢ (if ↑x.2 ≤ 1 / 2 then extend (eval F x.1) (2 * ↑x.2) else extend (eval G x.1) (2 * ↑x.2 - 1)) = if h : ↑x.2 ≤ 1 / 2 then ↑(eval F x.1) { val := 2 * ↑x.2, property := (_ : 2 * ↑x.2 ∈ I) } else ↑(eval G x.1) { val := 2 * ↑x.2 - 1, property := (_ : 2 * ↑x.2 - 1 ∈ I) } split_ifs <;> exact Path.extend_extends _ _ Qed
Path.Homotopy.hcomp_half X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ t : ↑I ⊢ 0 ≤ 1 / 2 norm_num X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ t : ↑I ⊢ 1 / 2 ≤ 1 norm_num ** X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ t : ↑I ⊢ (if ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2 ≤ 1 / 2 then extend (eval F (t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).1) (2 * ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2) else extend (eval G (t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).1) (2 * ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2 - 1)) = x₁ norm_num Qed
Path.Homotopic.hpath_hext X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₂ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ HEq (Quotient.mk (Homotopic.setoid x₀ x₁) p₁) (Quotient.mk (Homotopic.setoid x₂ x₃) p₂) obtain rfl : x₀ = x₂ := by convert hp 0 <;> simp X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₀ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ HEq (Quotient.mk (Homotopic.setoid x₀ x₁) p₁) (Quotient.mk (Homotopic.setoid x₀ x₃) p₂) obtain rfl : x₁ = x₃ := by convert hp 1 <;> simp X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ HEq (Quotient.mk (Homotopic.setoid x₀ x₁) p₁) (Quotient.mk (Homotopic.setoid x₀ x₁) p₂) rw [heq_iff_eq] X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ Quotient.mk (Homotopic.setoid x₀ x₁) p₁ = Quotient.mk (Homotopic.setoid x₀ x₁) p₂ congr case e_a X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ p₁ = p₂ ext t case e_a.a.h X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t t : ↑I ⊢ ↑p₁ t = ↑p₂ t exact hp t X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₂ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ x₀ = x₂ convert hp 0 <;> simp X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₀ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ x₁ = x₃ convert hp 1 <;> simp ** Qed
ContinuousMap.homotopic_const_iff X : Type u Y : Type v inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X inst✝ : Nonempty Y ⊢ Homotopic (const Y x₀) (const Y x₁) ↔ Joined x₀ x₁ inhabit Y X : Type u Y : Type v inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X inst✝ : Nonempty Y inhabited_h : Inhabited Y ⊢ Homotopic (const Y x₀) (const Y x₁) ↔ Joined x₀ x₁ refine ⟨fun ⟨H⟩ ↦ ⟨⟨(H.toContinuousMap.comp .prodSwap).curry default, ?_, ?_⟩⟩, fun ⟨p⟩ ↦ ⟨p.toHomotopyConst⟩⟩ <;> simp ** Qed
TopCat.isTopologicalBasis_cofiltered_limit J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} let D := limitConeInfi F J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} let E : C.pt ≅ D.pt := hC.conePointUniqueUpToIso (limitConeInfiIsLimit _) J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} have hE : Inducing E.hom := (TopCat.homeoOfIso E).inducing J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} convert isTopologicalBasis_iInf hT fun j (x : D.pt) => D.π.app j x using 1 case h.e'_3 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom ⊢ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} = {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} ext U0 case h.e'_3.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} ↔ U0 ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} constructor J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} convert this.inducing hE case h.e'_3.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt ⊢ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} = Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} ext U0 case h.e'_3.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt U0 : Set ↑C.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} ↔ U0 ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} constructor case h.e'_3.h.h.mp J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt U0 : Set ↑C.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} → U0 ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} rintro ⟨j, V, hV, rfl⟩ case h.e'_3.h.h.mp.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt j : J V : Set ↑(F.obj j) hV : V ∈ T j ⊢ ↑(C.π.app j) ⁻¹' V ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} refine' ⟨D.π.app j ⁻¹' V, ⟨j, V, hV, rfl⟩, rfl⟩ case h.e'_3.h.h.mpr J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt U0 : Set ↑C.pt ⊢ U0 ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} → U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} rintro ⟨W, ⟨j, V, hV, rfl⟩, rfl⟩ case h.e'_3.h.h.mpr.intro.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt j : J V : Set ↑(F.obj j) hV : V ∈ T j ⊢ ↑E.hom ⁻¹' (↑(D.π.app j) ⁻¹' V) ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} refine' ⟨j, V, hV, rfl⟩ case h.e'_3.h.mp J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} → U0 ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} rintro ⟨j, V, hV, rfl⟩ case h.e'_3.h.mp.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j ⊢ ↑(D.π.app j) ⁻¹' V ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} let U : ∀ i, Set (F.obj i) := fun i => if h : i = j then by rw [h]; exact V else Set.univ case h.e'_3.h.mp.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ↑(D.π.app j) ⁻¹' V ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} refine' ⟨U, {j}, _, _⟩ J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j i : J h : i = j ⊢ Set ↑(F.obj i) rw [h] J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j i : J h : i = j ⊢ Set ↑(F.obj j) exact V case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ∀ (i : J), i ∈ {j} → U i ∈ T i rintro i h case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i ∈ {j} ⊢ U i ∈ T i rw [Finset.mem_singleton] at h case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i = j ⊢ U i ∈ T i dsimp case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i = j ⊢ (if h : i = j then cast (_ : Set ↑(F.obj j) = Set ↑(F.obj i)) V else Set.univ) ∈ T i rw [dif_pos h] case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i = j ⊢ cast (_ : Set ↑(F.obj j) = Set ↑(F.obj i)) V ∈ T i subst h case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom i : J V : Set ↑(F.obj i) hV : V ∈ T i U : (i : J) → Set ↑(F.obj i) := fun i_1 => if h : i_1 = i then Eq.mpr (_ : Set ↑(F.obj i_1) = Set ↑(F.obj i)) V else Set.univ ⊢ cast (_ : Set ↑(F.obj i) = Set ↑(F.obj i)) V ∈ T i exact hV case h.e'_3.h.mp.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ↑(D.π.app j) ⁻¹' V = ⋂ i ∈ {j}, (fun x => ↑(D.π.app i) x) ⁻¹' U i dsimp case h.e'_3.h.mp.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ↑((limitConeInfi F).π.app j) ⁻¹' V = ⋂ i ∈ {j}, (fun x => ↑((limitConeInfi F).π.app i) x) ⁻¹' if h : i = j then cast (_ : Set ↑(F.obj j) = Set ↑(F.obj i)) V else Set.univ simp case h.e'_3.h.mpr J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt ⊢ U0 ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} → U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} rintro ⟨U, G, h1, h2⟩ case h.e'_3.h.mpr.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} obtain ⟨j, hj⟩ := IsCofiltered.inf_objs_exists G case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} let g : ∀ (e) (_he : e ∈ G), j ⟶ e := fun _ he => (hj he).some case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} let Vs : J → Set (F.obj j) := fun e => if h : e ∈ G then F.map (g e h) ⁻¹' U e else Set.univ case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} let V : Set (F.obj j) := ⋂ (e : J) (_he : e ∈ G), Vs e case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} refine' ⟨j, V, _, _⟩ case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ V ∈ T j have : ∀ (S : Set (Set (F.obj j))) (E : Finset J) (P : J → Set (F.obj j)) (_univ : Set.univ ∈ S) (_inter : ∀ A B : Set (F.obj j), A ∈ S → B ∈ S → A ∩ B ∈ S) (_cond : ∀ (e : J) (_he : e ∈ E), P e ∈ S), (⋂ (e) (_he : e ∈ E), P e) ∈ S := by intro S E induction E using Finset.induction_on with \| empty => intro P he _hh simpa \| @insert a E _ha hh1 => intro hh2 hh3 hh4 hh5 rw [Finset.set_biInter_insert] refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _) intro e he exact hh5 e (Finset.mem_insert_of_mem he) case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S ⊢ V ∈ T j refine' this _ _ _ (univ _) (inter _) _ case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S ⊢ ∀ (e : J), e ∈ G → Vs e ∈ T j intro e he case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S e : J he : e ∈ G ⊢ Vs e ∈ T j dsimp case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S e : J he : e ∈ G ⊢ (if h : e ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j ⟶ e)))) ⁻¹' U e else Set.univ) ∈ T j rw [dif_pos he] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S e : J he : e ∈ G ⊢ ↑(F.map (Nonempty.some (_ : Nonempty (j ⟶ e)))) ⁻¹' U e ∈ T j exact compat j e (g e he) (U e) (h1 e he) J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S intro S E J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) E : Finset J ⊢ ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S induction E using Finset.induction_on with \| empty => intro P he _hh simpa \| @insert a E _ha hh1 => intro hh2 hh3 hh4 hh5 rw [Finset.set_biInter_insert] refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _) intro e he exact hh5 e (Finset.mem_insert_of_mem he) case empty J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) ⊢ ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ ∅ → P e ∈ S) → ⋂ e ∈ ∅, P e ∈ S intro P he _hh case empty J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) P : J → Set ↑(F.obj j) he : Set.univ ∈ S _hh : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S ⊢ (∀ (e : J), e ∈ ∅ → P e ∈ S) → ⋂ e ∈ ∅, P e ∈ S simpa case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S ⊢ ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ insert a E → P e ∈ S) → ⋂ e ∈ insert a E, P e ∈ S intro hh2 hh3 hh4 hh5 case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S ⊢ ⋂ e ∈ insert a E, hh2 e ∈ S rw [Finset.set_biInter_insert] case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S ⊢ hh2 a ∩ ⋂ x ∈ E, hh2 x ∈ S refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _) case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S ⊢ ∀ (e : J), e ∈ E → hh2 e ∈ S intro e he case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S e : J he : e ∈ E ⊢ hh2 e ∈ S exact hh5 e (Finset.mem_insert_of_mem he) case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ U0 = ↑(D.π.app j) ⁻¹' V rw [h2] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i = ↑(D.π.app j) ⁻¹' V change _ = (D.π.app j)⁻¹' ⋂ (e : J) (_ : e ∈ G), Vs e case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i = ↑(D.π.app j) ⁻¹' ⋂ e ∈ G, Vs e rw [Set.preimage_iInter] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i = ⋂ i, ↑(D.π.app j) ⁻¹' ⋂ (_ : i ∈ G), Vs i apply congrArg case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ (fun i => ⋂ (_ : i ∈ G), (fun x => ↑(D.π.app i) x) ⁻¹' U i) = fun i => ↑(D.π.app j) ⁻¹' ⋂ (_ : i ∈ G), Vs i ext1 e case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J ⊢ ⋂ (_ : e ∈ G), (fun x => ↑(D.π.app e) x) ⁻¹' U e = ↑(D.π.app j) ⁻¹' ⋂ (_ : e ∈ G), Vs e erw [Set.preimage_iInter] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J ⊢ ⋂ (_ : e ∈ G), (fun x => ↑(D.π.app e) x) ⁻¹' U e = ⋂ (_ : e ∈ G), ↑(D.π.app j) ⁻¹' Vs e apply congrArg case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J ⊢ (fun x => (fun x => ↑(D.π.app e) x) ⁻¹' U e) = fun i => ↑(D.π.app j) ⁻¹' Vs e ext1 he case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ (fun x => ↑(D.π.app e) x) ⁻¹' U e = ↑(D.π.app j) ⁻¹' Vs e change (D.π.app e)⁻¹' U e = (D.π.app j) ⁻¹' if h : e ∈ G then F.map (g e h) ⁻¹' U e else Set.univ case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) ⁻¹' U e = ↑(D.π.app j) ⁻¹' if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ rw [dif_pos he, ← Set.preimage_comp] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) ⁻¹' U e = ↑(F.map (g e he)) ∘ ↑(D.π.app j) ⁻¹' U e apply congrFun case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ Set.preimage ↑(D.π.app e) = Set.preimage (↑(F.map (g e he)) ∘ ↑(D.π.app j)) apply congrArg case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) = ↑(F.map (g e he)) ∘ ↑(D.π.app j) rw [←coe_comp, D.w] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) = ↑(D.π.app e) rfl ** Qed
TopCat.of_isoOfHomeo X Y : TopCat f : ↑X ≃ₜ ↑Y ⊢ homeoOfIso (isoOfHomeo f) = f dsimp [homeoOfIso, isoOfHomeo] X Y : TopCat f : ↑X ≃ₜ ↑Y ⊢ Homeomorph.mk { toFun := ↑(Homeomorph.toContinuousMap f), invFun := ↑(Homeomorph.toContinuousMap (Homeomorph.symm f)), left_inv := (_ : ∀ (x : ↑X), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 x) = x) } = f ext case H X Y : TopCat f : ↑X ≃ₜ ↑Y x✝ : ↑X ⊢ ↑(Homeomorph.mk { toFun := ↑(Homeomorph.toContinuousMap f), invFun := ↑(Homeomorph.toContinuousMap (Homeomorph.symm f)), left_inv := (_ : ∀ (x : ↑X), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 x) = x) }) x✝ = ↑f x✝ rfl ** Qed
TopCat.of_homeoOfIso X Y : TopCat f : X ≅ Y ⊢ isoOfHomeo (homeoOfIso f) = f dsimp [homeoOfIso, isoOfHomeo] X Y : TopCat f : X ≅ Y ⊢ Iso.mk (Homeomorph.toContinuousMap (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) })) (Homeomorph.toContinuousMap (Homeomorph.symm (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) }))) = f ext case w.w X Y : TopCat f : X ≅ Y x✝ : (forget TopCat).obj X ⊢ ↑(Iso.mk (Homeomorph.toContinuousMap (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) })) (Homeomorph.toContinuousMap (Homeomorph.symm (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) })))).hom x✝ = ↑f.hom x✝ rfl ** Qed
TopCat.openEmbedding_iff_comp_isIso' X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso g ⊢ OpenEmbedding ((forget TopCat).map f ≫ (forget TopCat).map g) ↔ OpenEmbedding ↑f simp only [←Functor.map_comp] X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso g ⊢ OpenEmbedding ((forget TopCat).map (f ≫ g)) ↔ OpenEmbedding ↑f exact openEmbedding_iff_comp_isIso f g ** Qed
TopCat.openEmbedding_iff_isIso_comp X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ↑(f ≫ g) ↔ OpenEmbedding ↑g constructor case mp X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ↑(f ≫ g) → OpenEmbedding ↑g intro h case mp X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f h : OpenEmbedding ↑(f ≫ g) ⊢ OpenEmbedding ↑g convert h.comp (TopCat.homeoOfIso (asIso f).symm).openEmbedding case h.e'_5.h X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f h : OpenEmbedding ↑(f ≫ g) e_1✝ : (forget TopCat).obj Y = ↑Y ⊢ ↑g = ↑(f ≫ g) ∘ ↑(homeoOfIso (asIso f).symm) exact congrArg _ (IsIso.inv_hom_id_assoc f g).symm case mpr X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ↑g → OpenEmbedding ↑(f ≫ g) exact fun h => h.comp (TopCat.homeoOfIso (asIso f)).openEmbedding ** Qed
TopCat.openEmbedding_iff_isIso_comp' X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ((forget TopCat).map f ≫ (forget TopCat).map g) ↔ OpenEmbedding ↑g simp only [←Functor.map_comp] X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ((forget TopCat).map (f ≫ g)) ↔ OpenEmbedding ↑g exact openEmbedding_iff_isIso_comp f g ** Qed
ContinuousMap.coe_injective α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f✝ g✝ f g : C(α, β) h : ↑f = ↑g ⊢ f = g cases f case mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g✝ g : C(α, β) toFun✝ : α → β continuous_toFun✝ : Continuous toFun✝ h : ↑(mk toFun✝) = ↑g ⊢ mk toFun✝ = g cases g case mk.mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) toFun✝¹ : α → β continuous_toFun✝¹ : Continuous toFun✝¹ toFun✝ : α → β continuous_toFun✝ : Continuous toFun✝ h : ↑(mk toFun✝¹) = ↑(mk toFun✝) ⊢ mk toFun✝¹ = mk toFun✝ congr ** Qed
ContinuousMap.cancel_left α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f✝ g : C(α, β) f : C(β, γ) g₁ g₂ : C(α, β) hf : Injective ↑f h : comp f g₁ = comp f g₂ a : α ⊢ ↑f (↑g₁ a) = ↑f (↑g₂ a) rw [← comp_apply, h, comp_apply] ** Qed
ContinuousMap.liftCover_coe α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x i : ι x : ↑(S i) ⊢ ↑(liftCover S φ hφ hS) ↑x = ↑(φ i) x rw [liftCover, coe_mk, Set.liftCover_coe _] ** Qed
ContinuousMap.liftCover_restrict α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x i : ι ⊢ restrict (S i) (liftCover S φ hφ hS) = φ i ext case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x i : ι a✝ : ↑(S i) ⊢ ↑(restrict (S i) (liftCover S φ hφ hS)) a✝ = ↑(φ i) a✝ simp only [coe_restrict, Function.comp_apply, liftCover_coe] ** Qed
ContinuousMap.liftCover_coe' α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x A : Set (Set α) F : (s : Set α) → s ∈ A → C(↑s, β) hF : ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t), ↑(F s hs) { val := x, property := hxi } = ↑(F t ht) { val := x, property := hxj } hA : ∀ (x : α), ∃ i, i ∈ A ∧ i ∈ nhds x s : Set α hs : s ∈ A x : ↑s x' : ↑↑{ val := s, property := hs } := x ⊢ ↑(liftCover' A F hF hA) ↑x = ↑(F s hs) x delta liftCover' α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x A : Set (Set α) F : (s : Set α) → s ∈ A → C(↑s, β) hF : ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t), ↑(F s hs) { val := x, property := hxi } = ↑(F t ht) { val := x, property := hxj } hA : ∀ (x : α), ∃ i, i ∈ A ∧ i ∈ nhds x s : Set α hs : s ∈ A x : ↑s x' : ↑↑{ val := s, property := hs } := x ⊢ ↑(let S := Subtype.val; let F_1 := fun i => F ↑i (_ : ↑i ∈ A); liftCover S F_1 (_ : ∀ (i j : ↑A) (x : α) (hxi : x ∈ ↑i) (hxj : x ∈ ↑j), ↑(F ↑i (_ : ↑i ∈ A)) { val := x, property := hxi } = ↑(F ↑j (_ : ↑j ∈ A)) { val := x, property := hxj }) (_ : ∀ (x : α), ∃ i, ↑i ∈ nhds x)) ↑x = ↑(F s hs) x exact liftCover_coe x' ** Qed
Homeomorph.symm_comp_toContinuousMap α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ f : α ≃ₜ β g : β ≃ₜ γ ⊢ ContinuousMap.comp (toContinuousMap (Homeomorph.symm f)) (toContinuousMap f) = ContinuousMap.id α rw [← coe_trans, self_trans_symm, coe_refl] ** Qed
Homeomorph.toContinuousMap_comp_symm α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ f : α ≃ₜ β g : β ≃ₜ γ ⊢ ContinuousMap.comp (toContinuousMap f) (toContinuousMap (Homeomorph.symm f)) = ContinuousMap.id β rw [← coe_trans, symm_trans_self, coe_refl] ** Qed
Stonean.effectiveEpiFamily_tfae α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 1 → 2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 1 → 2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 2 → 3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 3 → 1 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b tfae_3_to_1 : (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_finish case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B ⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) intro case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B ✝ : EffectiveEpiFamily X π ⊢ Epi (Sigma.desc π) infer_instance case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) intro case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ✝ : EffectiveEpiFamily X π ⊢ Epi (Sigma.desc π) infer_instance case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b intro e case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Epi (Sigma.desc π) ⊢ ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b rw [epi_iff_surjective] at e case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) ⊢ ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b intro b case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) b : CoeSort.coe B ⊢ ∃ a x, ↑(π a) x = b obtain ⟨t, rfl⟩ := e b case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t let q := (coproductIsoCoproduct X).inv t case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t refine ⟨q.1, q.2, ?_⟩ case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t rw [← (coproductIsoCoproduct X).inv_hom_id_apply t] case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑(coproductIsoCoproduct X).1 (↑(coproductIsoCoproduct X).2 t)) show _ = ((coproductIsoCoproduct X).hom ≫ Sigma.desc π) ((coproductIsoCoproduct X).inv t) case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ↑(π q.fst) q.snd = ↑((coproductIsoCoproduct X).hom ≫ Sigma.desc π) (↑(coproductIsoCoproduct X).inv t) suffices : (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π case this α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π apply Eq.symm case this.h α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ finiteCoproduct.desc X π = (coproductIsoCoproduct X).hom ≫ Sigma.desc π rw [← Iso.inv_comp_eq] case this.h α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ (coproductIsoCoproduct X).inv ≫ finiteCoproduct.desc X π = Sigma.desc π apply colimit.hom_ext case this.h.w α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ∀ (j : Discrete α), colimit.ι (Discrete.functor X) j ≫ (coproductIsoCoproduct X).inv ≫ finiteCoproduct.desc X π = colimit.ι (Discrete.functor X) j ≫ Sigma.desc π rintro ⟨a⟩ case this.h.w.mk α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t a : α ⊢ colimit.ι (Discrete.functor X) { as := a } ≫ (coproductIsoCoproduct X).inv ≫ finiteCoproduct.desc X π = colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app, coproductIsoCoproduct, colimit.comp_coconePointUniqueUpToIso_inv_assoc] case this.h.w.mk α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t a : α ⊢ (finiteCoproduct.explicitCocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π = π a ext case this.h.w.mk.w α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t a : α x✝ : (forget Stonean).obj (X a) ⊢ ↑((finiteCoproduct.explicitCocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝ = ↑(π a) x✝ rfl case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t this : (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑((coproductIsoCoproduct X).hom ≫ Sigma.desc π) (↑(coproductIsoCoproduct X).inv t) rw [this] case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t this : (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑(coproductIsoCoproduct X).inv t) rfl case tfae_3_to_1 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b ⊢ (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π apply effectiveEpiFamily_of_jointly_surjective ** Qed
CategoryTheory.EffectiveEpiFamily.toCompHaus α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B H : EffectiveEpiFamily X π ⊢ EffectiveEpiFamily (fun x => Stonean.toCompHaus.obj (X x)) fun x => Stonean.toCompHaus.map (π x) refine' ((CompHaus.effectiveEpiFamily_tfae _ _).out 0 2).2 (fun b => _) α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B H : EffectiveEpiFamily X π b : ↑(Stonean.toCompHaus.obj B).toTop ⊢ ∃ a x, ↑(Stonean.toCompHaus.map (π a)) x = b exact (((effectiveEpiFamily_tfae _ _).out 0 2).1 H : ∀ _, ∃ _, _) _ ** Qed

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.

formal stringlengths 41 427k	informal stringclasses 1 value
Metric.isBounded_range_of_tendsto_cofinite_uniformity α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) ⊢ IsBounded (range f) rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with ⟨s, hsf, hs1⟩ case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} ⊢ IsBounded (range f) rw [← image_union_image_compl_eq_range] case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} ⊢ IsBounded (f '' ?m.498800 ∪ f '' ?m.498800ᶜ) α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} ⊢ Set β refine (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩) case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ : α s✝ t : Set α r : ℝ f : β → α hf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α) s : Set β hsf : Set.Finite s hs1 : ∀ (x : β × β), x ∈ sᶜ ×ˢ sᶜ → Prod.map f f x ∈ {p \| dist p.1 p.2 < 1} x : β hx : x ∈ sᶜ y : β hy : y ∈ sᶜ ⊢ dist (f x) (f y) ≤ 1 exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩) ** Qed
CauchySeq.isBounded_range α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x : α s t : Set α r : ℝ f : ℕ → α hf : CauchySeq f ⊢ Cauchy (map f Filter.cofinite) rwa [Nat.cofinite_eq_atTop] ** Qed
Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) have : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) := by rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left] exact fun x hx ↦ disjoint_left_comm.2 <\| tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx) α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x this : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) rcases ((((hasBasis_nhdsSet _).inf_principal _)).disjoint_iff ((basis_sets _).comap _)).1 this with ⟨U, ⟨hUo, hkU⟩, t, ht, hd⟩ case intro.intro.intro.intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t✝ : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x this : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) U : Set β hUo : IsOpen U hkU : k ⊆ U t : Set α ht : t ∈ cobounded α hd : Disjoint (U ∩ s) (f ⁻¹' id t) ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) refine ⟨U, hkU, hUo, (isBounded_compl_iff.2 ht).subset ?_⟩ case intro.intro.intro.intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t✝ : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x this : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) U : Set β hUo : IsOpen U hkU : k ⊆ U t : Set α ht : t ∈ cobounded α hd : Disjoint (U ∩ s) (f ⁻¹' id t) ⊢ f '' (U ∩ s) ⊆ tᶜ rwa [image_subset_iff, preimage_compl, subset_compl_iff_disjoint_right] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x ⊢ Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s✝ t : Set α r : ℝ inst✝ : TopologicalSpace β k s : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f s x ⊢ ∀ (x : β), x ∈ k → Disjoint (𝓝 x) (comap f (cobounded α) ⊓ 𝓟 s) exact fun x hx ↦ disjoint_left_comm.2 <\| tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx) ** Qed
Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s t : Set α r : ℝ inst✝ : TopologicalSpace β k : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousAt f x ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) simp_rw [← continuousWithinAt_univ] at hf α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s t : Set α r : ℝ inst✝ : TopologicalSpace β k : Set β f : β → α hk : IsCompact k hf : ∀ (x : β), x ∈ k → ContinuousWithinAt f univ x ⊢ ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) simpa only [inter_univ] using exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf ** Qed
Metric.isCompact_of_isClosed_isBounded α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α s t : Set α r : ℝ inst✝ : ProperSpace α hc : IsClosed s hb : IsBounded s ⊢ IsCompact s rcases eq_empty_or_nonempty s with (rfl \| ⟨x, -⟩) case inl α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x : α t : Set α r : ℝ inst✝ : ProperSpace α hc : IsClosed ∅ hb : IsBounded ∅ ⊢ IsCompact ∅ exact isCompact_empty case inr.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x✝ : α s t : Set α r : ℝ inst✝ : ProperSpace α hc : IsClosed s hb : IsBounded s x : α ⊢ IsCompact s rcases hb.subset_closedBall x with ⟨r, hr⟩ case inr.intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x✝ : α s t : Set α r✝ : ℝ inst✝ : ProperSpace α hc : IsClosed s hb : IsBounded s x : α r : ℝ hr : s ⊆ closedBall x r ⊢ IsCompact s exact (isCompact_closedBall x r).of_isClosed_subset hc hr ** Qed
Metric.diam_subsingleton α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α hs : Set.Subsingleton s ⊢ diam s = 0 simp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal] ** Qed
Metric.diam_pair α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α ⊢ diam {x, y} = dist x y simp only [diam, EMetric.diam_pair, dist_edist] ** Qed
Metric.diam_triple α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α ⊢ diam {x, y, z} = max (max (dist x y) (dist x z)) (dist y z) simp only [Metric.diam, EMetric.diam_triple, dist_edist] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α ⊢ ENNReal.toReal (max (max (edist x y) (edist x z)) (edist y z)) = max (max (ENNReal.toReal (edist x y)) (ENNReal.toReal (edist x z))) (ENNReal.toReal (edist y z)) rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt] ** Qed
Metric.dist_le_diam_of_mem' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : EMetric.diam s ≠ ⊤ hx : x ∈ s hy : y ∈ s ⊢ dist x y ≤ diam s rw [diam, dist_edist] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : EMetric.diam s ≠ ⊤ hx : x ∈ s hy : y ∈ s ⊢ ENNReal.toReal (edist x y) ≤ ENNReal.toReal (EMetric.diam s) rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : EMetric.diam s ≠ ⊤ hx : x ∈ s hy : y ∈ s ⊢ edist x y ≤ EMetric.diam s exact EMetric.edist_le_diam_of_mem hx hy ** Qed
Metric.ediam_univ_eq_top_iff_noncompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α s : Set α x y z : α inst✝ : ProperSpace α ⊢ EMetric.diam univ = ⊤ ↔ NoncompactSpace α rw [← not_compactSpace_iff, compactSpace_iff_isBounded_univ, isBounded_iff_ediam_ne_top, Classical.not_not] ** Qed
Metric.diam_univ_of_noncompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α s : Set α x y z : α inst✝¹ : ProperSpace α inst✝ : NoncompactSpace α ⊢ diam univ = 0 simp [diam] ** Qed
Metric.diam_eq_zero_of_unbounded α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α h : ¬IsBounded s ⊢ diam s = 0 rw [diam, ediam_of_unbounded h, ENNReal.top_toReal] ** Qed
Metric.diam_union α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ diam (s ∪ t) ≤ diam s + dist x y + diam t simp only [diam, dist_edist] α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ ENNReal.toReal (EMetric.diam (s ∪ t)) ≤ ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (edist x y) + ENNReal.toReal (EMetric.diam t) refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans (add_le_add_right ENNReal.toReal_add_le _) case refine_1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ EMetric.diam s + edist x y = ⊤ → EMetric.diam (s ∪ t) = ⊤ simp only [ENNReal.add_eq_top, edist_ne_top, or_false] case refine_1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ EMetric.diam s = ⊤ → EMetric.diam (s ∪ t) = ⊤ exact fun h ↦ top_unique <\| h ▸ EMetric.diam_mono (subset_union_left _ _) case refine_2 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α xs : x ∈ s yt : y ∈ t ⊢ EMetric.diam t = ⊤ → EMetric.diam (s ∪ t) = ⊤ exact fun h ↦ top_unique <\| h ▸ EMetric.diam_mono (subset_union_right _ _) ** Qed
Metric.diam_union' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α t : Set α h : Set.Nonempty (s ∩ t) ⊢ diam (s ∪ t) ≤ diam s + diam t rcases h with ⟨x, ⟨xs, xt⟩⟩ case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x✝ y z : α t : Set α x : α xs : x ∈ s xt : x ∈ t ⊢ diam (s ∪ t) ≤ diam s + diam t simpa using diam_union xs xt ** Qed
Metric.diam_le_of_subset_closedBall ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s : Set α x y z : α r : ℝ hr : 0 ≤ r h : s ⊆ closedBall x r a : α ha : a ∈ s b : α hb : b ∈ s ⊢ r + r = 2 * r simp [mul_two, mul_comm] Qed
IsComplete.nonempty_iInter_of_nonempty_biInter α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) ⊢ Set.Nonempty (⋂ n, s n) let u N := (h N).some α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) ⊢ Set.Nonempty (⋂ n, s n) have I : ∀ n N, n ≤ N → u N ∈ s n := by intro n N hn apply mem_of_subset_of_mem _ (h N).choose_spec intro x hx simp only [mem_iInter] at hx exact hx n hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n ⊢ Set.Nonempty (⋂ n, s n) have : CauchySeq u := by apply cauchySeq_of_le_tendsto_0 _ _ h' intro m n N hm hn exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn) α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u ⊢ Set.Nonempty (⋂ n, s n) obtain ⟨x, -, xlim⟩ : ∃ x ∈ s 0, Tendsto (fun n : ℕ => u n) atTop (𝓝 x) := cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) ⊢ Set.Nonempty (⋂ n, s n) refine' ⟨x, mem_iInter.2 fun n => _⟩ case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) n : ℕ ⊢ x ∈ s n apply (hs n).mem_of_tendsto xlim case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) n : ℕ ⊢ ∀ᶠ (x : ℕ) in atTop, u x ∈ s n filter_upwards [Ici_mem_atTop n] with p hp case h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n this : CauchySeq u x : α xlim : Tendsto (fun n => u n) atTop (𝓝 x) n p : ℕ hp : p ∈ Ici n ⊢ u p ∈ s n exact I n p hp α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) ⊢ ∀ (n N : ℕ), n ≤ N → u N ∈ s n intro n N hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N ⊢ u N ∈ s n apply mem_of_subset_of_mem _ (h N).choose_spec α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N ⊢ ⋂ n, ⋂ (_ : n ≤ N), s n ⊆ s n intro x hx α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N x : α hx : x ∈ ⋂ n, ⋂ (_ : n ≤ N), s n ⊢ x ∈ s n simp only [mem_iInter] at hx α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x✝ y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) n N : ℕ hn : n ≤ N x : α hx : ∀ (i : ℕ), i ≤ N → x ∈ s i ⊢ x ∈ s n exact hx n hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n ⊢ CauchySeq u apply cauchySeq_of_le_tendsto_0 _ _ h' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n ⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) ≤ diam (s N) intro m n N hm hn α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α s✝ : Set α x y z : α s : ℕ → Set α h0 : IsComplete (s 0) hs : ∀ (n : ℕ), IsClosed (s n) h's : ∀ (n : ℕ), IsBounded (s n) h : ∀ (N : ℕ), Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n) h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0) u : ℕ → α := fun N => Set.Nonempty.some (_ : Set.Nonempty (⋂ n, ⋂ (_ : n ≤ N), s n)) I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n m n N : ℕ hm : N ≤ m hn : N ≤ n ⊢ dist (u m) (u n) ≤ diam (s N) exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn) ** Qed
Metric.exists_isLocalMin_mem_ball α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝⁴ : PseudoMetricSpace α inst✝³ : ProperSpace α inst✝² : TopologicalSpace β inst✝¹ : ConditionallyCompleteLinearOrder β inst✝ : OrderTopology β f : α → β a z : α r : ℝ hf : ContinuousOn f (closedBall a r) hz : z ∈ closedBall a r hf1 : ∀ (z' : α), z' ∈ sphere a r → f z < f z' ⊢ ∃ z, z ∈ ball a r ∧ IsLocalMin f z simp_rw [← closedBall_diff_ball] at hf1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝⁴ : PseudoMetricSpace α inst✝³ : ProperSpace α inst✝² : TopologicalSpace β inst✝¹ : ConditionallyCompleteLinearOrder β inst✝ : OrderTopology β f : α → β a z : α r : ℝ hf : ContinuousOn f (closedBall a r) hz : z ∈ closedBall a r hf1 : ∀ (z' : α), z' ∈ closedBall a r \ ball a r → f z < f z' ⊢ ∃ z, z ∈ ball a r ∧ IsLocalMin f z exact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1 isOpen_ball ** Qed
Metric.cobounded_eq_cocompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α ⊢ cobounded α = cocompact α nontriviality α α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α ✝ : Nontrivial α ⊢ cobounded α = cocompact α inhabit α α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α ✝ : Nontrivial α inhabited_h : Inhabited α ⊢ cobounded α = cocompact α exact cobounded_le_cocompact.antisymm <\| (hasBasis_cobounded_compl_closedBall default).ge_iff.2 fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact ** Qed
comap_dist_left_atTop_eq_cocompact α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α x : α ⊢ comap (dist x) atTop = cocompact α simp [cobounded_eq_cocompact] ** Qed
MetricSpace.ext α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 m m' : MetricSpace α h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ m = m' cases m case mk α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 m' : MetricSpace α toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ mk eq_of_dist_eq_zero✝ = m' cases m' case mk.mk α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 toPseudoMetricSpace✝¹ : PseudoMetricSpace α eq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ mk eq_of_dist_eq_zero✝¹ = mk eq_of_dist_eq_zero✝ congr case mk.mk.e_toPseudoMetricSpace α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 toPseudoMetricSpace✝¹ : PseudoMetricSpace α eq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ toPseudoMetricSpace✝¹ = toPseudoMetricSpace✝ ext1 case mk.mk.e_toPseudoMetricSpace.h α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α✝ α : Type u_3 toPseudoMetricSpace✝¹ : PseudoMetricSpace α eq_of_dist_eq_zero✝¹ : ∀ {x y : α}, dist x y = 0 → x = y toPseudoMetricSpace✝ : PseudoMetricSpace α eq_of_dist_eq_zero✝ : ∀ {x y : α}, dist x y = 0 → x = y h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist ⊢ PseudoMetricSpace.toDist = PseudoMetricSpace.toDist assumption ** Qed
zero_eq_dist α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ 0 = dist x y ↔ x = y rw [eq_comm, dist_eq_zero] ** Qed
dist_ne_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ dist x y ≠ 0 ↔ x ≠ y simpa only [not_iff_not] using dist_eq_zero ** Qed
dist_le_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ dist x y ≤ 0 ↔ x = y simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y ** Qed
dist_pos α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ 0 < dist x y ↔ x ≠ y simpa only [not_le] using not_congr dist_le_zero ** Qed
eq_of_nndist_eq_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ nndist x y = 0 → x = y simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero] ** Qed
nndist_eq_zero α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ nndist x y = 0 ↔ x = y simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero] ** Qed
zero_eq_nndist α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x y : γ ⊢ 0 = nndist x y ↔ x = y simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist] ** Qed
Metric.subsingleton_closedBall α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ r : ℝ hr : r ≤ 0 ⊢ Set.Subsingleton (closedBall x r) rcases hr.lt_or_eq with (hr \| rfl) case inl α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ r : ℝ hr✝ : r ≤ 0 hr : r < 0 ⊢ Set.Subsingleton (closedBall x r) rw [closedBall_eq_empty.2 hr] case inl α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ r : ℝ hr✝ : r ≤ 0 hr : r < 0 ⊢ Set.Subsingleton ∅ exact subsingleton_empty case inr α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ hr : 0 ≤ 0 ⊢ Set.Subsingleton (closedBall x 0) rw [closedBall_zero] case inr α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x✝ : γ s : Set γ x : γ hr : 0 ≤ 0 ⊢ Set.Subsingleton {x} exact subsingleton_singleton ** Qed
Metric.uniformEmbedding_iff' α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α γ : Type w inst✝¹ : MetricSpace γ x : γ s : Set γ inst✝ : MetricSpace β f : γ → β ⊢ UniformEmbedding f ↔ (∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧ ∀ (δ : ℝ), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff] ** Qed
Metric.isClosed_of_pairwise_le_dist α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x : γ s✝ s : Set γ ε : ℝ hε : 0 < ε hs : Set.Pairwise s fun x y => ε ≤ dist x y ⊢ Set.Pairwise s fun x y => ¬(x, y) ∈ {p \| dist p.1 p.2 < ε} simpa using hs ** Qed
Metric.closedEmbedding_of_pairwise_le_dist α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α✝ γ : Type w inst✝² : MetricSpace γ x : γ s : Set γ α : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : DiscreteTopology α ε : ℝ hε : 0 < ε f : α → γ hf : Pairwise fun x y => ε ≤ dist (f x) (f y) ⊢ Pairwise fun x y => ¬(f x, f y) ∈ {p \| dist p.1 p.2 < ε} simpa using hf ** Qed
Metric.uniformEmbedding_bot_of_pairwise_le_dist α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x : γ s : Set γ β : Type u_3 ε : ℝ hε : 0 < ε f : β → α hf : Pairwise fun x y => ε ≤ dist (f x) (f y) ⊢ UniformSpace α infer_instance α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ : Type w inst✝ : MetricSpace γ x : γ s : Set γ β : Type u_3 ε : ℝ hε : 0 < ε f : β → α hf : Pairwise fun x y => ε ≤ dist (f x) (f y) ⊢ Pairwise fun x y => ¬(f x, f y) ∈ {p \| dist p.1 p.2 < ε} simpa using hf ** Qed
Metric.finite_isBounded_inter_isClosed α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α γ : Type w inst✝² : MetricSpace γ x : γ s✝ : Set γ inst✝¹ : ProperSpace α K s : Set α inst✝ : DiscreteTopology ↑s hK : IsBounded K hs : IsClosed s ⊢ Set.Finite (K ∩ s) refine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure) case refine_1 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α γ : Type w inst✝² : MetricSpace γ x : γ s✝ : Set γ inst✝¹ : ProperSpace α K s : Set α inst✝ : DiscreteTopology ↑s hK : IsBounded K hs : IsClosed s ⊢ IsCompact (closure K ∩ s) exact hK.isCompact_closure.inter_right hs case refine_2 α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝³ : PseudoMetricSpace α γ : Type w inst✝² : MetricSpace γ x : γ s✝ : Set γ inst✝¹ : ProperSpace α K s : Set α inst✝ : DiscreteTopology ↑s hK : IsBounded K hs : IsClosed s ⊢ DiscreteTopology ↑(closure K ∩ s) exact DiscreteTopology.of_subset inferInstance (Set.inter_subset_right _ s) ** Qed
MetricSpace.replaceUniformity_eq α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : UniformSpace γ m : MetricSpace γ H : 𝓤 γ = 𝓤 γ ⊢ replaceUniformity m H = m ext case h.dist.h.h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : UniformSpace γ m : MetricSpace γ H : 𝓤 γ = 𝓤 γ x✝¹ x✝ : γ ⊢ dist x✝¹ x✝ = dist x✝¹ x✝ rfl ** Qed
MetricSpace.replaceTopology_eq α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : TopologicalSpace γ m : MetricSpace γ H : U = UniformSpace.toTopologicalSpace ⊢ replaceTopology m H = m ext case h.dist.h.h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α γ✝ : Type w inst✝ : MetricSpace γ✝ γ : Type u_3 U : TopologicalSpace γ m : MetricSpace γ H : U = UniformSpace.toTopologicalSpace x✝¹ x✝ : γ ⊢ dist x✝¹ x✝ = dist x✝¹ x✝ rfl ** Qed
MetricSpace.replaceBornology_eq α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α✝ γ : Type w inst✝ : MetricSpace γ α : Type u_3 m : MetricSpace α B : Bornology α H : ∀ (s : Set α), IsBounded s ↔ IsBounded s ⊢ replaceBornology m H = m ext case h.dist.h.h α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α✝ γ : Type w inst✝ : MetricSpace γ α : Type u_3 m : MetricSpace α B : Bornology α H : ∀ (s : Set α), IsBounded s ↔ IsBounded s x✝¹ x✝ : α ⊢ dist x✝¹ x✝ = dist x✝¹ x✝ rfl ** Qed
Metric.secondCountable_of_countable_discretization α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ⊢ SecondCountableTopology α refine secondCountable_of_almost_dense_set fun ε ε0 => ?_ α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 ⊢ ∃ s, Set.Countable s ∧ ∀ (x : α), ∃ y, y ∈ s ∧ dist x y ≤ ε rcases H ε ε0 with ⟨β, fβ, F, hF⟩ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε ⊢ ∃ s, Set.Countable s ∧ ∀ (x : α), ∃ y, y ∈ s ∧ dist x y ≤ ε let Finv := rangeSplitting F case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F ⊢ ∃ s, Set.Countable s ∧ ∀ (x : α), ∃ y, y ∈ s ∧ dist x y ≤ ε refine ⟨range Finv, ⟨countable_range _, fun x => ?_⟩⟩ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F x : α ⊢ ∃ y, y ∈ range Finv ∧ dist x y ≤ ε let x' := Finv ⟨F x, mem_range_self _⟩ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F x : α x' : α := Finv { val := F x, property := (_ : F x ∈ range F) } ⊢ ∃ y, y ∈ range Finv ∧ dist x y ≤ ε have : F x' = F x := apply_rangeSplitting F _ case intro.intro.intro α✝ : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝² : PseudoMetricSpace α✝ γ : Type w inst✝¹ : MetricSpace γ α : Type u inst✝ : MetricSpace α H : ∀ (ε : ℝ), ε > 0 → ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε ε : ℝ ε0 : ε > 0 β : Type u_3 fβ : Encodable β F : α → β hF : ∀ (x y : α), F x = F y → dist x y ≤ ε Finv : ↑(range F) → α := rangeSplitting F x : α x' : α := Finv { val := F x, property := (_ : F x ∈ range F) } this : F x' = F x ⊢ ∃ y, y ∈ range Finv ∧ dist x y ≤ ε exact ⟨x', mem_range_self _, hF _ _ this.symm⟩ ** Qed
TopologicalSpace.OpenNhds.map_id_obj_unop X Y : TopCat f : X ⟶ Y x : ↑X U : (OpenNhds x)ᵒᵖ ⊢ (map (𝟙 X) x).obj U.unop = U.unop simp ** Qed
TopologicalSpace.OpenNhds.op_map_id_obj X Y : TopCat f : X ⟶ Y x : ↑X U : (OpenNhds x)ᵒᵖ ⊢ (map (𝟙 X) x).op.obj U = U simp ** Qed
TopologicalSpace.Closeds.mem_iInf ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 x : α s : ι → Closeds α ⊢ x ∈ iInf s ↔ ∀ (i : ι), x ∈ s i simp [iInf] ** Qed
TopologicalSpace.Closeds.coe_iInf ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α ⊢ ↑(⨅ i, s i) = ⋂ i, ↑(s i) ext case h ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α x✝ : α ⊢ x✝ ∈ ↑(⨅ i, s i) ↔ x✝ ∈ ⋂ i, ↑(s i) simp ** Qed
TopologicalSpace.Closeds.iInf_def ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α ⊢ ⨅ i, s i = { carrier := ⋂ i, ↑(s i), closed' := (_ : IsClosed (⋂ i, ↑(s i))) } ext1 case h ι✝ : Type u_1 α : Type u_2 β : Type u_3 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β ι : Sort u_4 s : ι → Closeds α ⊢ ↑(⨅ i, s i) = ↑{ carrier := ⋂ i, ↑(s i), closed' := (_ : IsClosed (⋂ i, ↑(s i))) } simp ** Qed
TopologicalSpace.Closeds.isAtom_iff ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α ⊢ IsAtom s ↔ ∃ x, s = singleton x have : IsAtom (s : Set α) ↔ IsAtom s := by refine' Closeds.gi.isAtom_iff' rfl (fun t ht => _) s obtain ⟨x, rfl⟩ := t.isAtom_iff.mp ht exact closure_singleton ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α this : IsAtom ↑s ↔ IsAtom s ⊢ IsAtom s ↔ ∃ x, s = singleton x simp only [← this, (s : Set α).isAtom_iff, SetLike.ext'_iff, Closeds.singleton_coe] ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α ⊢ IsAtom ↑s ↔ IsAtom s refine' Closeds.gi.isAtom_iff' rfl (fun t ht => _) s ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α t : Set α ht : IsAtom t ⊢ ↑(Closeds.closure t) = t obtain ⟨x, rfl⟩ := t.isAtom_iff.mp ht case intro ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Closeds α x : α ht : IsAtom {x} ⊢ ↑(Closeds.closure {x}) = {x} exact closure_singleton ** Qed
TopologicalSpace.Opens.isCoatom_iff ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Opens α ⊢ IsCoatom s ↔ ∃ x, s = Closeds.compl (Closeds.singleton x) rw [← s.compl_compl, ← isAtom_dual_iff_isCoatom] ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Opens α ⊢ IsAtom (↑toDual (Closeds.compl (compl s))) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x) change IsAtom (Closeds.complOrderIso α s.compl) ↔ _ ι : Type u_1 α : Type u_2 β : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : T1Space α s : Opens α ⊢ IsAtom (↑(Closeds.complOrderIso α) (compl s)) ↔ ∃ x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x) simp only [(Closeds.complOrderIso α).isAtom_iff, Closeds.isAtom_iff, Closeds.compl_bijective.injective.eq_iff] ** Qed
Cube.insertAt_boundary N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I H : (t₀ = 0 ∨ t₀ = 1) ∨ t ∈ boundary { j // j ≠ i } ⊢ ↑(insertAt i) (t₀, t) ∈ boundary N obtain H \| ⟨j, H⟩ := H case inl N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I H : t₀ = 0 ∨ t₀ = 1 ⊢ ↑(insertAt i) (t₀, t) ∈ boundary N use i case h N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I H : t₀ = 0 ∨ t₀ = 1 ⊢ ↑(insertAt i) (t₀, t) i = 0 ∨ ↑(insertAt i) (t₀, t) i = 1 rwa [funSplitAt_symm_apply, dif_pos rfl] case inr.intro N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I j : { j // j ≠ i } H : t j = 0 ∨ t j = 1 ⊢ ↑(insertAt i) (t₀, t) ∈ boundary N use j case h N : Type u_1 inst✝ : DecidableEq N i : N t₀ : ↑I t : { j // j ≠ i } → ↑I j : { j // j ≠ i } H : t j = 0 ∨ t j = 1 ⊢ ↑(insertAt i) (t₀, t) ↑j = 0 ∨ ↑(insertAt i) (t₀, t) ↑j = 1 rwa [funSplitAt_symm_apply, dif_neg j.prop, Subtype.coe_eta] ** Qed
GenLoop.copy_eq N : Type u_1 X : Type u_2 inst✝ : TopologicalSpace X x : X f : ↑(Ω^ N X x) g : (N → ↑I) → X h : g = ↑f ⊢ copy f g h = f ext x case H N : Type u_1 X : Type u_2 inst✝ : TopologicalSpace X x✝ : X f : ↑(Ω^ N X x✝) g : (N → ↑I) → X h : g = ↑f x : N → ↑I ⊢ ↑(copy f g h) x = ↑f x exact congr_fun h x ** Qed
GenLoop.to_from N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p : Ω (↑(Ω^ { j // j ≠ i } X x)) const ⊢ toLoop i (fromLoop i p) = p simp_rw [toLoop, fromLoop, ContinuousMap.comp_assoc, toContinuousMap_comp_symm, ContinuousMap.comp_id] N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p : Ω (↑(Ω^ { j // j ≠ i } X x)) const ⊢ { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }, source' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 0 = const), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 1 = const) } = p ext case a.h.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p : Ω (↑(Ω^ { j // j ≠ i } X x)) const x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑(↑{ toContinuousMap := ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }, source' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 0 = const), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t, property := (_ : (fun x_1 => x_1 ∈ Ω^ { j // j ≠ i } X x) (↑(ContinuousMap.curry (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))) t)) }) 1 = const) } x✝) y✝ = ↑(↑p x✝) y✝ rfl ** Qed
GenLoop.homotopicTo N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) ⊢ Homotopic p q → Path.Homotopic (toLoop i p) (toLoop i q) refine' Nonempty.map fun H => ⟨⟨⟨fun t => ⟨homotopyTo i H t, _⟩, _⟩, _, _⟩, _⟩ case refine'_3 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (x_1 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_1) = ↑(toLoop i p).toContinuousMap x_1 case refine'_4 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (x_1 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_1) = ↑(toLoop i q).toContinuousMap x_1 case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 iterate 2 intro; ext; erw [homotopyTo_apply, toLoop_apply]; swap case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 intro t y yH case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} ⊢ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_2) = ↑(toLoop i p).toContinuousMap x_2), map_one_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_2) = ↑(toLoop i q).toContinuousMap x_2) }.toContinuousMap (t, x_1)) y = ↑(toLoop i p).toContinuousMap y ∧ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (0, x_2) = ↑(toLoop i p).toContinuousMap x_2), map_one_left := (_ : ∀ (x_2 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_2) = ↑(toLoop i q).toContinuousMap x_2) }.toContinuousMap (t, x_1)) y = ↑(toLoop i q).toContinuousMap y constructor <;> ext <;> erw [homotopyTo_apply] case refine'_5.left.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i p).toContinuousMap y) y✝ case refine'_5.right.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i q).toContinuousMap y) y✝ apply H.eq_fst case refine'_5.left.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N case refine'_5.right.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i q).toContinuousMap y) y✝ on_goal 2 => apply H.eq_snd case refine'_5.left.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N case refine'_5.right.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N all_goals use i; rw [funSplitAt_symm_apply, dif_pos rfl]; exact yH case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I ⊢ ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x rintro y ⟨i, iH⟩ case refine'_1.intro N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(↑(homotopyTo i✝ H) t) y = x rw [homotopyTo_apply, H.eq_fst, p.2] case refine'_1.intro.a N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(Cube.insertAt i✝) (t.2, y) ∈ Cube.boundary N case refine'_1.intro.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(Cube.insertAt i✝) (t.2, y) ∈ Cube.boundary N all_goals apply Cube.insertAt_boundary; right; exact ⟨i, iH⟩ case refine'_1.intro.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ ↑(Cube.insertAt i✝) (t.2, y) ∈ Cube.boundary N apply Cube.insertAt_boundary case refine'_1.intro.hx.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ (t.2 = 0 ∨ t.2 = 1) ∨ y ∈ Cube.boundary { j // j ≠ i✝ } right case refine'_1.intro.hx.H.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i✝ : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t : ↑I × ↑I y : { j // j ≠ i✝ } → ↑I i : { j // j ≠ i✝ } iH : y i = 0 ∨ y i = 1 ⊢ y ∈ Cube.boundary { j // j ≠ i✝ } exact ⟨i, iH⟩ case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ Continuous fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) } continuity case refine'_4 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (x_1 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x_1) = ↑(toLoop i q).toContinuousMap x_1 case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := ?refine'_4 }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 intro case refine'_4 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I ⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x✝) = ↑(toLoop i q).toContinuousMap x✝ case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 ext case refine'_4.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑(ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }) (1, x✝)) y✝ = ↑(↑(toLoop i q).toContinuousMap x✝) y✝ case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 erw [homotopyTo_apply, toLoop_apply] case refine'_4.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((1, x✝).1, ↑(Cube.insertAt i) ((1, x✝).2, y✝)) = ↑q (↑(Cube.insertAt i) (x✝, y✝)) case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) case refine'_5 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) ⊢ ∀ (t x_1 : ↑I), x_1 ∈ {0, 1} → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i p).toContinuousMap x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ∀ (y : { j // j ≠ i } → ↑I), y ∈ Cube.boundary { j // j ≠ i } → ↑(↑(homotopyTo i H) t) y = x) }, map_zero_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (0, x_3) = ↑(toLoop i p).toContinuousMap x_3), map_one_left := (_ : ∀ (x_3 : ↑I), ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(homotopyTo i H) t, property := (_ : ↑(homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x) }) (1, x_3) = ↑(toLoop i q).toContinuousMap x_3) }.toContinuousMap (t, x_2)) x_1 = ↑(toLoop i q).toContinuousMap x_1 swap case refine'_3.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((0, x✝).1, ↑(Cube.insertAt i) ((0, x✝).2, y✝)) = ↑p (↑(Cube.insertAt i) (x✝, y✝)) apply H.apply_zero case refine'_4.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) x✝ : ↑I y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((1, x✝).1, ↑(Cube.insertAt i) ((1, x✝).2, y✝)) = ↑q (↑(Cube.insertAt i) (x✝, y✝)) apply H.apply_one case refine'_5.right.H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑H ((t, y).1, ↑(Cube.insertAt i) ((t, y).2, y✝)) = ↑(↑(toLoop i q).toContinuousMap y) y✝ apply H.eq_snd case refine'_5.right.H.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) ∈ Cube.boundary N use i case h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ↑(Cube.insertAt i) ((t, y).2, y✝) i = 0 ∨ ↑(Cube.insertAt i) ((t, y).2, y✝) i = 1 rw [funSplitAt_symm_apply, dif_pos rfl] case h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : ContinuousMap.HomotopyRel (↑p) (↑q) (Cube.boundary N) t y : ↑I yH : y ∈ {0, 1} y✝ : { j // j ≠ i } → ↑I ⊢ ((t, y).2, y✝).1 = 0 ∨ ((t, y).2, y✝).1 = 1 exact yH ** Qed
GenLoop.homotopicFrom N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) ⊢ Path.Homotopic (toLoop i p) (toLoop i q) → Homotopic p q refine' Nonempty.map fun H => ⟨⟨homotopyFrom i H, _, _⟩, _⟩ case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = ↑↑p x_1 case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 case refine'_3 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (t : ↑I) (x_1 : N → ↑I), x_1 ∈ Cube.boundary N → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑p x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑q x_1 pick_goal 3 case refine'_3 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (t : ↑I) (x_1 : N → ↑I), x_1 ∈ Cube.boundary N → ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑p x_1 ∧ ↑(ContinuousMap.mk fun x_2 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_2)) x_1 = ↑↑q x_1 rintro t y ⟨j, jH⟩ case refine'_3.intro N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 ⊢ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_1)) y = ↑↑p y ∧ ↑(ContinuousMap.mk fun x_1 => ContinuousMap.toFun { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1, map_one_left := ?refine'_2 }.toContinuousMap (t, x_1)) y = ↑↑q y erw [homotopyFrom_apply] case refine'_3.intro N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 ⊢ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑p y ∧ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑q y case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = ↑↑p x_1 case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 obtain rfl \| h := eq_or_ne j i case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = ↑↑p x_1 case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 all_goals intro apply (homotopyFrom_apply _ _ _).trans first \| rw [H.apply_zero] \| rw [H.apply_one] first \| apply congr_arg p \| apply congr_arg q apply (Cube.splitAt i).left_inv case refine'_3.intro.inl N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑p y ∧ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑q y constructor case refine'_3.intro.inl.left N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑p y rw [H.eq_fst] case refine'_3.intro.inl.left N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑(toLoop j p).toContinuousMap ((t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑p y case refine'_3.intro.inl.left.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (t, y).2 j ∈ {0, 1} exacts [congr_arg p ((Cube.splitAt j).left_inv _), jH] case refine'_3.intro.inl.right N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑H ((t, y).1, (t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑q y rw [H.eq_snd] case refine'_3.intro.inl.right N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (↑↑(↑(toLoop j q).toContinuousMap ((t, y).2 j)) fun j_1 => (t, y).2 ↑j_1) = ↑↑q y case refine'_3.intro.inl.right.hx N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N p q : ↑(Ω^ N X x) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 H : Path.Homotopy (toLoop j p) (toLoop j q) ⊢ (t, y).2 j ∈ {0, 1} exacts [congr_arg q ((Cube.splitAt j).left_inv _), jH] case refine'_3.intro.inr N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 h : j ≠ i ⊢ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑p y ∧ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = ↑↑q y rw [p.2 _ ⟨j, jH⟩, q.2 _ ⟨j, jH⟩] case refine'_3.intro.inr.right N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 h : j ≠ i ⊢ (↑↑(↑H ((t, y).1, (t, y).2 i)) fun j => (t, y).2 ↑j) = x apply boundary case refine'_3.intro.inr.right.a N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) t : ↑I y : N → ↑I j : N jH : y j = 0 ∨ y j = 1 h : j ≠ i ⊢ (fun j => (t, y).2 ↑j) ∈ Cube.boundary { j // ¬j = i } exact ⟨⟨j, h⟩, jH⟩ case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) ⊢ ∀ (x_1 : N → ↑I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = ↑↑q x_1 intro case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ ContinuousMap.toFun (homotopyFrom i H) (1, x✝) = ↑↑q x✝ apply (homotopyFrom_apply _ _ _).trans case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑H ((1, x✝).1, (1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ first \| rw [H.apply_zero] \| rw [H.apply_one] case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑(toLoop i q).toContinuousMap ((1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ first \| apply congr_arg p \| apply congr_arg q case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ ↑(Homeomorph.toContinuousMap (Cube.insertAt i)) ((1, x✝).2 i, fun j => (1, x✝).2 ↑j) = x✝ apply (Cube.splitAt i).left_inv case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑H ((0, x✝).1, (0, x✝).2 i)) fun j => (0, x✝).2 ↑j) = ↑↑p x✝ rw [H.apply_zero] case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑H ((1, x✝).1, (1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ rw [H.apply_one] case refine'_1 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑(toLoop i p).toContinuousMap ((0, x✝).2 i)) fun j => (0, x✝).2 ↑j) = ↑↑p x✝ apply congr_arg p case refine'_2 N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N p q : ↑(Ω^ N X x) H : Path.Homotopy (toLoop i p) (toLoop i q) x✝ : N → ↑I ⊢ (↑↑(↑(toLoop i q).toContinuousMap ((1, x✝).2 i)) fun j => (1, x✝).2 ↑j) = ↑↑q x✝ apply congr_arg q ** Qed
GenLoop.transAt_distrib N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) ⊢ transAt i (transAt j a b) (transAt j c d) = transAt j (transAt i a c) (transAt i b d) ext case H N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I ⊢ ↑(transAt i (transAt j a b) (transAt j c d)) y✝ = ↑(transAt j (transAt i a c) (transAt i b d)) y✝ simp_rw [transAt, coe_copy, Function.update_apply, if_neg h, if_neg h.symm] ** case neg N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 ⊢ ↑d (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) = ↑d (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) congr 1 case neg.h.e_6.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 ⊢ Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1)) = Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1)) ext1 case neg.h.e_6.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 x✝ : N ⊢ Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1)) x✝ = Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1)) x✝ simp only [Function.update, eq_rec_constant, dite_eq_ite] case neg.h.e_6.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 x✝ : N ⊢ (if x✝ = j then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1) else if x✝ = i then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1) else y✝ x✝) = if x✝ = i then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1) else if x✝ = j then Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1) else y✝ x✝ apply ite_ite_comm case neg.h.e_6.h.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i ≠ j a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 h✝ : ¬↑(y✝ j) ≤ 1 / 2 x✝ : N ⊢ x✝ = j → ¬x✝ = i rintro rfl case neg.h.e_6.h.h.h N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i : N a b c d : ↑(Ω^ N X x) y✝ : N → ↑I h✝¹ : ¬↑(y✝ i) ≤ 1 / 2 x✝ : N h : i ≠ x✝ h✝ : ¬↑(y✝ x✝) ≤ 1 / 2 ⊢ ¬x✝ = i exact h.symm Qed
HomotopyGroup.auxGroup_indep N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N ⊢ auxGroup i = auxGroup j by_cases h : i = j case neg N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j ⊢ auxGroup i = auxGroup j refine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _) case neg N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j ⊢ ∀ (a b c d : HomotopyGroup N X x), Mul.mul (Mul.mul a b) (Mul.mul c d) = Mul.mul (Mul.mul a c) (Mul.mul b d) rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩ case neg.mk.mk.mk.mk N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j a✝ : HomotopyGroup N X x a : ↑(Ω^ N X x) b✝ : HomotopyGroup N X x b : ↑(Ω^ N X x) c✝ : HomotopyGroup N X x c : ↑(Ω^ N X x) d✝ : HomotopyGroup N X x d : ↑(Ω^ N X x) ⊢ Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r b)) (Mul.mul (Quot.mk Setoid.r c) (Quot.mk Setoid.r d)) = Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d)) change Quotient.mk' _ = _ case neg.mk.mk.mk.mk N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j a✝ : HomotopyGroup N X x a : ↑(Ω^ N X x) b✝ : HomotopyGroup N X x b : ↑(Ω^ N X x) c✝ : HomotopyGroup N X x c : ↑(Ω^ N X x) d✝ : HomotopyGroup N X x d : ↑(Ω^ N X x) ⊢ Quotient.mk' (↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv d) (↑(loopHomeo j).toEquiv c)))) (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv b) (↑(loopHomeo j).toEquiv a)))))) = Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d)) apply congr_arg Quotient.mk' case neg.mk.mk.mk.mk N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : ¬i = j a✝ : HomotopyGroup N X x a : ↑(Ω^ N X x) b✝ : HomotopyGroup N X x b : ↑(Ω^ N X x) c✝ : HomotopyGroup N X x c : ↑(Ω^ N X x) d✝ : HomotopyGroup N X x d : ↑(Ω^ N X x) ⊢ ↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv d) (↑(loopHomeo j).toEquiv c)))) (↑(loopHomeo i).toEquiv (↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv b) (↑(loopHomeo j).toEquiv a))))) = ↑(loopHomeo j).symm (Path.trans (↑(loopHomeo j).toEquiv (↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv d) (↑(loopHomeo i).toEquiv b)))) (↑(loopHomeo j).toEquiv (↑(loopHomeo i).symm (Path.trans (↑(loopHomeo i).toEquiv c) (↑(loopHomeo i).toEquiv a))))) simp only [fromLoop_trans_toLoop, transAt_distrib h, coe_toEquiv, loopHomeo_apply, coe_symm_toEquiv, loopHomeo_symm_apply] case pos N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N h : i = j ⊢ auxGroup i = auxGroup j rw [h] ** Qed
HomotopyGroup.transAt_indep N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f g : ↑(Ω^ N X x) ⊢ Quotient.mk (Homotopic.setoid N x) (transAt i f g) = Quotient.mk (Homotopic.setoid N x) (transAt j f g) simp_rw [← fromLoop_trans_toLoop] ** N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f g : ↑(Ω^ N X x) m : (G : Type ?u.862554) → Group G → G → G → G := fun G x x_1 x_2 => x_1 * x_2 ⊢ Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.trans (toLoop i f) (toLoop i g))) = Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.trans (toLoop j f) (toLoop j g))) exact congr_fun₂ (congr_arg (m <\| HomotopyGroup N X x) <\| auxGroup_indep i j) ⟦g⟧ ⟦f⟧ Qed
HomotopyGroup.symmAt_indep N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f : ↑(Ω^ N X x) ⊢ Quotient.mk (Homotopic.setoid N x) (symmAt i f) = Quotient.mk (Homotopic.setoid N x) (symmAt j f) simp_rw [← fromLoop_symm_toLoop] N : Type u_1 X : Type u_2 inst✝¹ : TopologicalSpace X x : X inst✝ : DecidableEq N i j : N f : ↑(Ω^ N X x) inv : (G : Type ?u.864646) → Group G → G → G := fun G x x_1 => x_1⁻¹ ⊢ Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.symm (toLoop i f))) = Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.symm (toLoop j f))) exact congr_fun (congr_arg (inv <\| HomotopyGroup N X x) <\| auxGroup_indep i j) ⟦f⟧ ** Qed
HomotopyGroup.mul_spec ** N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ (fun x_1 x_2 => x_1 * x_2) (Quotient.mk (Homotopic.setoid N x) p) (Quotient.mk (Homotopic.setoid N x) q) = Quotient.mk (Homotopic.setoid N x) (transAt i q p) rw [transAt_indep _ q, ← fromLoop_trans_toLoop] N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ (fun x_1 x_2 => x_1 * x_2) (Quotient.mk (Homotopic.setoid N x) p) (Quotient.mk (Homotopic.setoid N x) q) = Quotient.mk (Homotopic.setoid N x) (fromLoop ?m.900147 (Path.trans (toLoop ?m.900147 q) (toLoop ?m.900147 p))) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ N apply Quotient.sound case a N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ ↑(loopHomeo (Classical.arbitrary N)).symm (Path.trans (↑(loopHomeo (Classical.arbitrary N)).toEquiv q) (↑(loopHomeo (Classical.arbitrary N)).toEquiv p)) ≈ fromLoop ?m.900147 (Path.trans (toLoop ?m.900147 q) (toLoop ?m.900147 p)) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p q : ↑(Ω^ N X x) ⊢ N rfl Qed
HomotopyGroup.inv_spec N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ (Quotient.mk (Homotopic.setoid N x) p)⁻¹ = Quotient.mk (Homotopic.setoid N x) (symmAt i p) rw [symmAt_indep _ p, ← fromLoop_symm_toLoop] N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ (Quotient.mk (Homotopic.setoid N x) p)⁻¹ = Quotient.mk (Homotopic.setoid N x) (fromLoop ?m.901306 (Path.symm (toLoop ?m.901306 p))) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ N apply Quotient.sound case a N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ ↑(loopHomeo (Classical.arbitrary N)).symm (Path.symm (↑(loopHomeo (Classical.arbitrary N)).toEquiv p)) ≈ fromLoop ?m.901306 (Path.symm (toLoop ?m.901306 p)) N : Type u_1 X : Type u_2 inst✝² : TopologicalSpace X x : X inst✝¹ : DecidableEq N inst✝ : Nonempty N i : N p : ↑(Ω^ N X x) ⊢ N rfl ** Qed
UniformSpaceCat.extension_comp_coe X : UniformSpaceCat Y : CpltSepUniformSpace f : toUniformSpace (CpltSepUniformSpace.of (Completion ↑X)) ⟶ toUniformSpace Y ⊢ extensionHom (completionHom X ≫ f) = f apply Subtype.eq case a X : UniformSpaceCat Y : CpltSepUniformSpace f : toUniformSpace (CpltSepUniformSpace.of (Completion ↑X)) ⟶ toUniformSpace Y ⊢ ↑(extensionHom (completionHom X ≫ f)) = ↑f funext x case a.h X : UniformSpaceCat Y : CpltSepUniformSpace f : toUniformSpace (CpltSepUniformSpace.of (Completion ↑X)) ⟶ toUniformSpace Y x : ↑(toUniformSpace (completionFunctor.obj X)) ⊢ ↑(extensionHom (completionHom X ≫ f)) x = ↑f x exact congr_fun (Completion.extension_comp_coe f.property) x ** Qed
Profinite.exists_clopen_of_cofiltered J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hB := TopCat.isTopologicalBasis_cofiltered_limit.{u, u} (F ⋙ Profinite.toTopCat) (Profinite.toTopCat.mapCone C) (isLimitOfPreserves _ hC) (fun j => {W \| IsClopen W}) ?_ (fun i => isClopen_univ) (fun i U1 U2 hU1 hU2 => hU1.inter hU2) ?_ case refine_3 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U hB : TopologicalSpace.IsTopologicalBasis {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W \| IsClopen W}) j) case refine_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)), V ∈ (fun j => {W \| IsClopen W}) j → ↑((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W \| IsClopen W}) i rotate_left case refine_3 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U hB : TopologicalSpace.IsTopologicalBasis {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V obtain ⟨S, hS, h⟩ := hB.open_eq_sUnion hU.1 case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U hB : TopologicalSpace.IsTopologicalBasis {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V clear hB case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let j : S → J := fun s => (hS s.2).choose case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let V : ∀ s : S, Set (F.obj (j s)) := fun s => (hS s.2).choose_spec.choose case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hV : ∀ s : S, IsClopen (V s) ∧ s.1 = C.π.app (j s) ⁻¹' V s := fun s => (hS s.2).choose_spec.choose_spec case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hUo : ∀ (i : ↑S), IsOpen ((fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have hsU : U ⊆ ⋃ (i : ↑S), (fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V have := hU.2.isCompact.elim_finite_subcover (fun s : S => C.π.app (j s) ⁻¹' V s) hUo hsU case refine_3.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i this : ∃ t, U ⊆ ⋃ i ∈ t, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V obtain ⟨G, hG⟩ := this case refine_3.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists (G.image j) case refine_3.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let f : ∀ (s : S) (_ : s ∈ G), j0 ⟶ j s := fun s hs => (hj0 (Finset.mem_image.mpr ⟨s, hs, rfl⟩)).some case refine_3.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V let W : S → Set (F.obj j0) := fun s => if hs : s ∈ G then F.map (f s hs) ⁻¹' V s else Set.univ case refine_3.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ ∃ j V x, U = ↑(C.π.app j) ⁻¹' V refine' ⟨j0, ⋃ (s : S) (_ : s ∈ G), W s, _, _⟩ case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W \| IsClopen W}) j) intro i case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U i : J ⊢ TopologicalSpace.IsTopologicalBasis ((fun j => {W \| IsClopen W}) i) change TopologicalSpace.IsTopologicalBasis {W : Set (F.obj i) \| IsClopen W} case refine_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U i : J ⊢ TopologicalSpace.IsTopologicalBasis {W \| IsClopen W} apply isTopologicalBasis_clopen case refine_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U ⊢ ∀ (i j : J) (f : i ⟶ j) (V : Set ↑((F ⋙ toTopCat).obj j)), V ∈ (fun j => {W \| IsClopen W}) j → ↑((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W \| IsClopen W}) i rintro i j f V (hV : IsClopen _) case refine_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U i j : J f : i ⟶ j V : Set ↑((F ⋙ toTopCat).obj j) hV : IsClopen V ⊢ ↑((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W \| IsClopen W}) i exact ⟨hV.1.preimage ((F ⋙ toTopCat).map f).continuous, hV.2.preimage ((F ⋙ toTopCat).map f).continuous⟩ case hUo J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s ⊢ ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) intro s case hUo J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s s : ↑S ⊢ IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) s) exact (hV s).1.1.preimage (C.π.app (j s)).continuous case hsU J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i dsimp only case hsU J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ U ⊆ ⋃ i, (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) rw [h] case hsU J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) ⊢ ⋃₀ S ⊆ ⋃ i, (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) rintro x ⟨T, hT, hx⟩ case hsU.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) x : ↑C.pt.toCompHaus.toTop T : Set ↑(toTopCat.mapCone C).pt hT : T ∈ S hx : x ∈ T ⊢ x ∈ ⋃ i, (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) refine' ⟨_, ⟨⟨T, hT⟩, rfl⟩, _⟩ case hsU.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) x : ↑C.pt.toCompHaus.toTop T : Set ↑(toTopCat.mapCone C).pt hT : T ∈ S hx : x ∈ T ⊢ x ∈ (fun i => (forget Profinite).map (C.π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V)) { val := T, property := hT } dsimp only [forget_map_eq_coe] case hsU.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) x : ↑C.pt.toCompHaus.toTop T : Set ↑(toTopCat.mapCone C).pt hT : T ∈ S hx : x ∈ T ⊢ x ∈ ↑(C.π.app (Exists.choose (_ : T ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ {W \| IsClopen W} ∧ T = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : T ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) rwa [← (hV ⟨T, hT⟩).2] case refine_3.intro.intro.intro.intro.refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ IsClopen (⋃ s ∈ G, W s) apply isClopen_biUnion_finset case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ ∀ (i : ↑S), i ∈ G → IsClopen (W i) intro s hs case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ s : ↑S hs : s ∈ G ⊢ IsClopen (W s) dsimp case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ s : ↑S hs : s ∈ G ⊢ IsClopen (if hs : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹' Exists.choose (_ : ∃ V, IsClopen V ∧ ↑s = ↑(C.π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ) rw [dif_pos hs] case refine_3.intro.intro.intro.intro.refine'_1.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ s : ↑S hs : s ∈ G ⊢ IsClopen (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))))) ⁻¹' Exists.choose (_ : ∃ V, IsClopen V ∧ ↑s = ↑(C.π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V)) exact ⟨(hV s).1.1.preimage (F.map _).continuous, (hV s).1.2.preimage (F.map _).continuous⟩ case refine_3.intro.intro.intro.intro.refine'_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ ⊢ U = ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s ext x case refine_3.intro.intro.intro.intro.refine'_2.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop ⊢ x ∈ U ↔ x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s constructor case refine_3.intro.intro.intro.intro.refine'_2.h.mp J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop ⊢ x ∈ U → x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s intro hx case refine_3.intro.intro.intro.intro.refine'_2.h.mp J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U ⊢ x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s simp_rw [Set.preimage_iUnion, Set.mem_iUnion] case refine_3.intro.intro.intro.intro.refine'_2.h.mp J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U ⊢ ∃ i i_1, x ∈ ↑(C.π.app j0) ⁻¹' if h : i ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ obtain ⟨_, ⟨s, rfl⟩, _, ⟨hs, rfl⟩, hh⟩ := hG hx case refine_3.intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U s : ↑S hs : s ∈ G hh : x ∈ (fun h => (fun s => ↑(C.π.app (j s)) ⁻¹' V s) s) hs ⊢ ∃ i i_1, x ∈ ↑(C.π.app j0) ⁻¹' if h : i ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ refine' ⟨s, hs, _⟩ case refine_3.intro.intro.intro.intro.refine'_2.h.mp.intro.intro.intro.intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ U s : ↑S hs : s ∈ G hh : x ∈ (fun h => (fun s => ↑(C.π.app (j s)) ⁻¹' V s) s) hs ⊢ x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ rwa [dif_pos hs, ← Set.preimage_comp, ← Profinite.coe_comp, ← Functor.map_comp, C.w] case refine_3.intro.intro.intro.intro.refine'_2.h.mpr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop ⊢ x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s → x ∈ U intro hx case refine_3.intro.intro.intro.intro.refine'_2.h.mpr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : x ∈ ↑(C.π.app j0) ⁻¹' ⋃ s ∈ G, W s ⊢ x ∈ U simp_rw [Set.preimage_iUnion, Set.mem_iUnion] at hx case refine_3.intro.intro.intro.intro.refine'_2.h.mpr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop hx : ∃ i i_1, x ∈ ↑(C.π.app j0) ⁻¹' if h : i ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j i)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑i = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑i ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ U obtain ⟨s, hs, hx⟩ := hx case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ U rw [h] case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ ⋃₀ S refine' ⟨s.1, s.2, _⟩ case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ ↑s rw [(hV s).2] case refine_3.intro.intro.intro.intro.refine'_2.h.mpr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F U : Set ↑C.pt.toCompHaus.toTop hC : IsLimit C hU : IsClopen U S : Set (Set ↑(toTopCat.mapCone C).pt) hS : S ⊆ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V} h : U = ⋃₀ S j : ↑S → J := fun s => Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}) V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop := fun s => Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ↑(C.π.app (j s)) ⁻¹' V s hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i) hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i G : Finset ↑S hG : U ⊆ ⋃ i ∈ G, (fun s => ↑(C.π.app (j s)) ⁻¹' V s) i j0 : J hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X) f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s)) W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ↑(F.map (f s hs)) ⁻¹' V s else Set.univ x : ↑C.pt.toCompHaus.toTop s : ↑S hs : s ∈ G hx : x ∈ ↑(C.π.app j0) ⁻¹' if h : s ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j s)))) ⁻¹' Exists.choose (_ : ∃ V, V ∈ (fun j => {W \| IsClopen W}) (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V})) ∧ ↑s = ↑((toTopCat.mapCone C).π.app (Exists.choose (_ : ↑s ∈ {U \| ∃ j V, V ∈ (fun j => {W \| IsClopen W}) j ∧ U = ↑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹' V) else Set.univ ⊢ x ∈ ↑(C.π.app (j s)) ⁻¹' V s rwa [dif_pos hs, ← Set.preimage_comp, ← Profinite.coe_comp, ← Functor.map_comp, C.w] at hx ** Qed
Profinite.exists_locallyConstant_fin_two J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let U := f ⁻¹' {0} J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g have hU : IsClopen U := f.isLocallyConstant.isClopen_fiber _ J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j, V, hV, h⟩ := exists_clopen_of_cofiltered C hC hU case intro.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g use j, LocallyConstant.ofClopen hV case h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ f = LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.ofClopen hV) apply LocallyConstant.locallyConstant_eq_of_fiber_zero_eq case h.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ↑f ⁻¹' {0} = ↑(LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.ofClopen hV)) ⁻¹' {0} erw [LocallyConstant.coe_comap _ _ (C.π.app j).continuous] case h.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ↑f ⁻¹' {0} = ↑(LocallyConstant.ofClopen hV) ∘ ↑(C.π.app j) ⁻¹' {0} conv_rhs => rw [Set.preimage_comp] case h.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) U : Set ↑C.pt.toCompHaus.toTop := ↑f ⁻¹' {0} hU : IsClopen U j : J V : Set ↑(F.obj j).toCompHaus.toTop hV : IsClopen V h : U = ↑(C.π.app j) ⁻¹' V ⊢ ↑f ⁻¹' {0} = ↑(C.π.app j) ⁻¹' (↑(LocallyConstant.ofClopen hV) ⁻¹' {0}) erw [LocallyConstant.ofClopen_fiber_zero hV, ← h] ** Qed
Profinite.exists_locallyConstant_finite_aux J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g cases nonempty_fintype α case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let ff := (f.map ι).flip case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g have hff := fun a : α => exists_locallyConstant_fin_two _ hC (ff a) case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) hff : ∀ (a : α), ∃ j g, ff a = LocallyConstant.comap (↑(C.π.app j)) g ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g choose j g h using hff case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let G : Finset J := Finset.univ.image j case intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists G case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g have hj : ∀ a, j a ∈ (Finset.univ.image j : Finset J) := by intro a simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply] case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let fs : ∀ a : α, j0 ⟶ j a := fun a => (hj0 (hj a)).some case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let gg : α → LocallyConstant (F.obj j0) (Fin 2) := fun a => (g a).comap (F.map (fs _)) case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g let ggg := LocallyConstant.unflip gg case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ ∃ j g, LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨j0, ggg, _⟩ case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j0)) ggg have : f.map ι = LocallyConstant.unflip (f.map ι).flip := by simp case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) ⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j0)) ggg rw [this] case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.comap (↑(C.π.app j0)) ggg clear this case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.comap (↑(C.π.app j0)) ggg have : LocallyConstant.comap (C.π.app j0) ggg = LocallyConstant.unflip (LocallyConstant.comap (C.π.app j0) ggg).flip := by simp case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.comap (↑(C.π.app j0)) ggg = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.comap (↑(C.π.app j0)) ggg rw [this] case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg this : LocallyConstant.comap (↑(C.π.app j0)) ggg = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) clear this case intro.intro J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) congr 1 case intro.intro.e_f J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.flip (LocallyConstant.map ι f) = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) ext1 a case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ LocallyConstant.flip (LocallyConstant.map ι f) a = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) a change ff a = _ case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ ff a = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) a rw [h] case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ LocallyConstant.comap (↑(C.π.app (j a))) (g a) = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg) a dsimp case intro.intro.e_f.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α ⊢ LocallyConstant.comap (↑(C.π.app (j a))) (g a) = LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) (LocallyConstant.unflip fun a => LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))) a ext1 x case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(LocallyConstant.comap (↑(C.π.app (j a))) (g a)) x = ↑(LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) (LocallyConstant.unflip fun a => LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))) a) x erw [LocallyConstant.coe_comap _ _ (C.π.app (j a)).continuous] case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ (↑(g a) ∘ ↑(C.π.app (j a))) x = ↑(LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) (LocallyConstant.unflip fun a => LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))) a) x dsimp [LocallyConstant.flip, LocallyConstant.unflip] case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = ↑(LocallyConstant.comap ↑(C.π.app j0) { toFun := fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x, isLocallyConstant := (_ : IsLocallyConstant fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x) }) x a erw [LocallyConstant.coe_comap _ _ (C.π.app j0).continuous] case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = (↑{ toFun := fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x, isLocallyConstant := (_ : IsLocallyConstant fun x a => ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) x) } ∘ ↑(C.π.app j0)) x a dsimp case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = ↑(LocallyConstant.comap (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)) (↑(C.π.app j0) x) rw [LocallyConstant.coe_comap _ _ _] J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) ⊢ ∀ (a : α), j a ∈ Finset.image j Finset.univ intro a J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) a : α ⊢ j a ∈ Finset.image j Finset.univ simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply] J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) simp J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg ⊢ LocallyConstant.comap (↑(C.π.app j0)) ggg = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap (↑(C.π.app j0)) ggg)) simp case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = (↑(g a) ∘ ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (↑(C.π.app j0) x) dsimp case intro.intro.e_f.h.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ ↑(g a) (↑(C.π.app (j a)) x) = ↑(g a) (↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) (↑(C.π.app j0) x)) congr! 1 case intro.intro.e_f.h.h.h.e'_6.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt e_2✝ : ↑(F.obj (j a)).toCompHaus.toTop = (forget Profinite).obj (F.obj (j a)) ⊢ ↑(C.π.app (j a)) x = ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) (↑(C.π.app j0) x) change _ = (C.π.app j0 ≫ F.map (fs a)) x case intro.intro.e_f.h.h.h.e'_6.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt e_2✝ : ↑(F.obj (j a)).toCompHaus.toTop = (forget Profinite).obj (F.obj (j a)) ⊢ ↑(C.π.app (j a)) x = ↑(C.π.app j0 ≫ F.map (fs a)) x rw [C.w] case intro.intro.e_f.h.h.h.e'_6.h J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt e_2✝ : ↑(F.obj (j a)).toCompHaus.toTop = (forget Profinite).obj (F.obj (j a)) ⊢ ↑(C.π.app (j a)) x = ↑(C.π.app (j a)) x rfl J : Type u inst✝² : SmallCategory J inst✝¹ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝ : Finite α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α val✝ : Fintype α ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1 ff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f) j : α → J g : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2) h : ∀ (a : α), ff a = LocallyConstant.comap (↑(C.π.app (j a))) (g a) G : Finset J := Finset.image j Finset.univ j0 : J hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X) hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ fs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a)) gg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) := fun a => LocallyConstant.comap (↑(F.map (fs a))) (g a) ggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg a : α x : (forget Profinite).obj C.pt ⊢ Continuous ↑(F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) exact (F.map _).continuous ** Qed
Profinite.exists_locallyConstant_finite_nonempty J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g inhabit α J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j, gg, h⟩ := exists_locallyConstant_finite_aux _ hC f case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let σ : (α → Fin 2) → α := fun f => if h : ∃ a : α, ι a = f then h.choose else default case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨j, gg.map σ, _⟩ case intro.intro J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default ⊢ f = LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.map σ gg) ext x case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = ↑(LocallyConstant.comap (↑(C.π.app j)) (LocallyConstant.map σ gg)) x erw [LocallyConstant.coe_comap _ _ (C.π.app j).continuous] case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = (↑(LocallyConstant.map σ gg) ∘ ↑(C.π.app j)) x dsimp case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg (↑(C.π.app j) x) then Exists.choose h else default have h1 : ι (f x) = gg (C.π.app j x) := by change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _ erw [h, LocallyConstant.coe_comap _ _ (C.π.app j).continuous] rfl case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) ⊢ ↑f x = if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg (↑(C.π.app j) x) then Exists.choose h else default have h2 : ∃ a : α, ι a = gg (C.π.app j x) := ⟨f x, h1⟩ case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ↑f x = if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg (↑(C.π.app j) x) then Exists.choose h else default erw [dif_pos h2] case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ↑f x = Exists.choose h2 apply_fun ι J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ι (↑f x) = ↑gg (↑(C.π.app j) x) change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _ J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ ↑(LocallyConstant.map (fun a b => if a = b then 0 else 1) f) x = ↑gg (↑(C.π.app j) x) erw [h, LocallyConstant.coe_comap _ _ (C.π.app j).continuous] J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop ⊢ (↑gg ∘ ↑(C.π.app j)) x = ↑gg (↑(C.π.app j) x) rfl case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ι (↑f x) = ι (Exists.choose h2) rw [h2.choose_spec] case intro.intro.h J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ ι (↑f x) = ↑gg (↑(C.π.app j) x) exact h1 case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) ⊢ Function.Injective ι intro a b hh case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b ⊢ a = b have hhh := congr_fun hh a case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hhh : ι a a = ι b a ⊢ a = b dsimp at hhh case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hhh : (if a = a then 0 else 1) = if b = a then 0 else 1 ⊢ a = b rw [if_pos rfl] at hhh case intro.intro.h.inj J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hhh : 0 = if b = a then 0 else 1 ⊢ a = b split_ifs at hhh with hh1 case pos J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hh1 : b = a hhh : 0 = 0 ⊢ a = b exact hh1.symm case neg J : Type u inst✝³ : SmallCategory J inst✝² : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 inst✝¹ : Finite α inst✝ : Nonempty α hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α inhabited_h : Inhabited α j : J gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (↑(C.π.app j)) gg ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1 σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default x : ↑C.pt.toCompHaus.toTop h1 : ι (↑f x) = ↑gg (↑(C.π.app j) x) h2 : ∃ a, ι a = ↑gg (↑(C.π.app j) x) a b : α hh : ι a = ι b hh1 : ¬b = a hhh : 0 = 1 ⊢ a = b exact False.elim (bot_ne_top hhh) ** Qed
Profinite.exists_locallyConstant J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let S := f.discreteQuotient J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let ff : S → α := f.lift J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g cases isEmpty_or_nonempty S case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) ⊢ ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop simp only [← not_nonempty_iff, ← not_forall] case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) ⊢ ¬∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop intro h case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop ⊢ False haveI : ∀ j : J, Nonempty ((F ⋙ Profinite.toTopCat).obj j) := h case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) ⊢ False haveI : ∀ j : J, T2Space ((F ⋙ Profinite.toTopCat).obj j) := fun j => (inferInstance : T2Space (F.obj j)) case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) ⊢ False haveI : ∀ j : J, CompactSpace ((F ⋙ Profinite.toTopCat).obj j) := fun j => (inferInstance : CompactSpace (F.obj j)) case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) ⊢ False have cond := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} (F ⋙ Profinite.toTopCat) case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt ⊢ False suffices : Nonempty C.pt case inl J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝² : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝¹ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt this : Nonempty ↑C.pt.toCompHaus.toTop ⊢ False case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt ⊢ Nonempty ↑C.pt.toCompHaus.toTop exact IsEmpty.false (S.proj this.some) case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt ⊢ Nonempty ↑C.pt.toCompHaus.toTop let D := Profinite.toTopCat.mapCone C case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C ⊢ Nonempty ↑C.pt.toCompHaus.toTop have hD : IsLimit D := isLimitOfPreserves Profinite.toTopCat hC case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C hD : IsLimit D ⊢ Nonempty ↑C.pt.toCompHaus.toTop have CD := (hD.conePointUniqueUpToIso (TopCat.limitConeIsLimit.{u, u} _)).inv case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) h : ∀ (x : J), Nonempty ↑(F.obj x).toCompHaus.toTop this✝¹ : ∀ (j : J), Nonempty ↑((F ⋙ toTopCat).obj j) this✝ : ∀ (j : J), T2Space ↑((F ⋙ toTopCat).obj j) this : ∀ (j : J), CompactSpace ↑((F ⋙ toTopCat).obj j) cond : Nonempty ↑(TopCat.limitCone (F ⋙ toTopCat)).pt D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C hD : IsLimit D CD : (TopCat.limitCone (F ⋙ toTopCat)).pt ⟶ D.pt ⊢ Nonempty ↑C.pt.toCompHaus.toTop exact cond.map CD J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' this.imp fun j hj => _ J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop ⊢ ∃ g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨⟨hj.elim, fun A => _⟩, _⟩ case refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α ⊢ IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A) suffices : (fun a ↦ IsEmpty.elim hj a) ⁻¹' A = ∅ case this J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α ⊢ (fun a => IsEmpty.elim hj a) ⁻¹' A = ∅ exact @Set.eq_empty_of_isEmpty _ hj _ case refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this✝ : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α this : (fun a => IsEmpty.elim hj a) ⁻¹' A = ∅ ⊢ IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A) rw [this] case refine'_1 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this✝ : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop A : Set α this : (fun a => IsEmpty.elim hj a) ⁻¹' A = ∅ ⊢ IsOpen ∅ exact isOpen_empty case refine'_2 J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop ⊢ f = LocallyConstant.comap ↑(C.π.app j) { toFun := fun a => IsEmpty.elim hj a, isLocallyConstant := (_ : ∀ (A : Set α), IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A)) } ext x case refine'_2.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : IsEmpty (Quotient S.toSetoid) this : ∃ j, IsEmpty ↑(F.obj j).toCompHaus.toTop j : J hj : IsEmpty ↑(F.obj j).toCompHaus.toTop x : ↑C.pt.toCompHaus.toTop ⊢ ↑f x = ↑(LocallyConstant.comap ↑(C.π.app j) { toFun := fun a => IsEmpty.elim hj a, isLocallyConstant := (_ : ∀ (A : Set α), IsOpen ((fun a => IsEmpty.elim hj a) ⁻¹' A)) }) x exact hj.elim' (C.π.app j x) case inr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g let f' : LocallyConstant C.pt S := ⟨S.proj, S.proj_isLocallyConstant⟩ case inr J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g obtain ⟨j, g', hj⟩ := exists_locallyConstant_finite_nonempty _ hC f' case inr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) hj : f' = LocallyConstant.comap (↑(C.π.app j)) g' ⊢ ∃ j g, f = LocallyConstant.comap (↑(C.π.app j)) g refine' ⟨j, ⟨ff ∘ g', g'.isLocallyConstant.comp _⟩, _⟩ case inr.intro.intro J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) hj : f' = LocallyConstant.comap (↑(C.π.app j)) g' ⊢ f = LocallyConstant.comap ↑(C.π.app j) { toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) } ext1 t case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) hj : f' = LocallyConstant.comap (↑(C.π.app j)) g' t : ↑C.pt.toCompHaus.toTop ⊢ ↑f t = ↑(LocallyConstant.comap ↑(C.π.app j) { toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) }) t apply_fun fun e => e t at hj case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : ↑f' t = ↑(LocallyConstant.comap (↑(C.π.app j)) g') t ⊢ ↑f t = ↑(LocallyConstant.comap ↑(C.π.app j) { toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) }) t erw [LocallyConstant.coe_comap _ _ (C.π.app j).continuous] at hj ⊢ case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : ↑f' t = (↑g' ∘ ↑(C.π.app j)) t ⊢ ↑f t = (↑{ toFun := ff ∘ ↑g', isLocallyConstant := (_ : IsLocallyConstant (ff ∘ g'.toFun)) } ∘ ↑(C.π.app j)) t dsimp at hj ⊢ case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : DiscreteQuotient.proj (LocallyConstant.discreteQuotient f) t = ↑g' (↑(C.π.app j) t) ⊢ ↑f t = ↑(LocallyConstant.lift f) (↑g' (↑(C.π.app j) t)) rw [← hj] case inr.intro.intro.h J : Type u inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ Profinite C : Cone F α : Type u_1 hC : IsLimit C f : LocallyConstant (↑C.pt.toCompHaus.toTop) α S : DiscreteQuotient ↑C.pt.toCompHaus.toTop := LocallyConstant.discreteQuotient f ff : Quotient S.toSetoid → α := ↑(LocallyConstant.lift f) h✝ : Nonempty (Quotient S.toSetoid) f' : LocallyConstant (↑C.pt.toCompHaus.toTop) (Quotient S.toSetoid) := { toFun := DiscreteQuotient.proj S, isLocallyConstant := (_ : IsLocallyConstant (DiscreteQuotient.proj S)) } j : J g' : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (Quotient S.toSetoid) t : ↑C.pt.toCompHaus.toTop hj : DiscreteQuotient.proj (LocallyConstant.discreteQuotient f) t = ↑g' (↑(C.π.app j) t) ⊢ ↑f t = ↑(LocallyConstant.lift f) (DiscreteQuotient.proj (LocallyConstant.discreteQuotient f) t) rfl ** Qed
ContinuousMap.polynomial_comp_attachBound X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] ⊢ comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f) = ↑(↑(Polynomial.aeval f) g) ext case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] a✝ : X ⊢ ↑(comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f)) a✝ = ↑↑(↑(Polynomial.aeval f) g) a✝ simp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe, Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe, Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply] case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] a✝ : X ⊢ Polynomial.eval (↑(↑(attachBound ↑f) a✝)) g = Polynomial.eval (↑↑f a✝) g erw [ContinuousMap.attachBound_apply_coe] ** Qed
ContinuousMap.polynomial_comp_attachBound_mem X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] ⊢ comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f) ∈ A rw [polynomial_comp_attachBound] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } g : ℝ[X] ⊢ ↑(↑(Polynomial.aeval f) g) ∈ A apply SetLike.coe_mem ** Qed
ContinuousMap.comp_attachBound_mem_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) ⊢ comp p (attachBound ↑f) ∈ Subalgebra.topologicalClosure A have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure := continuousMap_mem_polynomialFunctions_closure _ _ p X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ comp p (attachBound ↑f) ∈ Subalgebra.topologicalClosure A have frequently_mem_polynomials := mem_closure_iff_frequently.mp mem_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ comp p (attachBound ↑f) ∈ Subalgebra.topologicalClosure A apply mem_closure_iff_frequently.mpr X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ ∃ᶠ (x : C(X, ℝ)) in nhds (comp p (attachBound ↑f)), x ∈ ↑A refine' ((compRightContinuousMap ℝ (attachBound (f : C(X, ℝ)))).continuousAt p).tendsto.frequently_map _ _ frequently_mem_polynomials X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) ⊢ ∀ (x : C(↑(Set.Icc (-‖↑f‖) ‖↑f‖), ℝ)), x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) → ↑(compRightContinuousMap ℝ (attachBound ↑f)) x ∈ ↑A rintro _ ⟨g, ⟨-, rfl⟩⟩ case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) g : ℝ[X] ⊢ ↑(compRightContinuousMap ℝ (attachBound ↑f)) (↑↑(Polynomial.toContinuousMapOnAlgHom (Set.Icc (-‖f‖) ‖f‖)) g) ∈ ↑A simp only [SetLike.mem_coe, AlgHom.coe_toRingHom, compRightContinuousMap_apply, Polynomial.toContinuousMapOnAlgHom_apply] case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) mem_closure : p ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) frequently_mem_polynomials : ∃ᶠ (x : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) in nhds p, x ∈ ↑(polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)) g : ℝ[X] ⊢ comp (Polynomial.toContinuousMapOn g (Set.Icc (-‖f‖) ‖f‖)) (attachBound ↑f) ∈ A apply polynomial_comp_attachBound_mem ** Qed
ContinuousMap.abs_mem_subalgebra_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } ⊢ abs ↑f ∈ Subalgebra.topologicalClosure A let f' := attachBound (f : C(X, ℝ)) X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } f' : C(X, ↑(Set.Icc (-‖↑f‖) ‖↑f‖)) := attachBound ↑f ⊢ abs ↑f ∈ Subalgebra.topologicalClosure A let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => \|(x : ℝ)\| } X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } f' : C(X, ↑(Set.Icc (-‖↑f‖) ‖↑f‖)) := attachBound ↑f abs : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) := mk fun x => \|↑x\| ⊢ ContinuousMap.abs ↑f ∈ Subalgebra.topologicalClosure A change abs.comp f' ∈ A.topologicalClosure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f : { x // x ∈ A } f' : C(X, ↑(Set.Icc (-‖↑f‖) ‖↑f‖)) := attachBound ↑f abs : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ) := mk fun x => \|↑x\| ⊢ comp abs f' ∈ Subalgebra.topologicalClosure A apply comp_attachBound_mem_closure ** Qed
ContinuousMap.inf_mem_subalgebra_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ ↑f ⊓ ↑g ∈ Subalgebra.topologicalClosure A rw [inf_eq_half_smul_add_sub_abs_sub' ℝ] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ 2⁻¹ • (↑f + ↑g - \|↑g - ↑f\|) ∈ Subalgebra.topologicalClosure A refine' A.topologicalClosure.smul_mem (A.topologicalClosure.sub_mem (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property) (A.le_topologicalClosure g.property)) _) _ X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ \|↑g - ↑f\| ∈ Subalgebra.topologicalClosure A exact_mod_cast abs_mem_subalgebra_closure A _ ** Qed
ContinuousMap.inf_mem_closed_subalgebra X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑f ⊓ ↑g ∈ A convert inf_mem_subalgebra_closure A f g case h.e'_5 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ A = Subalgebra.topologicalClosure A apply SetLike.ext' case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑A = ↑(Subalgebra.topologicalClosure A) symm case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑(Subalgebra.topologicalClosure A) = ↑A erw [closure_eq_iff_isClosed] case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ IsClosed ↑A exact h ** Qed
ContinuousMap.sup_mem_subalgebra_closure X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ ↑f ⊔ ↑g ∈ Subalgebra.topologicalClosure A rw [sup_eq_half_smul_add_add_abs_sub' ℝ] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ 2⁻¹ • (↑f + ↑g + \|↑g - ↑f\|) ∈ Subalgebra.topologicalClosure A refine' A.topologicalClosure.smul_mem (A.topologicalClosure.add_mem (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property) (A.le_topologicalClosure g.property)) _) _ X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) f g : { x // x ∈ A } ⊢ \|↑g - ↑f\| ∈ Subalgebra.topologicalClosure A exact_mod_cast abs_mem_subalgebra_closure A _ ** Qed
ContinuousMap.sup_mem_closed_subalgebra X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑f ⊔ ↑g ∈ A convert sup_mem_subalgebra_closure A f g case h.e'_5 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ A = Subalgebra.topologicalClosure A apply SetLike.ext' case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑A = ↑(Subalgebra.topologicalClosure A) symm case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ ↑(Subalgebra.topologicalClosure A) = ↑A erw [closure_eq_iff_isClosed] case h.e'_5.h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) h : IsClosed ↑A f g : { x // x ∈ A } ⊢ IsClosed ↑A exact h ** Qed
ContinuousMap.sublattice_closure_eq_top X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L ⊢ closure L = ⊤ apply eq_top_iff.mpr X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L ⊢ ⊤ ≤ closure L rintro f - X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ⊢ f ∈ closure L refine' Filter.Frequently.mem_closure ((Filter.HasBasis.frequently_iff Metric.nhds_basis_ball).mpr fun ε pos => _) X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ x, x ∈ Metric.ball f ε ∧ x ∈ L simp only [exists_prop, Metric.mem_ball] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ x, dist x f < ε ∧ x ∈ L by_cases nX : Nonempty X case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L case neg X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : ¬Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L swap case neg X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : ¬Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L exact ⟨nA.some, (dist_lt_iff pos).mpr fun x => False.elim (nX ⟨x⟩), nA.choose_spec⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : Set.SeparatesPointsStrongly L f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L dsimp only [Set.SeparatesPointsStrongly] at sep case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X ⊢ ∃ x, dist x f < ε ∧ x ∈ L choose g hg w₁ w₂ using sep f case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y ⊢ ∃ x, dist x f < ε ∧ x ∈ L let U : X → X → Set X := fun x y => {z \| f z - ε < g x y z} case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ⊢ ∃ x, dist x f < ε ∧ x ∈ L let ys : ∀ _, Finset X := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ⊢ ∃ x, dist x f < ε ∧ x ∈ L let ys_w : ∀ x, ⋃ y ∈ ys x, U x y = ⊤ := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose_spec case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ⊢ ∃ x, dist x f < ε ∧ x ∈ L have ys_nonempty : ∀ x, (ys x).Nonempty := fun x => Set.nonempty_of_union_eq_top_of_nonempty _ _ nX (ys_w x) case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) ⊢ ∃ x, dist x f < ε ∧ x ∈ L let h : ∀ _, L := fun x => ⟨(ys x).sup' (ys_nonempty x) fun y => (g x y : C(X, ℝ)), Finset.sup'_mem _ sup_mem _ _ _ fun y _ => hg x y⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } ⊢ ∃ x, dist x f < ε ∧ x ∈ L have lt_h : ∀ x z, f z - ε < (h x : X → ℝ) z := by intro x z obtain ⟨y, ym, zm⟩ := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z dsimp simp only [Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff] exact ⟨y, ym, zm⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z ⊢ ∃ x, dist x f < ε ∧ x ∈ L have h_eq : ∀ x, (h x : X → ℝ) x = f x := by intro x; simp [w₁] case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x ⊢ ∃ x, dist x f < ε ∧ x ∈ L let W : ∀ _, Set X := fun x => {z \| (h x : X → ℝ) z < f z + ε} case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x ⊢ ∃ x, dist x f < ε ∧ x ∈ L let xs : Finset X := (CompactSpace.elim_nhds_subcover W W_nhd).choose case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) ⊢ ∃ x, dist x f < ε ∧ x ∈ L let xs_w : ⋃ x ∈ xs, W x = ⊤ := (CompactSpace.elim_nhds_subcover W W_nhd).choose_spec case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) ⊢ ∃ x, dist x f < ε ∧ x ∈ L have xs_nonempty : xs.Nonempty := Set.nonempty_of_union_eq_top_of_nonempty _ _ nX xs_w case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs ⊢ ∃ x, dist x f < ε ∧ x ∈ L let k : (L : Type _) := ⟨xs.inf' xs_nonempty fun x => (h x : C(X, ℝ)), Finset.inf'_mem _ inf_mem _ _ _ fun x _ => (h x).2⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } ⊢ ∃ x, dist x f < ε ∧ x ∈ L refine' ⟨k.1, _, k.2⟩ case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } ⊢ dist (↑k) f < ε rw [dist_lt_iff pos] case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } ⊢ ∀ (x : X), dist (↑↑k x) (↑f x) < ε intro z case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ dist (↑↑k z) (↑f z) < ε rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by intros; simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm]] case pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑↑k z < ↑f z + ε ∧ ↑f z - ε < ↑↑k z fconstructor X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} ⊢ ∀ (x y : X), U x y ∈ 𝓝 y intro x y X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ U x y ∈ 𝓝 y refine' IsOpen.mem_nhds _ _ case refine'_1 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ IsOpen (U x y) apply isOpen_lt <;> continuity case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ y ∈ U x y rw [Set.mem_setOf_eq, w₂] case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} x y : X ⊢ ↑f y - ε < ↑f y exact sub_lt_self _ pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } ⊢ ∀ (x z : X), ↑f z - ε < ↑↑(h x) z intro x z X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z : X ⊢ ↑f z - ε < ↑↑(h x) z obtain ⟨y, ym, zm⟩ := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z y : X ym : y ∈ fun i => i ∈ (ys x).val zm : z ∈ U x y ⊢ ↑f z - ε < ↑↑(h x) z dsimp case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z y : X ym : y ∈ fun i => i ∈ (ys x).val zm : z ∈ U x y ⊢ ↑f z - ε < ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z simp only [Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff] case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } x z y : X ym : y ∈ fun i => i ∈ (ys x).val zm : z ∈ U x y ⊢ ∃ b, b ∈ Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤) ∧ ↑f z - ε < ↑(g x b) z exact ⟨y, ym, zm⟩ X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z ⊢ ∀ (x : X), ↑↑(h x) x = ↑f x intro x X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z x : X ⊢ ↑↑(h x) x = ↑f x simp [w₁] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} ⊢ ∀ (x : X), W x ∈ 𝓝 x intro x X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ W x ∈ 𝓝 x refine' IsOpen.mem_nhds _ _ case refine'_1 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ IsOpen (W x) apply isOpen_lt (continuous_set_coe _ _) case refine'_1 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ Continuous fun b => ↑f b + ε continuity case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ x ∈ W x dsimp only [Set.mem_setOf_eq] case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) x < ↑f x + ε rw [h_eq] case refine'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} x : X ⊢ ↑f x < ↑f x + ε exact lt_add_of_pos_right _ pos X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ∀ (a b ε : ℝ), dist a b < ε ↔ a < b + ε ∧ b - ε < a intros X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X a✝ b✝ ε✝ : ℝ ⊢ dist a✝ b✝ < ε✝ ↔ a✝ < b✝ + ε✝ ∧ b✝ - ε✝ < a✝ simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm] case pos.left X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑↑k z < ↑f z + ε dsimp case pos.left X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑(Finset.inf' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤)) xs_nonempty fun x => Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε simp only [Finset.inf'_lt_iff, ContinuousMap.inf'_apply] case pos.left X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ∃ i, i ∈ Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤) ∧ ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑f z - ε < ↑(g i x) z} = ⊤)) (_ : Finset.Nonempty (ys i)) fun y => g i y) z < ↑f z + ε exact Set.exists_set_mem_of_union_eq_top _ _ xs_w z case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑f z - ε < ↑↑k z dsimp case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ↑f z - ε < ↑(Finset.inf' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤)) xs_nonempty fun x => Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z simp only [Finset.lt_inf'_iff, ContinuousMap.inf'_apply] case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z : X ⊢ ∀ (i : X), i ∈ Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z < ↑f z + ε} = ⊤) → ↑f z - ε < ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x ∈ t, {z \| ↑f z - ε < ↑(g i x) z} = ⊤)) (_ : Finset.Nonempty (ys i)) fun y => g i y) z rintro x - case pos.right X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X L : Set C(X, ℝ) nA : Set.Nonempty L inf_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊓ g ∈ L sup_mem : ∀ (f : C(X, ℝ)), f ∈ L → ∀ (g : C(X, ℝ)), g ∈ L → f ⊔ g ∈ L sep : ∀ (v : X → ℝ) (x y : X), ∃ f, f ∈ L ∧ ↑f x = v x ∧ ↑f y = v y f : C(X, ℝ) ε : ℝ pos : 0 < ε nX : Nonempty X g : X → X → C(X, ℝ) hg : ∀ (x y : X), g x y ∈ L w₁ : ∀ (x y : X), ↑(g x y) x = ↑f x w₂ : ∀ (x y : X), ↑(g x y) y = ↑f y U : X → X → Set X := fun x y => {z \| ↑f z - ε < ↑(g x y) z} U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y ys : X → Finset X := fun x => Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, U x x_1 = ⊤) ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)) ys_nonempty : ∀ (x : X), Finset.Nonempty (ys x) h : X → ↑L := fun x => { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y, property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) ∈ L) } lt_h : ∀ (x z : X), ↑f z - ε < ↑↑(h x) z h_eq : ∀ (x : X), ↑↑(h x) x = ↑f x W : X → Set X := fun x => {z \| ↑↑(h x) z < ↑f z + ε} W_nhd : ∀ (x : X), W x ∈ 𝓝 x xs : Finset X := Exists.choose (_ : ∃ t, ⋃ x ∈ t, W x = ⊤) xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd) xs_nonempty : Finset.Nonempty xs k : ↑L := { val := Finset.inf' xs xs_nonempty fun x => ↑(h x), property := (_ : (Finset.inf' xs xs_nonempty fun x => ↑(h x)) ∈ L) } z x : X ⊢ ↑f z - ε < ↑(Finset.sup' (Exists.choose (_ : ∃ t, ⋃ x_1 ∈ t, {z \| ↑f z - ε < ↑(g x x_1) z} = ⊤)) (_ : Finset.Nonempty (ys x)) fun y => g x y) z apply lt_h ** Qed
ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A ⊢ Subalgebra.topologicalClosure A = ⊤ apply SetLike.ext' case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A ⊢ ↑(Subalgebra.topologicalClosure A) = ↑⊤ let L := A.topologicalClosure case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A L : Subalgebra ℝ C(X, ℝ) := Subalgebra.topologicalClosure A ⊢ ↑(Subalgebra.topologicalClosure A) = ↑⊤ have n : Set.Nonempty (L : Set C(X, ℝ)) := ⟨(1 : C(X, ℝ)), A.le_topologicalClosure A.one_mem⟩ case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A L : Subalgebra ℝ C(X, ℝ) := Subalgebra.topologicalClosure A n : Set.Nonempty ↑L ⊢ ↑(Subalgebra.topologicalClosure A) = ↑⊤ convert sublattice_closure_eq_top (L : Set C(X, ℝ)) n (fun f fm g gm => inf_mem_closed_subalgebra L A.isClosed_topologicalClosure ⟨f, fm⟩ ⟨g, gm⟩) (fun f fm g gm => sup_mem_closed_subalgebra L A.isClosed_topologicalClosure ⟨f, fm⟩ ⟨g, gm⟩) (Subalgebra.SeparatesPoints.strongly (Subalgebra.separatesPoints_monotone A.le_topologicalClosure w)) case h.e'_2 X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A L : Subalgebra ℝ C(X, ℝ) := Subalgebra.topologicalClosure A n : Set.Nonempty ↑L ⊢ ↑(Subalgebra.topologicalClosure A) = closure ↑L simp ** Qed
ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : C(X, ℝ) ⊢ f ∈ Subalgebra.topologicalClosure A rw [subalgebra_topologicalClosure_eq_top_of_separatesPoints A w] X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : C(X, ℝ) ⊢ f ∈ ⊤ simp ** Qed
ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε ⊢ ∃ g, ‖↑g - f‖ < ε have w := mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f) X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∃ᶠ (x : C(X, ℝ)) in 𝓝 f, x ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε rw [Metric.nhds_basis_ball.frequently_iff] at w X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε obtain ⟨g, H, m⟩ := w ε pos case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A g : C(X, ℝ) H : g ∈ Metric.ball f ε m : g ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε rw [Metric.mem_ball, dist_eq_norm] at H case intro.intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w✝ : Subalgebra.SeparatesPoints A f : C(X, ℝ) ε : ℝ pos : 0 < ε w : ∀ (i : ℝ), 0 < i → ∃ x, x ∈ Metric.ball f i ∧ x ∈ ↑A g : C(X, ℝ) H : ‖g - f‖ < ε m : g ∈ ↑A ⊢ ∃ g, ‖↑g - f‖ < ε exact ⟨⟨g, m⟩, H⟩ ** Qed
ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : X → ℝ c : Continuous f ε : ℝ pos : 0 < ε ⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuousMap_of_separatesPoints A w ⟨f, c⟩ ε pos case intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : X → ℝ c : Continuous f ε : ℝ pos : 0 < ε g : { x // x ∈ A } b : ‖↑g - mk f‖ < ε ⊢ ∃ g, ∀ (x : X), ‖↑↑g x - f x‖ < ε use g case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : Subalgebra ℝ C(X, ℝ) w : Subalgebra.SeparatesPoints A f : X → ℝ c : Continuous f ε : ℝ pos : 0 < ε g : { x // x ∈ A } b : ‖↑g - mk f‖ < ε ⊢ ∀ (x : X), ‖↑↑g x - f x‖ < ε rwa [norm_lt_iff _ pos] at b ** Qed
Subalgebra.SeparatesPoints.isROrC_to_real 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra ⊢ SeparatesPoints (comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) intro x₁ x₂ hx 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) ∧ f x₁ ≠ f x₂ obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx case intro.intro.intro.intro 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) ∧ f x₁ ≠ f x₂ let F : C(X, 𝕜) := f - const _ (f x₂) case intro.intro.intro.intro 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) ∧ f x₁ ≠ f x₂ refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_normSq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩ 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) ⊢ ↑f x₂ • 1 = const X (↑f x₂) ext1 case h 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) a✝ : X ⊢ ↑(↑f x₂ • 1) a✝ = ↑(const X (↑f x₂)) a✝ simp only [coe_smul, coe_one, smul_apply, one_apply, Algebra.id.smul_eq_mul, mul_one, const_apply] case intro.intro.intro.intro.refine'_1 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ comp (ContinuousMap.mk ↑normSq) F ∈ ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (restrictScalars ℝ A.toSubalgebra)) rw [SetLike.mem_coe, Subalgebra.mem_comap] case intro.intro.intro.intro.refine'_1 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ ↑(AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk ↑normSq) F) ∈ restrictScalars ℝ A.toSubalgebra convert (A.restrictScalars ℝ).mul_mem hFA (star_mem hFA : star F ∈ A) ** case h.e'_4 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ ↑(AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk ↑normSq) F) = F * star F ext1 case h.e'_4.h 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A a✝ : X ⊢ ↑(↑(AlgHom.compLeftContinuous ℝ ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk ↑normSq) F)) a✝ = ↑(F * star F) a✝ exact (IsROrC.mul_conj (K := 𝕜) _).symm case intro.intro.intro.intro.refine'_2 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A ⊢ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₁ ≠ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₂ have : f x₁ - f x₂ ≠ 0 := sub_ne_zero.mpr hf case intro.intro.intro.intro.refine'_2 𝕜 : Type u_1 X : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : SeparatesPoints A.toSubalgebra x₁ x₂ : X hx : x₁ ≠ x₂ f : C(X, 𝕜) hfA : f ∈ ↑A.toSubalgebra hf : (fun f => ↑f) f x₁ ≠ (fun f => ↑f) f x₂ F : C(X, 𝕜) := f - const X (↑f x₂) hFA : F ∈ A this : ↑f x₁ - ↑f x₂ ≠ 0 ⊢ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₁ ≠ (fun f => ↑f) (comp (ContinuousMap.mk ↑normSq) F) x₂ simpa only [comp_apply, coe_sub, coe_const, sub_apply, coe_mk, sub_self, map_zero, Ne.def, normSq_eq_zero, const_apply] using this Qed
ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra ⊢ StarSubalgebra.topologicalClosure A = ⊤ rw [StarSubalgebra.eq_top_iff] 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra ⊢ ∀ (x : C(X, 𝕜)), x ∈ StarSubalgebra.topologicalClosure A let I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ofRealClm.compLeftContinuous ℝ X 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm ⊢ ∀ (x : C(X, 𝕜)), x ∈ StarSubalgebra.topologicalClosure A have key : LinearMap.range I ≤ (A.toSubmodule.restrictScalars ℝ).topologicalClosure := by let A₀ : Submodule ℝ C(X, ℝ) := (A.toSubmodule.restrictScalars ℝ).comap I have SW : A₀.topologicalClosure = ⊤ := haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real congr_arg Subalgebra.toSubmodule this rw [← Submodule.map_top, ← SW] have h₁ := A₀.topologicalClosure_map ((@ofRealClm 𝕜 _).compLeftContinuousCompact X) have h₂ := (A.toSubmodule.restrictScalars ℝ).map_comap_le I exact h₁.trans (Submodule.topologicalClosure_mono h₂) 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) ⊢ ∀ (x : C(X, 𝕜)), x ∈ StarSubalgebra.topologicalClosure A intro f 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) ⊢ f ∈ StarSubalgebra.topologicalClosure A let f_re : C(X, ℝ) := (⟨IsROrC.re, IsROrC.reClm.continuous⟩ : C(𝕜, ℝ)).comp f 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f ⊢ f ∈ StarSubalgebra.topologicalClosure A let f_im : C(X, ℝ) := (⟨IsROrC.im, IsROrC.imClm.continuous⟩ : C(𝕜, ℝ)).comp f 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f ⊢ f ∈ StarSubalgebra.topologicalClosure A have h_f_re : I f_re ∈ A.topologicalClosure := key ⟨f_re, rfl⟩ 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A ⊢ f ∈ StarSubalgebra.topologicalClosure A have h_f_im : I f_im ∈ A.topologicalClosure := key ⟨f_im, rfl⟩ 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A ⊢ f ∈ StarSubalgebra.topologicalClosure A have := A.topologicalClosure.add_mem h_f_re (A.topologicalClosure.smul_mem h_f_im IsROrC.I) 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ (StarSubalgebra.topologicalClosure A).toSubalgebra ⊢ f ∈ StarSubalgebra.topologicalClosure A rw [StarSubalgebra.mem_toSubalgebra] at this 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A ⊢ f ∈ StarSubalgebra.topologicalClosure A convert this case h.e'_4 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A ⊢ f = ↑I f_re + IsROrC.I • ↑I f_im ext case h.e'_4.h 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A a✝ : X ⊢ ↑f a✝ = ↑(↑I f_re + IsROrC.I • ↑I f_im) a✝ apply Eq.symm case h.e'_4.h.h 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm key : LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) f : C(X, 𝕜) f_re : C(X, ℝ) := comp (mk ↑re) f f_im : C(X, ℝ) := comp (mk ↑im) f h_f_re : ↑I f_re ∈ StarSubalgebra.topologicalClosure A h_f_im : ↑I f_im ∈ StarSubalgebra.topologicalClosure A this : ↑I f_re + IsROrC.I • ↑I f_im ∈ StarSubalgebra.topologicalClosure A a✝ : X ⊢ ↑(↑I f_re + IsROrC.I • ↑I f_im) a✝ = ↑f a✝ simp [mul_comm IsROrC.I _] 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm ⊢ LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) let A₀ : Submodule ℝ C(X, ℝ) := (A.toSubmodule.restrictScalars ℝ).comap I 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) ⊢ LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) have SW : A₀.topologicalClosure = ⊤ := haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real congr_arg Subalgebra.toSubmodule this 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ ⊢ LinearMap.range I ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) rw [← Submodule.map_top, ← SW] 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ ⊢ Submodule.map I (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) have h₁ := A₀.topologicalClosure_map ((@ofRealClm 𝕜 _).compLeftContinuousCompact X) 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ h₁ : Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) A₀) ⊢ Submodule.map I (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) have h₂ := (A.toSubmodule.restrictScalars ℝ).map_comap_le I 𝕜 : Type u_1 X : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : TopologicalSpace X inst✝ : CompactSpace X A : StarSubalgebra 𝕜 C(X, 𝕜) hA : Subalgebra.SeparatesPoints A.toSubalgebra I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealClm A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) SW : Submodule.topologicalClosure A₀ = ⊤ h₁ : Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) A₀) h₂ : Submodule.map I (Submodule.comap I (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra))) ≤ Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra) ⊢ Submodule.map I (Submodule.topologicalClosure A₀) ≤ Submodule.topologicalClosure (Submodule.restrictScalars ℝ (↑Subalgebra.toSubmodule A.toSubalgebra)) exact h₁.trans (Submodule.topologicalClosure_mono h₂) ** Qed
ContinuousMap.algHom_ext_map_X A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ φ = ψ suffices (⊤ : Subalgebra ℝ C(s, ℝ)) ≤ AlgHom.equalizer φ ψ from AlgHom.ext fun x => this (by trivial) A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ ⊤ ≤ AlgHom.equalizer φ ψ rw [← polynomialFunctions.topologicalClosure s] A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ Subalgebra.topologicalClosure (polynomialFunctions s) ≤ AlgHom.equalizer φ ψ exact Subalgebra.topologicalClosure_minimal (polynomialFunctions s) (polynomialFunctions.le_equalizer s φ ψ h) (isClosed_eq hφ hψ) A : Type u_1 inst✝⁴ : Ring A inst✝³ : Algebra ℝ A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set ℝ inst✝ : CompactSpace ↑s φ ψ : C(↑s, ℝ) →ₐ[ℝ] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) this : ⊤ ≤ AlgHom.equalizer φ ψ x : C(↑s, ℝ) ⊢ x ∈ ⊤ trivial ** Qed
ContinuousMap.starAlgHom_ext_map_X 𝕜 : Type u_1 A : Type u_2 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : Ring A inst✝⁴ : StarRing A inst✝³ : Algebra 𝕜 A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set 𝕜 inst✝ : CompactSpace ↑s φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ φ = ψ suffices (⊤ : StarSubalgebra 𝕜 C(s, 𝕜)) ≤ StarAlgHom.equalizer φ ψ from StarAlgHom.ext fun x => this mem_top 𝕜 : Type u_1 A : Type u_2 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : Ring A inst✝⁴ : StarRing A inst✝³ : Algebra 𝕜 A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set 𝕜 inst✝ : CompactSpace ↑s φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ ⊤ ≤ StarAlgHom.equalizer φ ψ rw [← polynomialFunctions.starClosure_topologicalClosure s] 𝕜 : Type u_1 A : Type u_2 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : Ring A inst✝⁴ : StarRing A inst✝³ : Algebra 𝕜 A inst✝² : TopologicalSpace A inst✝¹ : T2Space A s : Set 𝕜 inst✝ : CompactSpace ↑s φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A hφ : Continuous ↑φ hψ : Continuous ↑ψ h : ↑φ (↑(toContinuousMapOnAlgHom s) X) = ↑ψ (↑(toContinuousMapOnAlgHom s) X) ⊢ topologicalClosure (Subalgebra.starClosure (polynomialFunctions s)) ≤ StarAlgHom.equalizer φ ψ exact StarSubalgebra.topologicalClosure_minimal (polynomialFunctions.starClosure_le_equalizer s φ ψ h) (isClosed_eq hφ hψ) ** Qed
Profinite.effectiveEpiFamily_tfae α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_have 1 → 2 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_have 2 → 3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_have 3 → 1 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b tfae_3_to_1 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] tfae_finish case tfae_1_to_2 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B ⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) intro case tfae_1_to_2 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B ✝ : EffectiveEpiFamily X π ⊢ Epi (Sigma.desc π) infer_instance case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b intro e case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Epi (Sigma.desc π) ⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b rw [epi_iff_surjective] at e case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) ⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b let i : ∐ X ≅ finiteCoproduct X := (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _) case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) ⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b intro b case tfae_2_to_3 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) b : ↑B.toCompHaus.toTop ⊢ ∃ a x, ↑(π a) x = b obtain ⟨t, rfl⟩ := e b case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t let q := i.hom t case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t refine ⟨q.1, q.2, ?_⟩ case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t have : t = i.inv (i.hom t) case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t rw [this] case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t)) show _ = (i.inv ≫ Sigma.desc π) (i.hom t) case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) suffices : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π rw [Iso.inv_comp_eq] case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π apply colimit.hom_ext case this.w α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ∀ (j : Discrete α), colimit.ι (Discrete.functor X) j ≫ Sigma.desc π = colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π rintro ⟨a⟩ case this.w.mk α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α ⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π = colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app, colimit.comp_coconePointUniqueUpToIso_hom_assoc] case this.w.mk α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α ⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π ext case this.w.mk.w α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α x✝ : (forget Profinite).obj (X a) ⊢ ↑(π a) x✝ = ↑((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝ rfl case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑i.inv (↑i.hom t) show t = (i.hom ≫ i.inv) t case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑(i.hom ≫ i.inv) t simp only [i.hom_inv_id] case this α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑(𝟙 (∐ X)) t rfl case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this✝ : t = ↑i.inv (↑i.hom t) this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) rw [this] case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget Profinite).obj (∐ fun b => X b) q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t this✝ : t = ↑i.inv (↑i.hom t) this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑i.hom t) rfl case tfae_3_to_1 α : Type inst✝ : Fintype α B : Profinite X : α → Profinite π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b ⊢ (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π apply effectiveEpiFamily_of_jointly_surjective ** Qed
Path.Homotopy.eval_zero X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ ⊢ eval F 0 = p₀ ext t case a.h X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ t : ↑I ⊢ ↑(eval F 0) t = ↑p₀ t simp [eval] ** Qed
Path.Homotopy.eval_one X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ ⊢ eval F 1 = p₁ ext t case a.h X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ p₁ : Path x₀ x₁ F : Homotopy p₀ p₁ t : ↑I ⊢ ↑(eval F 1) t = ↑p₁ t simp [eval] ** Qed
Path.Homotopy.hcomp_apply ** X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ x : ↑I × ↑I ⊢ (if ↑x.2 ≤ 1 / 2 then extend (eval F x.1) (2 * ↑x.2) else extend (eval G x.1) (2 * ↑x.2 - 1)) = if h : ↑x.2 ≤ 1 / 2 then ↑(eval F x.1) { val := 2 * ↑x.2, property := (_ : 2 * ↑x.2 ∈ I) } else ↑(eval G x.1) { val := 2 * ↑x.2 - 1, property := (_ : 2 * ↑x.2 - 1 ∈ I) } split_ifs <;> exact Path.extend_extends _ _ Qed
Path.Homotopy.hcomp_half X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ t : ↑I ⊢ 0 ≤ 1 / 2 norm_num X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ t : ↑I ⊢ 1 / 2 ≤ 1 norm_num ** X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₀ q₀ : Path x₀ x₁ p₁ q₁ : Path x₁ x₂ F : Homotopy p₀ q₀ G : Homotopy p₁ q₁ t : ↑I ⊢ (if ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2 ≤ 1 / 2 then extend (eval F (t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).1) (2 * ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2) else extend (eval G (t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).1) (2 * ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2 - 1)) = x₁ norm_num Qed
Path.Homotopic.hpath_hext X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₂ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ HEq (Quotient.mk (Homotopic.setoid x₀ x₁) p₁) (Quotient.mk (Homotopic.setoid x₂ x₃) p₂) obtain rfl : x₀ = x₂ := by convert hp 0 <;> simp X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₀ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ HEq (Quotient.mk (Homotopic.setoid x₀ x₁) p₁) (Quotient.mk (Homotopic.setoid x₀ x₃) p₂) obtain rfl : x₁ = x₃ := by convert hp 1 <;> simp X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ HEq (Quotient.mk (Homotopic.setoid x₀ x₁) p₁) (Quotient.mk (Homotopic.setoid x₀ x₁) p₂) rw [heq_iff_eq] X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ Quotient.mk (Homotopic.setoid x₀ x₁) p₁ = Quotient.mk (Homotopic.setoid x₀ x₁) p₂ congr case e_a X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ p₁ = p₂ ext t case e_a.a.h X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ : X p₁ p₂ : Path x₀ x₁ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t t : ↑I ⊢ ↑p₁ t = ↑p₂ t exact hp t X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₂ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ x₀ = x₂ convert hp 0 <;> simp X : Type u Y : Type v inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x₀ x₁ x₃ : X p₁ : Path x₀ x₁ p₂ : Path x₀ x₃ hp : ∀ (t : ↑I), ↑p₁ t = ↑p₂ t ⊢ x₁ = x₃ convert hp 1 <;> simp ** Qed
ContinuousMap.homotopic_const_iff X : Type u Y : Type v inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X inst✝ : Nonempty Y ⊢ Homotopic (const Y x₀) (const Y x₁) ↔ Joined x₀ x₁ inhabit Y X : Type u Y : Type v inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y x₀ x₁ x₂ x₃ : X inst✝ : Nonempty Y inhabited_h : Inhabited Y ⊢ Homotopic (const Y x₀) (const Y x₁) ↔ Joined x₀ x₁ refine ⟨fun ⟨H⟩ ↦ ⟨⟨(H.toContinuousMap.comp .prodSwap).curry default, ?_, ?_⟩⟩, fun ⟨p⟩ ↦ ⟨p.toHomotopyConst⟩⟩ <;> simp ** Qed
TopCat.isTopologicalBasis_cofiltered_limit J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} let D := limitConeInfi F J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} let E : C.pt ≅ D.pt := hC.conePointUniqueUpToIso (limitConeInfiIsLimit _) J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} have hE : Inducing E.hom := (TopCat.homeoOfIso E).inducing J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} convert isTopologicalBasis_iInf hT fun j (x : D.pt) => D.π.app j x using 1 case h.e'_3 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom ⊢ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} = {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} ext U0 case h.e'_3.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} ↔ U0 ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} constructor J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} ⊢ IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} convert this.inducing hE case h.e'_3.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt ⊢ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} = Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} ext U0 case h.e'_3.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt U0 : Set ↑C.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} ↔ U0 ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} constructor case h.e'_3.h.h.mp J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt U0 : Set ↑C.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} → U0 ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} rintro ⟨j, V, hV, rfl⟩ case h.e'_3.h.h.mp.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt j : J V : Set ↑(F.obj j) hV : V ∈ T j ⊢ ↑(C.π.app j) ⁻¹' V ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} refine' ⟨D.π.app j ⁻¹' V, ⟨j, V, hV, rfl⟩, rfl⟩ case h.e'_3.h.h.mpr J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt U0 : Set ↑C.pt ⊢ U0 ∈ Set.preimage ↑E.hom '' {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} → U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} rintro ⟨W, ⟨j, V, hV, rfl⟩, rfl⟩ case h.e'_3.h.h.mpr.intro.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom this : IsTopologicalBasis {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} e_1✝ : ↑C.pt = (forget TopCat).obj C.pt j : J V : Set ↑(F.obj j) hV : V ∈ T j ⊢ ↑E.hom ⁻¹' (↑(D.π.app j) ⁻¹' V) ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(C.π.app j) ⁻¹' V} refine' ⟨j, V, hV, rfl⟩ case h.e'_3.h.mp J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} → U0 ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} rintro ⟨j, V, hV, rfl⟩ case h.e'_3.h.mp.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j ⊢ ↑(D.π.app j) ⁻¹' V ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} let U : ∀ i, Set (F.obj i) := fun i => if h : i = j then by rw [h]; exact V else Set.univ case h.e'_3.h.mp.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ↑(D.π.app j) ⁻¹' V ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} refine' ⟨U, {j}, _, _⟩ J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j i : J h : i = j ⊢ Set ↑(F.obj i) rw [h] J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j i : J h : i = j ⊢ Set ↑(F.obj j) exact V case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ∀ (i : J), i ∈ {j} → U i ∈ T i rintro i h case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i ∈ {j} ⊢ U i ∈ T i rw [Finset.mem_singleton] at h case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i = j ⊢ U i ∈ T i dsimp case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i = j ⊢ (if h : i = j then cast (_ : Set ↑(F.obj j) = Set ↑(F.obj i)) V else Set.univ) ∈ T i rw [dif_pos h] case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ i : J h : i = j ⊢ cast (_ : Set ↑(F.obj j) = Set ↑(F.obj i)) V ∈ T i subst h case h.e'_3.h.mp.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom i : J V : Set ↑(F.obj i) hV : V ∈ T i U : (i : J) → Set ↑(F.obj i) := fun i_1 => if h : i_1 = i then Eq.mpr (_ : Set ↑(F.obj i_1) = Set ↑(F.obj i)) V else Set.univ ⊢ cast (_ : Set ↑(F.obj i) = Set ↑(F.obj i)) V ∈ T i exact hV case h.e'_3.h.mp.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ↑(D.π.app j) ⁻¹' V = ⋂ i ∈ {j}, (fun x => ↑(D.π.app i) x) ⁻¹' U i dsimp case h.e'_3.h.mp.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom j : J V : Set ↑(F.obj j) hV : V ∈ T j U : (i : J) → Set ↑(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set ↑(F.obj i) = Set ↑(F.obj j)) V else Set.univ ⊢ ↑((limitConeInfi F).π.app j) ⁻¹' V = ⋂ i ∈ {j}, (fun x => ↑((limitConeInfi F).π.app i) x) ⁻¹' if h : i = j then cast (_ : Set ↑(F.obj j) = Set ↑(F.obj i)) V else Set.univ simp case h.e'_3.h.mpr J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt ⊢ U0 ∈ {S \| ∃ U F_1, (∀ (i : J), i ∈ F_1 → U i ∈ T i) ∧ S = ⋂ i ∈ F_1, (fun x => ↑(D.π.app i) x) ⁻¹' U i} → U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} rintro ⟨U, G, h1, h2⟩ case h.e'_3.h.mpr.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} obtain ⟨j, hj⟩ := IsCofiltered.inf_objs_exists G case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} let g : ∀ (e) (_he : e ∈ G), j ⟶ e := fun _ he => (hj he).some case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} let Vs : J → Set (F.obj j) := fun e => if h : e ∈ G then F.map (g e h) ⁻¹' U e else Set.univ case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} let V : Set (F.obj j) := ⋂ (e : J) (_he : e ∈ G), Vs e case h.e'_3.h.mpr.intro.intro.intro.intro J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ U0 ∈ {U \| ∃ j V, V ∈ T j ∧ U = ↑(D.π.app j) ⁻¹' V} refine' ⟨j, V, _, _⟩ case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ V ∈ T j have : ∀ (S : Set (Set (F.obj j))) (E : Finset J) (P : J → Set (F.obj j)) (_univ : Set.univ ∈ S) (_inter : ∀ A B : Set (F.obj j), A ∈ S → B ∈ S → A ∩ B ∈ S) (_cond : ∀ (e : J) (_he : e ∈ E), P e ∈ S), (⋂ (e) (_he : e ∈ E), P e) ∈ S := by intro S E induction E using Finset.induction_on with \| empty => intro P he _hh simpa \| @insert a E _ha hh1 => intro hh2 hh3 hh4 hh5 rw [Finset.set_biInter_insert] refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _) intro e he exact hh5 e (Finset.mem_insert_of_mem he) case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S ⊢ V ∈ T j refine' this _ _ _ (univ _) (inter _) _ case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S ⊢ ∀ (e : J), e ∈ G → Vs e ∈ T j intro e he case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S e : J he : e ∈ G ⊢ Vs e ∈ T j dsimp case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S e : J he : e ∈ G ⊢ (if h : e ∈ G then ↑(F.map (Nonempty.some (_ : Nonempty (j ⟶ e)))) ⁻¹' U e else Set.univ) ∈ T j rw [dif_pos he] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e this : ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S e : J he : e ∈ G ⊢ ↑(F.map (Nonempty.some (_ : Nonempty (j ⟶ e)))) ⁻¹' U e ∈ T j exact compat j e (g e he) (U e) (h1 e he) J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S intro S E J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) E : Finset J ⊢ ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S induction E using Finset.induction_on with \| empty => intro P he _hh simpa \| @insert a E _ha hh1 => intro hh2 hh3 hh4 hh5 rw [Finset.set_biInter_insert] refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _) intro e he exact hh5 e (Finset.mem_insert_of_mem he) case empty J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) ⊢ ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ ∅ → P e ∈ S) → ⋂ e ∈ ∅, P e ∈ S intro P he _hh case empty J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) P : J → Set ↑(F.obj j) he : Set.univ ∈ S _hh : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S ⊢ (∀ (e : J), e ∈ ∅ → P e ∈ S) → ⋂ e ∈ ∅, P e ∈ S simpa case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S ⊢ ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ insert a E → P e ∈ S) → ⋂ e ∈ insert a E, P e ∈ S intro hh2 hh3 hh4 hh5 case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S ⊢ ⋂ e ∈ insert a E, hh2 e ∈ S rw [Finset.set_biInter_insert] case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S ⊢ hh2 a ∩ ⋂ x ∈ E, hh2 x ∈ S refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _) case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S ⊢ ∀ (e : J), e ∈ E → hh2 e ∈ S intro e he case insert J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E✝ : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E✝.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e S : Set (Set ↑(F.obj j)) a : J E : Finset J _ha : ¬a ∈ E hh1 : ∀ (P : J → Set ↑(F.obj j)), Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ (e : J), e ∈ E → P e ∈ S) → ⋂ e ∈ E, P e ∈ S hh2 : J → Set ↑(F.obj j) hh3 : Set.univ ∈ S hh4 : ∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S hh5 : ∀ (e : J), e ∈ insert a E → hh2 e ∈ S e : J he : e ∈ E ⊢ hh2 e ∈ S exact hh5 e (Finset.mem_insert_of_mem he) case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ U0 = ↑(D.π.app j) ⁻¹' V rw [h2] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i = ↑(D.π.app j) ⁻¹' V change _ = (D.π.app j)⁻¹' ⋂ (e : J) (_ : e ∈ G), Vs e case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i = ↑(D.π.app j) ⁻¹' ⋂ e ∈ G, Vs e rw [Set.preimage_iInter] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2 J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i = ⋂ i, ↑(D.π.app j) ⁻¹' ⋂ (_ : i ∈ G), Vs i apply congrArg case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e ⊢ (fun i => ⋂ (_ : i ∈ G), (fun x => ↑(D.π.app i) x) ⁻¹' U i) = fun i => ↑(D.π.app j) ⁻¹' ⋂ (_ : i ∈ G), Vs i ext1 e case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J ⊢ ⋂ (_ : e ∈ G), (fun x => ↑(D.π.app e) x) ⁻¹' U e = ↑(D.π.app j) ⁻¹' ⋂ (_ : e ∈ G), Vs e erw [Set.preimage_iInter] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J ⊢ ⋂ (_ : e ∈ G), (fun x => ↑(D.π.app e) x) ⁻¹' U e = ⋂ (_ : e ∈ G), ↑(D.π.app j) ⁻¹' Vs e apply congrArg case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J ⊢ (fun x => (fun x => ↑(D.π.app e) x) ⁻¹' U e) = fun i => ↑(D.π.app j) ⁻¹' Vs e ext1 he case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ (fun x => ↑(D.π.app e) x) ⁻¹' U e = ↑(D.π.app j) ⁻¹' Vs e change (D.π.app e)⁻¹' U e = (D.π.app j) ⁻¹' if h : e ∈ G then F.map (g e h) ⁻¹' U e else Set.univ case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) ⁻¹' U e = ↑(D.π.app j) ⁻¹' if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ rw [dif_pos he, ← Set.preimage_comp] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) ⁻¹' U e = ↑(F.map (g e he)) ∘ ↑(D.π.app j) ⁻¹' U e apply congrFun case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ Set.preimage ↑(D.π.app e) = Set.preimage (↑(F.map (g e he)) ∘ ↑(D.π.app j)) apply congrArg case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) = ↑(F.map (g e he)) ∘ ↑(D.π.app j) rw [←coe_comp, D.w] case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h.h J : Type v inst✝¹ : SmallCategory J inst✝ : IsCofiltered J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C T : (j : J) → Set (Set ↑(F.obj j)) hT : ∀ (j : J), IsTopologicalBasis (T j) univ : ∀ (i : J), Set.univ ∈ T i inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i compat : ∀ (i j : J) (f : i ⟶ j) (V : Set ↑(F.obj j)), V ∈ T j → ↑(F.map f) ⁻¹' V ∈ T i D : Cone F := limitConeInfi F E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F) hE : Inducing ↑E.hom U0 : Set ↑D.pt U : (i : J) → Set ↑(F.obj i) G : Finset J h1 : ∀ (i : J), i ∈ G → U i ∈ T i h2 : U0 = ⋂ i ∈ G, (fun x => ↑(D.π.app i) x) ⁻¹' U i j : J hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X) g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x)) Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ↑(F.map (g e h)) ⁻¹' U e else Set.univ V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e e : J he : e ∈ G ⊢ ↑(D.π.app e) = ↑(D.π.app e) rfl ** Qed
TopCat.of_isoOfHomeo X Y : TopCat f : ↑X ≃ₜ ↑Y ⊢ homeoOfIso (isoOfHomeo f) = f dsimp [homeoOfIso, isoOfHomeo] X Y : TopCat f : ↑X ≃ₜ ↑Y ⊢ Homeomorph.mk { toFun := ↑(Homeomorph.toContinuousMap f), invFun := ↑(Homeomorph.toContinuousMap (Homeomorph.symm f)), left_inv := (_ : ∀ (x : ↑X), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 x) = x) } = f ext case H X Y : TopCat f : ↑X ≃ₜ ↑Y x✝ : ↑X ⊢ ↑(Homeomorph.mk { toFun := ↑(Homeomorph.toContinuousMap f), invFun := ↑(Homeomorph.toContinuousMap (Homeomorph.symm f)), left_inv := (_ : ∀ (x : ↑X), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).1 (↑(Iso.mk (Homeomorph.toContinuousMap f) (Homeomorph.toContinuousMap (Homeomorph.symm f))).2 x) = x) }) x✝ = ↑f x✝ rfl ** Qed
TopCat.of_homeoOfIso X Y : TopCat f : X ≅ Y ⊢ isoOfHomeo (homeoOfIso f) = f dsimp [homeoOfIso, isoOfHomeo] X Y : TopCat f : X ≅ Y ⊢ Iso.mk (Homeomorph.toContinuousMap (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) })) (Homeomorph.toContinuousMap (Homeomorph.symm (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) }))) = f ext case w.w X Y : TopCat f : X ≅ Y x✝ : (forget TopCat).obj X ⊢ ↑(Iso.mk (Homeomorph.toContinuousMap (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) })) (Homeomorph.toContinuousMap (Homeomorph.symm (Homeomorph.mk { toFun := ↑f.hom, invFun := ↑f.inv, left_inv := (_ : ∀ (x : ↑X), ↑f.2 (↑f.1 x) = x), right_inv := (_ : ∀ (x : ↑Y), ↑f.1 (↑f.2 x) = x) })))).hom x✝ = ↑f.hom x✝ rfl ** Qed
TopCat.openEmbedding_iff_comp_isIso' X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso g ⊢ OpenEmbedding ((forget TopCat).map f ≫ (forget TopCat).map g) ↔ OpenEmbedding ↑f simp only [←Functor.map_comp] X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso g ⊢ OpenEmbedding ((forget TopCat).map (f ≫ g)) ↔ OpenEmbedding ↑f exact openEmbedding_iff_comp_isIso f g ** Qed
TopCat.openEmbedding_iff_isIso_comp X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ↑(f ≫ g) ↔ OpenEmbedding ↑g constructor case mp X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ↑(f ≫ g) → OpenEmbedding ↑g intro h case mp X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f h : OpenEmbedding ↑(f ≫ g) ⊢ OpenEmbedding ↑g convert h.comp (TopCat.homeoOfIso (asIso f).symm).openEmbedding case h.e'_5.h X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f h : OpenEmbedding ↑(f ≫ g) e_1✝ : (forget TopCat).obj Y = ↑Y ⊢ ↑g = ↑(f ≫ g) ∘ ↑(homeoOfIso (asIso f).symm) exact congrArg _ (IsIso.inv_hom_id_assoc f g).symm case mpr X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ↑g → OpenEmbedding ↑(f ≫ g) exact fun h => h.comp (TopCat.homeoOfIso (asIso f)).openEmbedding ** Qed
TopCat.openEmbedding_iff_isIso_comp' X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ((forget TopCat).map f ≫ (forget TopCat).map g) ↔ OpenEmbedding ↑g simp only [←Functor.map_comp] X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z inst✝ : IsIso f ⊢ OpenEmbedding ((forget TopCat).map (f ≫ g)) ↔ OpenEmbedding ↑g exact openEmbedding_iff_isIso_comp f g ** Qed
ContinuousMap.coe_injective α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f✝ g✝ f g : C(α, β) h : ↑f = ↑g ⊢ f = g cases f case mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g✝ g : C(α, β) toFun✝ : α → β continuous_toFun✝ : Continuous toFun✝ h : ↑(mk toFun✝) = ↑g ⊢ mk toFun✝ = g cases g case mk.mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) toFun✝¹ : α → β continuous_toFun✝¹ : Continuous toFun✝¹ toFun✝ : α → β continuous_toFun✝ : Continuous toFun✝ h : ↑(mk toFun✝¹) = ↑(mk toFun✝) ⊢ mk toFun✝¹ = mk toFun✝ congr ** Qed
ContinuousMap.cancel_left α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f✝ g : C(α, β) f : C(β, γ) g₁ g₂ : C(α, β) hf : Injective ↑f h : comp f g₁ = comp f g₂ a : α ⊢ ↑f (↑g₁ a) = ↑f (↑g₂ a) rw [← comp_apply, h, comp_apply] ** Qed
ContinuousMap.liftCover_coe α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x i : ι x : ↑(S i) ⊢ ↑(liftCover S φ hφ hS) ↑x = ↑(φ i) x rw [liftCover, coe_mk, Set.liftCover_coe _] ** Qed
ContinuousMap.liftCover_restrict α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x i : ι ⊢ restrict (S i) (liftCover S φ hφ hS) = φ i ext case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x i : ι a✝ : ↑(S i) ⊢ ↑(restrict (S i) (liftCover S φ hφ hS)) a✝ = ↑(φ i) a✝ simp only [coe_restrict, Function.comp_apply, liftCover_coe] ** Qed
ContinuousMap.liftCover_coe' α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x A : Set (Set α) F : (s : Set α) → s ∈ A → C(↑s, β) hF : ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t), ↑(F s hs) { val := x, property := hxi } = ↑(F t ht) { val := x, property := hxj } hA : ∀ (x : α), ∃ i, i ∈ A ∧ i ∈ nhds x s : Set α hs : s ∈ A x : ↑s x' : ↑↑{ val := s, property := hs } := x ⊢ ↑(liftCover' A F hF hA) ↑x = ↑(F s hs) x delta liftCover' α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : C(α, β) ι : Type u_5 S : ι → Set α φ : (i : ι) → C(↑(S i), β) hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), ↑(φ i) { val := x, property := hxi } = ↑(φ j) { val := x, property := hxj } hS : ∀ (x : α), ∃ i, S i ∈ nhds x A : Set (Set α) F : (s : Set α) → s ∈ A → C(↑s, β) hF : ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t), ↑(F s hs) { val := x, property := hxi } = ↑(F t ht) { val := x, property := hxj } hA : ∀ (x : α), ∃ i, i ∈ A ∧ i ∈ nhds x s : Set α hs : s ∈ A x : ↑s x' : ↑↑{ val := s, property := hs } := x ⊢ ↑(let S := Subtype.val; let F_1 := fun i => F ↑i (_ : ↑i ∈ A); liftCover S F_1 (_ : ∀ (i j : ↑A) (x : α) (hxi : x ∈ ↑i) (hxj : x ∈ ↑j), ↑(F ↑i (_ : ↑i ∈ A)) { val := x, property := hxi } = ↑(F ↑j (_ : ↑j ∈ A)) { val := x, property := hxj }) (_ : ∀ (x : α), ∃ i, ↑i ∈ nhds x)) ↑x = ↑(F s hs) x exact liftCover_coe x' ** Qed
Homeomorph.symm_comp_toContinuousMap α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ f : α ≃ₜ β g : β ≃ₜ γ ⊢ ContinuousMap.comp (toContinuousMap (Homeomorph.symm f)) (toContinuousMap f) = ContinuousMap.id α rw [← coe_trans, self_trans_symm, coe_refl] ** Qed
Homeomorph.toContinuousMap_comp_symm α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ f : α ≃ₜ β g : β ≃ₜ γ ⊢ ContinuousMap.comp (toContinuousMap f) (toContinuousMap (Homeomorph.symm f)) = ContinuousMap.id β rw [← coe_trans, symm_trans_self, coe_refl] ** Qed
Stonean.effectiveEpiFamily_tfae α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 1 → 2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 1 → 2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 2 → 3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_have 3 → 1 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b tfae_3_to_1 : (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b] tfae_finish case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B ⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) intro case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B ✝ : EffectiveEpiFamily X π ⊢ Epi (Sigma.desc π) infer_instance case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) intro case tfae_1_to_2 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ✝ : EffectiveEpiFamily X π ⊢ Epi (Sigma.desc π) infer_instance case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b intro e case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Epi (Sigma.desc π) ⊢ ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b rw [epi_iff_surjective] at e case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) ⊢ ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b intro b case tfae_2_to_3 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) b : CoeSort.coe B ⊢ ∃ a x, ↑(π a) x = b obtain ⟨t, rfl⟩ := e b case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t let q := (coproductIsoCoproduct X).inv t case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t refine ⟨q.1, q.2, ?_⟩ case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t rw [← (coproductIsoCoproduct X).inv_hom_id_apply t] case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑(coproductIsoCoproduct X).1 (↑(coproductIsoCoproduct X).2 t)) show _ = ((coproductIsoCoproduct X).hom ≫ Sigma.desc π) ((coproductIsoCoproduct X).inv t) case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ↑(π q.fst) q.snd = ↑((coproductIsoCoproduct X).hom ≫ Sigma.desc π) (↑(coproductIsoCoproduct X).inv t) suffices : (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π case this α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π apply Eq.symm case this.h α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ finiteCoproduct.desc X π = (coproductIsoCoproduct X).hom ≫ Sigma.desc π rw [← Iso.inv_comp_eq] case this.h α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ (coproductIsoCoproduct X).inv ≫ finiteCoproduct.desc X π = Sigma.desc π apply colimit.hom_ext case this.h.w α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t ⊢ ∀ (j : Discrete α), colimit.ι (Discrete.functor X) j ≫ (coproductIsoCoproduct X).inv ≫ finiteCoproduct.desc X π = colimit.ι (Discrete.functor X) j ≫ Sigma.desc π rintro ⟨a⟩ case this.h.w.mk α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t a : α ⊢ colimit.ι (Discrete.functor X) { as := a } ≫ (coproductIsoCoproduct X).inv ≫ finiteCoproduct.desc X π = colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app, coproductIsoCoproduct, colimit.comp_coconePointUniqueUpToIso_inv_assoc] case this.h.w.mk α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t a : α ⊢ (finiteCoproduct.explicitCocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π = π a ext case this.h.w.mk.w α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t a : α x✝ : (forget Stonean).obj (X a) ⊢ ↑((finiteCoproduct.explicitCocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝ = ↑(π a) x✝ rfl case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t this : (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑((coproductIsoCoproduct X).hom ≫ Sigma.desc π) (↑(coproductIsoCoproduct X).inv t) rw [this] case tfae_2_to_3.intro α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) t : CoeSort.coe (∐ fun b => X b) q : (fun x => CoeSort.coe (finiteCoproduct X)) t := ↑(coproductIsoCoproduct X).inv t this : (coproductIsoCoproduct X).hom ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑(coproductIsoCoproduct X).inv t) rfl case tfae_3_to_1 α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b ⊢ (∀ (b : CoeSort.coe B), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π apply effectiveEpiFamily_of_jointly_surjective ** Qed
CategoryTheory.EffectiveEpiFamily.toCompHaus α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B H : EffectiveEpiFamily X π ⊢ EffectiveEpiFamily (fun x => Stonean.toCompHaus.obj (X x)) fun x => Stonean.toCompHaus.map (π x) refine' ((CompHaus.effectiveEpiFamily_tfae _ _).out 0 2).2 (fun b => _) α✝ : Type inst✝¹ : Fintype α✝ B✝ : Stonean X✝ : α✝ → Stonean π✝ : (a : α✝) → X✝ a ⟶ B✝ surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b α : Type inst✝ : Fintype α B : Stonean X : α → Stonean π : (a : α) → X a ⟶ B H : EffectiveEpiFamily X π b : ↑(Stonean.toCompHaus.obj B).toTop ⊢ ∃ a x, ↑(Stonean.toCompHaus.map (π a)) x = b exact (((effectiveEpiFamily_tfae _ _).out 0 2).1 H : ∀ _, ∃ _, _) _ ** Qed