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cauchySeq_of_edist_le_of_tsum_ne_top ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0∞ hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n hd : tsum d ≠ ⊤ ⊢ CauchySeq f ** lift d to ℕ → NNReal using fun i => ENNReal.ne_top_of_tsum_ne_top hd i ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0 hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ (fun i => ↑(d i)) n hd : ∑' (i : ℕ), ↑(d i) ≠ ⊤ ⊢ CauchySeq f ** rw [ENNReal.tsum_coe_ne_top_iff_summable] at hd ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0 hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ (fun i => ↑(d i)) n hd : Summable fun i => d i ⊢ CauchySeq f ** exact cauchySeq_of_edist_le_of_summable d hf hd ** Qed
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EMetric.diam_closure ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set α ⊢ diam (closure s) = diam s ** refine' le_antisymm (diam_le fun x hx y hy => _) (diam_mono subset_closure) ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set α x : α hx : x ∈ closure s y : α hy : y ∈ closure s ⊢ edist x y ≤ diam s ** have : edist x y ∈ closure (Iic (diam s)) :=
map_mem_closure₂ continuous_edist hx hy fun x hx y hy => edist_le_diam_of_mem hx hy ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set α x : α hx : x ∈ closure s y : α hy : y ∈ closure s this : edist x y ∈ closure (Iic (diam s)) ⊢ edist x y ≤ diam s ** rwa [closure_Iic] at this ** Qed
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Metric.diam_closure ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 inst✝¹ : PseudoEMetricSpace α✝ α : Type u_4 inst✝ : PseudoMetricSpace α s : Set α ⊢ diam (closure s) = diam s ** simp only [Metric.diam, EMetric.diam_closure] ** Qed
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isClosed_setOf_lipschitzOnWith ** α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 inst✝² : PseudoEMetricSpace α✝ α : Type u_4 β : Type u_5 inst✝¹ : PseudoEMetricSpace α inst✝ : PseudoEMetricSpace β K : ℝ≥0 s : Set α ⊢ IsClosed {f | LipschitzOnWith K f s} ** simp only [LipschitzOnWith, setOf_forall] ** α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 inst✝² : PseudoEMetricSpace α✝ α : Type u_4 β : Type u_5 inst✝¹ : PseudoEMetricSpace α inst✝ : PseudoEMetricSpace β K : ℝ≥0 s : Set α ⊢ IsClosed (⋂ i ∈ s, ⋂ i_1 ∈ s, {x | edist (x i) (x i_1) ≤ ↑K * edist i i_1}) ** refine' isClosed_biInter fun x _ => isClosed_biInter fun y _ => isClosed_le _ _ ** case refine'_1 α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 inst✝² : PseudoEMetricSpace α✝ α : Type u_4 β : Type u_5 inst✝¹ : PseudoEMetricSpace α inst✝ : PseudoEMetricSpace β K : ℝ≥0 s : Set α x : α x✝¹ : x ∈ s y : α x✝ : y ∈ s ⊢ Continuous fun x_1 => edist (x_1 x) (x_1 y) case refine'_2 α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 inst✝² : PseudoEMetricSpace α✝ α : Type u_4 β : Type u_5 inst✝¹ : PseudoEMetricSpace α inst✝ : PseudoEMetricSpace β K : ℝ≥0 s : Set α x : α x✝¹ : x ∈ s y : α x✝ : y ∈ s ⊢ Continuous fun x_1 => ↑K * edist x y ** exacts [.edist (continuous_apply x) (continuous_apply y), continuous_const] ** Qed
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isClosed_setOf_lipschitzWith ** α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 inst✝² : PseudoEMetricSpace α✝ α : Type u_4 β : Type u_5 inst✝¹ : PseudoEMetricSpace α inst✝ : PseudoEMetricSpace β K : ℝ≥0 ⊢ IsClosed {f | LipschitzWith K f} ** simp only [← lipschitzOn_univ, isClosed_setOf_lipschitzOnWith] ** Qed
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Real.ediam_eq ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s ⊢ EMetric.diam s = ENNReal.ofReal (sSup s - sInf s) ** rcases eq_empty_or_nonempty s with (rfl | hne) ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s hne : Set.Nonempty s ⊢ EMetric.diam s = ENNReal.ofReal (sSup s - sInf s) ** refine' le_antisymm (Metric.ediam_le_of_forall_dist_le fun x hx y hy => _) _ ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α h : Bornology.IsBounded ∅ ⊢ EMetric.diam ∅ = ENNReal.ofReal (sSup ∅ - sInf ∅) ** simp ** case inr.refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s hne : Set.Nonempty s x : ℝ hx : x ∈ s y : ℝ hy : y ∈ s ⊢ dist x y ≤ sSup s - sInf s ** have := Real.subset_Icc_sInf_sSup_of_isBounded h ** case inr.refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s hne : Set.Nonempty s x : ℝ hx : x ∈ s y : ℝ hy : y ∈ s this : s ⊆ Icc (sInf s) (sSup s) ⊢ dist x y ≤ sSup s - sInf s ** exact Real.dist_le_of_mem_Icc (this hx) (this hy) ** case inr.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s hne : Set.Nonempty s ⊢ ENNReal.ofReal (sSup s - sInf s) ≤ EMetric.diam s ** apply ENNReal.ofReal_le_of_le_toReal ** case inr.refine'_2.h α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s hne : Set.Nonempty s ⊢ sSup s - sInf s ≤ ENNReal.toReal (EMetric.diam s) ** rw [← Metric.diam, ← Metric.diam_closure] ** case inr.refine'_2.h α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s hne : Set.Nonempty s ⊢ sSup s - sInf s ≤ Metric.diam (closure s) ** have h' := Real.isBounded_iff_bddBelow_bddAbove.1 h ** case inr.refine'_2.h α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s hne : Set.Nonempty s h' : BddBelow s ∧ BddAbove s ⊢ sSup s - sInf s ≤ Metric.diam (closure s) ** calc sSup s - sInf s ≤ dist (sSup s) (sInf s) := le_abs_self _
_ ≤ Metric.diam (closure s) := dist_le_diam_of_mem h.closure (csSup_mem_closure hne h'.2)
(csInf_mem_closure hne h'.1) ** Qed
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Real.diam_eq ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s ⊢ Metric.diam s = sSup s - sInf s ** rw [Metric.diam, Real.ediam_eq h, ENNReal.toReal_ofReal] ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : Bornology.IsBounded s ⊢ 0 ≤ sSup s - sInf s ** rw [Real.isBounded_iff_bddBelow_bddAbove] at h ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α s : Set ℝ h : BddBelow s ∧ BddAbove s ⊢ 0 ≤ sSup s - sInf s ** exact sub_nonneg.2 (Real.sInf_le_sSup s h.1 h.2) ** Qed
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Real.ediam_Ioo ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ ⊢ EMetric.diam (Ioo a b) = ENNReal.ofReal (b - a) ** rcases le_or_lt b a with (h | h) ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : b ≤ a ⊢ EMetric.diam (Ioo a b) = ENNReal.ofReal (b - a) ** simp [h] ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : a < b ⊢ EMetric.diam (Ioo a b) = ENNReal.ofReal (b - a) ** rw [Real.ediam_eq (isBounded_Ioo _ _), csSup_Ioo h, csInf_Ioo h] ** Qed
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Real.ediam_Icc ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ ⊢ EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) ** rcases le_or_lt a b with (h | h) ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : a ≤ b ⊢ EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) ** rw [Real.ediam_eq (isBounded_Icc _ _), csSup_Icc h, csInf_Icc h] ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : b < a ⊢ EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) ** simp [h, h.le] ** Qed
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Real.diam_Icc ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : a ≤ b ⊢ Metric.diam (Icc a b) = b - a ** simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] ** Qed
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Real.diam_Ico ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : a ≤ b ⊢ Metric.diam (Ico a b) = b - a ** simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] ** Qed
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Real.diam_Ioc ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : a ≤ b ⊢ Metric.diam (Ioc a b) = b - a ** simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] ** Qed
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Real.diam_Ioo ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α a b : ℝ h : a ≤ b ⊢ Metric.diam (Ioo a b) = b - a ** simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] ** Qed
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edist_le_tsum_of_edist_le_of_tendsto ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0∞ hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n a : α ha : Tendsto f atTop (𝓝 a) n : ℕ ⊢ edist (f n) a ≤ ∑' (m : ℕ), d (n + m) ** refine' le_of_tendsto (tendsto_const_nhds.edist ha) (mem_atTop_sets.2 ⟨n, fun m hnm => _⟩) ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0∞ hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n a : α ha : Tendsto f atTop (𝓝 a) n m : ℕ hnm : m ≥ n ⊢ m ∈ {x | (fun c => edist (f n) (f c) ≤ ∑' (m : ℕ), d (n + m)) x} ** refine' le_trans (edist_le_Ico_sum_of_edist_le hnm fun _ _ => hf _) _ ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0∞ hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n a : α ha : Tendsto f atTop (𝓝 a) n m : ℕ hnm : m ≥ n ⊢ ∑ i in Finset.Ico n m, d i ≤ ∑' (m : ℕ), d (n + m) ** rw [Finset.sum_Ico_eq_sum_range] ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0∞ hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n a : α ha : Tendsto f atTop (𝓝 a) n m : ℕ hnm : m ≥ n ⊢ ∑ k in Finset.range (m - n), d (n + k) ≤ ∑' (m : ℕ), d (n + m) ** exact sum_le_tsum _ (fun _ _ => zero_le _) ENNReal.summable ** Qed
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edist_le_tsum_of_edist_le_of_tendsto₀ ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝ : PseudoEMetricSpace α f : ℕ → α d : ℕ → ℝ≥0∞ hf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ d n a : α ha : Tendsto f atTop (𝓝 a) ⊢ edist (f 0) a ≤ ∑' (m : ℕ), d m ** simpa using edist_le_tsum_of_edist_le_of_tendsto d hf ha 0 ** Qed
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ContinuousMap.idealOfSet_closed ** X : Type u_1 R : Type u_2 inst✝⁴ : TopologicalSpace X inst✝³ : Semiring R inst✝² : TopologicalSpace R inst✝¹ : TopologicalSemiring R inst✝ : T2Space R s : Set X ⊢ IsClosed ↑(idealOfSet R s) ** simp only [idealOfSet, Submodule.coe_set_mk, Set.setOf_forall] ** X : Type u_1 R : Type u_2 inst✝⁴ : TopologicalSpace X inst✝³ : Semiring R inst✝² : TopologicalSpace R inst✝¹ : TopologicalSemiring R inst✝ : T2Space R s : Set X ⊢ IsClosed ↑{ toAddSubsemigroup := { carrier := ⋂ i ∈ sᶜ, {x | ↑x i = 0}, add_mem' := (_ : ∀ {a b : C(X, R)}, a ∈ ⋂ i ∈ sᶜ, {x | ↑x i = 0} → b ∈ ⋂ i ∈ sᶜ, {x | ↑x i = 0} → a + b ∈ ⋂ i ∈ sᶜ, {x | ↑x i = 0}) }, zero_mem' := (_ : 0 ∈ { carrier := ⋂ i ∈ sᶜ, {x | ↑x i = 0}, add_mem' := (_ : ∀ {a b : C(X, R)}, a ∈ ⋂ i ∈ sᶜ, {x | ↑x i = 0} → b ∈ ⋂ i ∈ sᶜ, {x | ↑x i = 0} → a + b ∈ ⋂ i ∈ sᶜ, {x | ↑x i = 0}) }.carrier) } ** exact isClosed_iInter fun x => isClosed_iInter fun _ =>
isClosed_eq (continuous_eval_const x) continuous_const ** Qed
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ContinuousMap.mem_idealOfSet ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set X f : C(X, R) ⊢ f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → ↑f x = 0 ** convert Iff.rfl ** Qed
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ContinuousMap.not_mem_idealOfSet ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set X f : C(X, R) ⊢ ¬f ∈ idealOfSet R s ↔ ∃ x, x ∈ sᶜ ∧ ↑f x ≠ 0 ** simp_rw [mem_idealOfSet] ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set X f : C(X, R) ⊢ (¬∀ ⦃x : X⦄, x ∈ sᶜ → ↑f x = 0) ↔ ∃ x, x ∈ sᶜ ∧ ↑f x ≠ 0 ** push_neg ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R s : Set X f : C(X, R) ⊢ (Exists fun ⦃x⦄ => x ∈ sᶜ ∧ ↑f x ≠ 0) ↔ ∃ x, x ∈ sᶜ ∧ ↑f x ≠ 0 ** rfl ** Qed
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ContinuousMap.not_mem_setOfIdeal ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) x : X ⊢ ¬x ∈ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → ↑f x = 0 ** rw [← Set.mem_compl_iff, setOfIdeal, compl_compl, Set.mem_setOf] ** Qed
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ContinuousMap.mem_setOfIdeal ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) x : X ⊢ x ∈ setOfIdeal I ↔ ∃ f, f ∈ I ∧ ↑f x ≠ 0 ** simp_rw [setOfIdeal, Set.mem_compl_iff, Set.mem_setOf] ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) x : X ⊢ (¬∀ (f : C(X, R)), f ∈ I → ↑f x = 0) ↔ ∃ f, f ∈ I ∧ ↑f x ≠ 0 ** push_neg ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) x : X ⊢ (∃ f, f ∈ I ∧ ↑f x ≠ 0) ↔ ∃ f, f ∈ I ∧ ↑f x ≠ 0 ** rfl ** Qed
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ContinuousMap.setOfIdeal_open ** X : Type u_1 R : Type u_2 inst✝⁴ : TopologicalSpace X inst✝³ : Semiring R inst✝² : TopologicalSpace R inst✝¹ : TopologicalSemiring R inst✝ : T2Space R I : Ideal C(X, R) ⊢ IsOpen (setOfIdeal I) ** simp only [setOfIdeal, Set.setOf_forall, isOpen_compl_iff] ** X : Type u_1 R : Type u_2 inst✝⁴ : TopologicalSpace X inst✝³ : Semiring R inst✝² : TopologicalSpace R inst✝¹ : TopologicalSemiring R inst✝ : T2Space R I : Ideal C(X, R) ⊢ IsClosed (⋂ i ∈ I, {x | ↑i x = 0}) ** exact
isClosed_iInter fun f =>
isClosed_iInter fun _ => isClosed_eq (map_continuous f) continuous_const ** Qed
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ContinuousMap.idealOfEmpty_eq_bot ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R f : C(X, R) ⊢ f ∈ idealOfSet R ∅ ↔ f ∈ ⊥ ** simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
FunLike.ext_iff, zero_apply] ** Qed
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ContinuousMap.mem_idealOfSet_compl_singleton ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R x : X f : C(X, R) ⊢ f ∈ idealOfSet R {x}ᶜ ↔ ↑f x = 0 ** simp only [mem_idealOfSet, compl_compl, Set.mem_singleton_iff, forall_eq] ** Qed
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ContinuousMap.ideal_gc ** X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R ⊢ GaloisConnection setOfIdeal (idealOfSet R) ** refine' fun I s => ⟨fun h f hf => _, fun h x hx => _⟩ ** case refine'_1 X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) s : Set X h : setOfIdeal I ≤ s f : C(X, R) hf : f ∈ I ⊢ f ∈ idealOfSet R s ** by_contra h' ** case refine'_1 X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) s : Set X h : setOfIdeal I ≤ s f : C(X, R) hf : f ∈ I h' : ¬f ∈ idealOfSet R s ⊢ False ** rcases not_mem_idealOfSet.mp h' with ⟨x, hx, hfx⟩ ** case refine'_1.intro.intro X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) s : Set X h : setOfIdeal I ≤ s f : C(X, R) hf : f ∈ I h' : ¬f ∈ idealOfSet R s x : X hx : x ∈ sᶜ hfx : ↑f x ≠ 0 ⊢ False ** exact hfx (not_mem_setOfIdeal.mp (mt (@h x) hx) hf) ** case refine'_2 X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) s : Set X h : I ≤ idealOfSet R s x : X hx : x ∈ setOfIdeal I ⊢ x ∈ s ** obtain ⟨f, hf, hfx⟩ := mem_setOfIdeal.mp hx ** case refine'_2.intro.intro X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) s : Set X h : I ≤ idealOfSet R s x : X hx : x ∈ setOfIdeal I f : C(X, R) hf : f ∈ I hfx : ↑f x ≠ 0 ⊢ x ∈ s ** by_contra hx' ** case refine'_2.intro.intro X : Type u_1 R : Type u_2 inst✝³ : TopologicalSpace X inst✝² : Semiring R inst✝¹ : TopologicalSpace R inst✝ : TopologicalSemiring R I : Ideal C(X, R) s : Set X h : I ≤ idealOfSet R s x : X hx : x ∈ setOfIdeal I f : C(X, R) hf : f ∈ I hfx : ↑f x ≠ 0 hx' : ¬x ∈ s ⊢ False ** exact not_mem_idealOfSet.mpr ⟨x, hx', hfx⟩ (h hf) ** Qed
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ContinuousMap.exists_mul_le_one_eqOn_ge ** X : Type u_1 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 inst✝ : TopologicalSpace X f : C(X, ℝ≥0) c : ℝ≥0 hc : 0 < c x : X hx : x ∈ {x | c ≤ ↑f x} ⊢ ↑(mk (↑f ⊔ ↑(const X c))⁻¹ * f) x = OfNat.ofNat 1 x ** simpa only [coe_const, ge_iff_le, mul_apply, coe_mk, Pi.inv_apply, Pi.sup_apply,
Function.const_apply, sup_eq_left.mpr (Set.mem_setOf.mp hx), ne_eq, Pi.one_apply]
using inv_mul_cancel (hc.trans_le hx).ne' ** Qed
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ContinuousMap.idealOfSet_ofIdeal_eq_closure ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) ⊢ idealOfSet 𝕜 (setOfIdeal I) = Ideal.closure I ** refine' le_antisymm _
((idealOfSet_closed 𝕜 <| setOfIdeal I).closure_subset_iff.mpr fun f hf x hx =>
not_mem_setOfIdeal.mp hx hf) ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) ⊢ idealOfSet 𝕜 (setOfIdeal I) ≤ Ideal.closure I ** refine' (fun f hf => Metric.mem_closure_iff.mpr fun ε hε => _) ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ hε : ε > 0 ⊢ ∃ b, b ∈ ↑I ∧ dist f b < ε ** lift ε to ℝ≥0 using hε.lt.le ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : ↑ε > 0 ⊢ ∃ b, b ∈ ↑I ∧ dist f b < ↑ε ** replace hε := show (0 : ℝ≥0) < ε from hε ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε ⊢ ∃ b, b ∈ ↑I ∧ dist f b < ↑ε ** simp_rw [dist_nndist] ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε ⊢ ∃ b, b ∈ ↑I ∧ ↑(nndist f b) < ↑ε ** norm_cast ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε ⊢ ∃ b, b ∈ ↑I ∧ nndist f b < ε ** set t := {x : X | ε / 2 ≤ ‖f x‖₊} ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ⊢ ∃ b, b ∈ ↑I ∧ nndist f b < ε ** have ht : IsClosed t := isClosed_le continuous_const (map_continuous f).nnnorm ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t ⊢ ∃ b, b ∈ ↑I ∧ nndist f b < ε ** have htI : Disjoint t (setOfIdeal I)ᶜ := by
refine' Set.subset_compl_iff_disjoint_left.mp fun x hx => _
simpa only [Set.mem_setOf, Set.mem_compl_iff, not_le] using
(nnnorm_eq_zero.mpr (mem_idealOfSet.mp hf hx)).trans_lt (half_pos hε) ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ this : ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t → 0 < ↑g' x ⊢ ∃ g, comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I ∧ (∀ (x : X), ↑g x ≤ 1) ∧ Set.EqOn (↑g) 1 t ** obtain ⟨g', hI', hgt'⟩ := this ** case intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g' : C(X, ℝ≥0) hI' : comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I hgt' : ∀ (x : X), x ∈ t → 0 < ↑g' x ⊢ ∃ g, comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I ∧ (∀ (x : X), ↑g x ≤ 1) ∧ Set.EqOn (↑g) 1 t ** obtain ⟨c, hc, hgc'⟩ : ∃ (c : _) (_ : 0 < c), ∀ y : X, y ∈ t → c ≤ g' y :=
t.eq_empty_or_nonempty.elim
(fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy)⟩) fun ht' =>
let ⟨x, hx, hx'⟩ := ht.isCompact.exists_forall_le ht' (map_continuous g').continuousOn
⟨g' x, hgt' x hx, hx'⟩ ** case intro.intro.intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g' : C(X, ℝ≥0) hI' : comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I hgt' : ∀ (x : X), x ∈ t → 0 < ↑g' x c : ℝ≥0 hc : 0 < c hgc' : ∀ (y : X), y ∈ t → c ≤ ↑g' y ⊢ ∃ g, comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I ∧ (∀ (x : X), ↑g x ≤ 1) ∧ Set.EqOn (↑g) 1 t ** obtain ⟨g, hg, hgc⟩ := exists_mul_le_one_eqOn_ge g' hc ** case intro.intro.intro.intro.intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g' : C(X, ℝ≥0) hI' : comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I hgt' : ∀ (x : X), x ∈ t → 0 < ↑g' x c : ℝ≥0 hc : 0 < c hgc' : ∀ (y : X), y ∈ t → c ≤ ↑g' y g : C(X, ℝ≥0) hg : ∀ (x : X), ↑(g * g') x ≤ 1 hgc : Set.EqOn (↑(g * g')) 1 {x | c ≤ ↑g' x} ⊢ ∃ g, comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I ∧ (∀ (x : X), ↑g x ≤ 1) ∧ Set.EqOn (↑g) 1 t ** refine' ⟨g * g', _, hg, hgc.mono hgc'⟩ ** case intro.intro.intro.intro.intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g' : C(X, ℝ≥0) hI' : comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I hgt' : ∀ (x : X), x ∈ t → 0 < ↑g' x c : ℝ≥0 hc : 0 < c hgc' : ∀ (y : X), y ∈ t → c ≤ ↑g' y g : C(X, ℝ≥0) hg : ∀ (x : X), ↑(g * g') x ≤ 1 hgc : Set.EqOn (↑(g * g')) 1 {x | c ≤ ↑g' x} ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) (g * g') ∈ I ** convert I.mul_mem_left ((algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) hI' ** case h.e'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g' : C(X, ℝ≥0) hI' : comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I hgt' : ∀ (x : X), x ∈ t → 0 < ↑g' x c : ℝ≥0 hc : 0 < c hgc' : ∀ (y : X), y ∈ t → c ≤ ↑g' y g : C(X, ℝ≥0) hg : ∀ (x : X), ↑(g * g') x ≤ 1 hgc : Set.EqOn (↑(g * g')) 1 {x | c ≤ ↑g' x} ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) (g * g') = comp (↑(algebraMapClm ℝ≥0 𝕜)) g * comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ** ext ** case h.e'_4.h X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g' : C(X, ℝ≥0) hI' : comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I hgt' : ∀ (x : X), x ∈ t → 0 < ↑g' x c : ℝ≥0 hc : 0 < c hgc' : ∀ (y : X), y ∈ t → c ≤ ↑g' y g : C(X, ℝ≥0) hg : ∀ (x : X), ↑(g * g') x ≤ 1 hgc : Set.EqOn (↑(g * g')) 1 {x | c ≤ ↑g' x} a✝ : X ⊢ ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) (g * g')) a✝ = ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g * comp (↑(algebraMapClm ℝ≥0 𝕜)) g') a✝ ** simp only [algebraMapClm_coe, comp_apply, mul_apply, ContinuousMap.coe_coe, map_mul] ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t ⊢ Disjoint t (setOfIdeal I)ᶜ ** refine' Set.subset_compl_iff_disjoint_left.mp fun x hx => _ ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t x : X hx : x ∈ (setOfIdeal I)ᶜ ⊢ x ∈ tᶜ ** simpa only [Set.mem_setOf, Set.mem_compl_iff, not_le] using
(nnnorm_eq_zero.mpr (mem_idealOfSet.mp hf hx)).trans_lt (half_pos hε) ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ this : ∃ g, comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I ∧ (∀ (x : X), ↑g x ≤ 1) ∧ Set.EqOn (↑g) 1 t ⊢ ∃ b, b ∈ ↑I ∧ nndist f b < ε ** obtain ⟨g, hgI, hg, hgt⟩ := this ** case intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t ⊢ ∃ b, b ∈ ↑I ∧ nndist f b < ε ** refine' ⟨f * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, _⟩ ** case intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t ⊢ nndist f (f * comp (↑(algebraMapClm ℝ≥0 𝕜)) g) < ε ** rw [nndist_eq_nnnorm] ** case intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t ⊢ ‖f - f * comp (↑(algebraMapClm ℝ≥0 𝕜)) g‖₊ < ε ** refine' (nnnorm_lt_iff _ hε).2 fun x => _ ** case intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X ⊢ ‖↑(f - f * comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ < ε ** simp only [coe_sub, coe_mul, Pi.sub_apply, Pi.mul_apply] ** case intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X ⊢ ‖↑f x - ↑f x * ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ < ε ** by_cases hx : x ∈ t ** case pos X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : x ∈ t ⊢ ‖↑f x - ↑f x * ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ < ε ** simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapClm_apply,
map_one, mul_one, sub_self, nnnorm_zero] using hε ** case neg X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : ¬x ∈ t ⊢ ‖↑f x - ↑f x * ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ < ε ** refine' lt_of_le_of_lt _ (half_lt_self hε) ** case neg X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : ¬x ∈ t ⊢ ‖↑f x - ↑f x * ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ ≤ ε / 2 ** have :=
calc
‖((1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
‖1 - algebraMap ℝ≥0 𝕜 (g x)‖₊ := by
simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
Pi.one_apply, Function.comp_apply, algebraMapClm_apply]
_ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, ge_iff_le,
NNReal.coe_sub (hg x), NNReal.coe_one, sub_smul, one_smul]
_ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self ** case neg X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : ¬x ∈ t this : ‖↑(1 - comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ ≤ 1 ⊢ ‖↑f x - ↑f x * ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ ≤ ε / 2 ** calc
‖f x - f x * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
by simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
_ ≤ ε / 2 * ‖(1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
((nnnorm_mul_le _ _).trans
(mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
_ ≤ ε / 2 := by simpa only [mul_one] using mul_le_mul_left' this _ ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : ¬x ∈ t ⊢ ‖↑(1 - comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ = ‖1 - ↑(algebraMap ℝ≥0 𝕜) (↑g x)‖₊ ** simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
Pi.one_apply, Function.comp_apply, algebraMapClm_apply] ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : ¬x ∈ t ⊢ ‖1 - ↑(algebraMap ℝ≥0 𝕜) (↑g x)‖₊ = ‖↑(algebraMap ℝ≥0 𝕜) (1 - ↑g x)‖₊ ** simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, ge_iff_le,
NNReal.coe_sub (hg x), NNReal.coe_one, sub_smul, one_smul] ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : ¬x ∈ t this : ‖↑(1 - comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ ≤ 1 ⊢ ‖↑f x - ↑f x * ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ = ‖↑f x * ↑(1 - comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ ** simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one] ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g : C(X, ℝ≥0) hgI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hg : ∀ (x : X), ↑g x ≤ 1 hgt : Set.EqOn (↑g) 1 t x : X hx : ¬x ∈ t this : ‖↑(1 - comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ ≤ 1 ⊢ ε / 2 * ‖↑(1 - comp (↑(algebraMapClm ℝ≥0 𝕜)) g) x‖₊ ≤ ε / 2 ** simpa only [mul_one] using mul_le_mul_left' this _ ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ ⊢ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t → 0 < ↑g' x ** refine' ht.isCompact.induction_on _ _ _ _ ** case refine'_1 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ ⊢ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ ∅ → 0 < ↑g' x ** refine' ⟨0, _, fun x hx => False.elim hx⟩ ** case refine'_1 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) 0 ∈ I ** convert I.zero_mem ** case h.e'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) 0 = 0 ** ext ** case h.e'_4.h X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ a✝ : X ⊢ ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) 0) a✝ = ↑0 a✝ ** simp only [comp_apply, zero_apply, ContinuousMap.coe_coe, map_zero] ** case refine'_2 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ ⊢ ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t → 0 < ↑g' x) → ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ s → 0 < ↑g' x ** rintro s₁ s₂ hs ⟨g, hI, hgt⟩ ** case refine'_2.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X hs : s₁ ⊆ s₂ g : C(X, ℝ≥0) hI : comp (↑(algebraMapClm ℝ≥0 𝕜)) g ∈ I hgt : ∀ (x : X), x ∈ s₂ → 0 < ↑g x ⊢ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ s₁ → 0 < ↑g' x ** exact ⟨g, hI, fun x hx => hgt x (hs hx)⟩ ** case refine'_3 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ ⊢ ∀ ⦃s t : Set X⦄, (∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ s → 0 < ↑g' x) → (∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t → 0 < ↑g' x) → ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ s ∪ t → 0 < ↑g' x ** rintro s₁ s₂ ⟨g₁, hI₁, hgt₁⟩ ⟨g₂, hI₂, hgt₂⟩ ** case refine'_3.intro.intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x ⊢ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ s₁ ∪ s₂ → 0 < ↑g' x ** refine' ⟨g₁ + g₂, _, fun x hx => _⟩ ** case refine'_3.intro.intro.intro.intro.refine'_1 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) (g₁ + g₂) ∈ I ** convert I.add_mem hI₁ hI₂ ** case h.e'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) (g₁ + g₂) = comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ + comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ** ext y ** case h.e'_4.h X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x y : X ⊢ ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) (g₁ + g₂)) y = ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ + comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂) y ** simp only [coe_add, Pi.add_apply, map_add, coe_comp, Function.comp_apply,
ContinuousMap.coe_coe] ** case refine'_3.intro.intro.intro.intro.refine'_2 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x x : X hx : x ∈ s₁ ∪ s₂ ⊢ 0 < ↑(g₁ + g₂) x ** rcases hx with (hx | hx) ** case refine'_3.intro.intro.intro.intro.refine'_2.inl X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x x : X hx : x ∈ s₁ ⊢ 0 < ↑(g₁ + g₂) x case refine'_3.intro.intro.intro.intro.refine'_2.inr X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x x : X hx : x ∈ s₂ ⊢ 0 < ↑(g₁ + g₂) x ** simpa only [zero_add] using add_lt_add_of_lt_of_le (hgt₁ x hx) zero_le' ** case refine'_3.intro.intro.intro.intro.refine'_2.inr X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ s₁ s₂ : Set X g₁ : C(X, ℝ≥0) hI₁ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₁ ∈ I hgt₁ : ∀ (x : X), x ∈ s₁ → 0 < ↑g₁ x g₂ : C(X, ℝ≥0) hI₂ : comp (↑(algebraMapClm ℝ≥0 𝕜)) g₂ ∈ I hgt₂ : ∀ (x : X), x ∈ s₂ → 0 < ↑g₂ x x : X hx : x ∈ s₂ ⊢ 0 < ↑(g₁ + g₂) x ** simpa only [zero_add] using add_lt_add_of_le_of_lt zero_le' (hgt₂ x hx) ** case refine'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ ⊢ ∀ (x : X), x ∈ t → ∃ t_1, t_1 ∈ nhdsWithin x t ∧ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t_1 → 0 < ↑g' x ** intro x hx ** case refine'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X hx : x ∈ t ⊢ ∃ t_1, t_1 ∈ nhdsWithin x t ∧ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t_1 → 0 < ↑g' x ** replace hx := htI.subset_compl_right hx ** case refine'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X hx : x ∈ (setOfIdeal I)ᶜᶜ ⊢ ∃ t_1, t_1 ∈ nhdsWithin x t ∧ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t_1 → 0 < ↑g' x ** rw [compl_compl, mem_setOfIdeal] at hx ** case refine'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X hx : ∃ f, f ∈ I ∧ ↑f x ≠ 0 ⊢ ∃ t_1, t_1 ∈ nhdsWithin x t ∧ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t_1 → 0 < ↑g' x ** obtain ⟨g, hI, hgx⟩ := hx ** case refine'_4.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X g : C(X, 𝕜) hI : g ∈ I hgx : ↑g x ≠ 0 ⊢ ∃ t_1, t_1 ∈ nhdsWithin x t ∧ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t_1 → 0 < ↑g' x ** have := (map_continuous g).continuousAt.eventually_ne hgx ** case refine'_4.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X g : C(X, 𝕜) hI : g ∈ I hgx : ↑g x ≠ 0 this : ∀ᶠ (z : X) in nhds x, ↑g z ≠ 0 ⊢ ∃ t_1, t_1 ∈ nhdsWithin x t ∧ ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ (x : X), x ∈ t_1 → 0 < ↑g' x ** refine'
⟨{y : X | g y ≠ 0} ∩ t,
mem_nhdsWithin_iff_exists_mem_nhds_inter.mpr ⟨_, this, Set.Subset.rfl⟩,
⟨⟨fun x => ‖g x‖₊ ^ 2, (map_continuous g).nnnorm.pow 2⟩, _, fun x hx =>
pow_pos (norm_pos_iff.mpr hx.1) 2⟩⟩ ** case refine'_4.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X g : C(X, 𝕜) hI : g ∈ I hgx : ↑g x ≠ 0 this : ∀ᶠ (z : X) in nhds x, ↑g z ≠ 0 ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) (mk fun x => ‖↑g x‖₊ ^ 2) ∈ I ** convert I.mul_mem_left (star g) hI ** case h.e'_4 X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X g : C(X, 𝕜) hI : g ∈ I hgx : ↑g x ≠ 0 this : ∀ᶠ (z : X) in nhds x, ↑g z ≠ 0 ⊢ comp (↑(algebraMapClm ℝ≥0 𝕜)) (mk fun x => ‖↑g x‖₊ ^ 2) = star g * g ** ext ** case h.e'_4.h X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X g : C(X, 𝕜) hI : g ∈ I hgx : ↑g x ≠ 0 this : ∀ᶠ (z : X) in nhds x, ↑g z ≠ 0 a✝ : X ⊢ ↑(comp (↑(algebraMapClm ℝ≥0 𝕜)) (mk fun x => ‖↑g x‖₊ ^ 2)) a✝ = ↑(star g * g) a✝ ** simp only [comp_apply, ContinuousMap.coe_coe, coe_mk, algebraMapClm_toFun, map_pow,
mul_apply, star_apply, star_def] ** case h.e'_4.h X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X g : C(X, 𝕜) hI : g ∈ I hgx : ↑g x ≠ 0 this : ∀ᶠ (z : X) in nhds x, ↑g z ≠ 0 a✝ : X ⊢ ↑(algebraMap ℝ≥0 𝕜) ‖↑g a✝‖₊ ^ 2 = ↑(starRingEnd 𝕜) (↑g a✝) * ↑g a✝ ** simp only [normSq_eq_def', IsROrC.conj_mul, ofReal_pow] ** case h.e'_4.h X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ x : X g : C(X, 𝕜) hI : g ∈ I hgx : ↑g x ≠ 0 this : ∀ᶠ (z : X) in nhds x, ↑g z ≠ 0 a✝ : X ⊢ ↑(algebraMap ℝ≥0 𝕜) ‖↑g a✝‖₊ ^ 2 = ↑‖↑g a✝‖ ^ 2 ** rfl ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) f : C(X, 𝕜) hf : f ∈ idealOfSet 𝕜 (setOfIdeal I) ε : ℝ≥0 hε : 0 < ε t : Set X := {x | ε / 2 ≤ ‖↑f x‖₊} ht : IsClosed t htI : Disjoint t (setOfIdeal I)ᶜ g' : C(X, ℝ≥0) hI' : comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I hgt' : ∀ (x : X), x ∈ t → 0 < ↑g' x ht' : t = ∅ y : X hy : y ∈ t ⊢ False ** rwa [ht'] at hy ** Qed
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ContinuousMap.setOfIdeal_ofSet_eq_interior ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Set X ⊢ setOfIdeal (idealOfSet 𝕜 s) = interior s ** refine'
Set.Subset.antisymm
((setOfIdeal_open (idealOfSet 𝕜 s)).subset_interior_iff.mpr fun x hx =>
let ⟨f, hf, hfx⟩ := mem_setOfIdeal.mp hx
Set.not_mem_compl_iff.mp (mt (@hf x) hfx))
fun x hx => _ ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Set X x : X hx : x ∈ interior s ⊢ x ∈ setOfIdeal (idealOfSet 𝕜 s) ** rw [← compl_compl (interior s), ← closure_compl] at hx ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Set X x : X hx : x ∈ (closure sᶜ)ᶜ ⊢ x ∈ setOfIdeal (idealOfSet 𝕜 s) ** simp_rw [mem_setOfIdeal, mem_idealOfSet] ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Set X x : X hx : x ∈ (closure sᶜ)ᶜ ⊢ ∃ f, (∀ ⦃x : X⦄, x ∈ sᶜ → ↑f x = 0) ∧ ↑f x ≠ 0 ** obtain ⟨g, hgs, hgx : Set.EqOn g 1 {x}, -⟩ :=
exists_continuous_zero_one_of_closed isClosed_closure isClosed_singleton
(Set.disjoint_singleton_right.mpr hx) ** case intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Set X x : X hx : x ∈ (closure sᶜ)ᶜ g : C(X, ℝ) hgs : Set.EqOn (↑g) 0 (closure sᶜ) hgx : Set.EqOn (↑g) 1 {x} ⊢ ∃ f, (∀ ⦃x : X⦄, x ∈ sᶜ → ↑f x = 0) ∧ ↑f x ≠ 0 ** exact
⟨⟨fun x => g x, continuous_ofReal.comp (map_continuous g)⟩, by
simpa only [coe_mk, ofReal_eq_zero] using fun x hx => hgs (subset_closure hx), by
simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, IsROrC.ofReal_one] using
one_ne_zero⟩ ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Set X x : X hx : x ∈ (closure sᶜ)ᶜ g : C(X, ℝ) hgs : Set.EqOn (↑g) 0 (closure sᶜ) hgx : Set.EqOn (↑g) 1 {x} ⊢ ∀ ⦃x : X⦄, x ∈ sᶜ → ↑(mk fun x => ↑(↑g x)) x = 0 ** simpa only [coe_mk, ofReal_eq_zero] using fun x hx => hgs (subset_closure hx) ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Set X x : X hx : x ∈ (closure sᶜ)ᶜ g : C(X, ℝ) hgs : Set.EqOn (↑g) 0 (closure sᶜ) hgx : Set.EqOn (↑g) 1 {x} ⊢ ↑(mk fun x => ↑(↑g x)) x ≠ 0 ** simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, IsROrC.ofReal_one] using
one_ne_zero ** Qed
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ContinuousMap.idealOfSet_isMaximal_iff ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Opens X ⊢ Ideal.IsMaximal (idealOfSet 𝕜 ↑s) ↔ IsCoatom s ** rw [Ideal.isMaximal_def] ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Opens X ⊢ IsCoatom (idealOfSet 𝕜 ↑s) ↔ IsCoatom s ** refine' (idealOpensGI X 𝕜).isCoatom_iff (fun I hI => _) s ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Opens X I : Ideal C(X, 𝕜) hI : IsCoatom I ⊢ idealOfSet 𝕜 ↑(opensOfIdeal I) = I ** rw [← Ideal.isMaximal_def] at hI ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X s : Opens X I : Ideal C(X, 𝕜) hI : Ideal.IsMaximal I ⊢ idealOfSet 𝕜 ↑(opensOfIdeal I) = I ** exact idealOfSet_ofIdeal_isClosed inferInstance ** Qed
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ContinuousMap.setOfIdeal_eq_compl_singleton ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : Ideal.IsMaximal I ⊢ ∃ x, setOfIdeal I = {x}ᶜ ** have h : (idealOfSet 𝕜 (setOfIdeal I)).IsMaximal :=
(idealOfSet_ofIdeal_isClosed (inferInstance : IsClosed (I : Set C(X, 𝕜)))).symm ▸ hI ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : Ideal.IsMaximal I h : Ideal.IsMaximal (idealOfSet 𝕜 (setOfIdeal I)) ⊢ ∃ x, setOfIdeal I = {x}ᶜ ** obtain ⟨x, hx⟩ := Opens.isCoatom_iff.1 ((idealOfSet_isMaximal_iff 𝕜 (opensOfIdeal I)).1 h) ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : Ideal.IsMaximal I h : Ideal.IsMaximal (idealOfSet 𝕜 (setOfIdeal I)) x : X hx : opensOfIdeal I = Closeds.compl (Closeds.singleton x) ⊢ ∃ x, setOfIdeal I = {x}ᶜ ** refine ⟨x, congr_arg (fun (s : Opens X) => (s : Set X)) hx⟩ ** Qed
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ContinuousMap.ideal_isMaximal_iff ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : IsClosed ↑I ⊢ Ideal.IsMaximal I ↔ ∃ x, idealOfSet 𝕜 {x}ᶜ = I ** refine'
⟨_, fun h =>
let ⟨x, hx⟩ := h
hx ▸ idealOf_compl_singleton_isMaximal 𝕜 x⟩ ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : IsClosed ↑I ⊢ Ideal.IsMaximal I → ∃ x, idealOfSet 𝕜 {x}ᶜ = I ** intro hI' ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : IsClosed ↑I hI' : Ideal.IsMaximal I ⊢ ∃ x, idealOfSet 𝕜 {x}ᶜ = I ** obtain ⟨x, hx⟩ := setOfIdeal_eq_compl_singleton I ** case intro X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : IsClosed ↑I hI' : Ideal.IsMaximal I x : X hx : setOfIdeal I = {x}ᶜ ⊢ ∃ x, idealOfSet 𝕜 {x}ᶜ = I ** exact
⟨x, by
simpa only [idealOfSet_ofIdeal_eq_closure, I.closure_eq_of_isClosed hI] using
congr_arg (idealOfSet 𝕜) hx.symm⟩ ** X : Type u_1 𝕜 : Type u_2 inst✝³ : IsROrC 𝕜 inst✝² : TopologicalSpace X inst✝¹ : CompactSpace X inst✝ : T2Space X I : Ideal C(X, 𝕜) hI : IsClosed ↑I hI' : Ideal.IsMaximal I x : X hx : setOfIdeal I = {x}ᶜ ⊢ idealOfSet 𝕜 {x}ᶜ = I ** simpa only [idealOfSet_ofIdeal_eq_closure, I.closure_eq_of_isClosed hI] using
congr_arg (idealOfSet 𝕜) hx.symm ** Qed
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WeakDual.CharacterSpace.continuousMapEval_bijective ** X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 ⊢ Bijective ↑(continuousMapEval X 𝕜) ** refine' ⟨fun x y hxy => _, fun φ => _⟩ ** case refine'_1 X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 x y : X hxy : ↑(continuousMapEval X 𝕜) x = ↑(continuousMapEval X 𝕜) y ⊢ x = y ** contrapose! hxy ** case refine'_1 X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 x y : X hxy : x ≠ y ⊢ ↑(continuousMapEval X 𝕜) x ≠ ↑(continuousMapEval X 𝕜) y ** rcases exists_continuous_zero_one_of_closed (isClosed_singleton : _root_.IsClosed {x})
(isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
⟨f, fx, fy, -⟩ ** case refine'_1.intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 x y : X hxy : x ≠ y f : C(X, ℝ) fx : Set.EqOn (↑f) 0 {x} fy : Set.EqOn (↑f) 1 {y} ⊢ ↑(continuousMapEval X 𝕜) x ≠ ↑(continuousMapEval X 𝕜) y ** rw [FunLike.ne_iff] ** case refine'_1.intro.intro.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 x y : X hxy : x ≠ y f : C(X, ℝ) fx : Set.EqOn (↑f) 0 {x} fy : Set.EqOn (↑f) 1 {y} ⊢ ∃ a, ↑(↑(continuousMapEval X 𝕜) x) a ≠ ↑(↑(continuousMapEval X 𝕜) y) a ** use (⟨fun (x : ℝ) => (x : 𝕜), IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f ** case h X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 x y : X hxy : x ≠ y f : C(X, ℝ) fx : Set.EqOn (↑f) 0 {x} fy : Set.EqOn (↑f) 1 {y} ⊢ ↑(↑(continuousMapEval X 𝕜) x) (ContinuousMap.comp (mk fun x => ↑x) f) ≠ ↑(↑(continuousMapEval X 𝕜) y) (ContinuousMap.comp (mk fun x => ↑x) f) ** simpa only [continuousMapEval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
IsROrC.ofReal_inj] using
((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y) ** case refine'_2 X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 φ : ↑(characterSpace 𝕜 C(X, 𝕜)) ⊢ ∃ a, ↑(continuousMapEval X 𝕜) a = φ ** obtain ⟨x, hx⟩ := (ideal_isMaximal_iff (RingHom.ker φ)).mp inferInstance ** case refine'_2.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 φ : ↑(characterSpace 𝕜 C(X, 𝕜)) x : X hx : idealOfSet 𝕜 {x}ᶜ = RingHom.ker φ ⊢ ∃ a, ↑(continuousMapEval X 𝕜) a = φ ** refine' ⟨x, CharacterSpace.ext_ker <| Ideal.ext fun f => _⟩ ** case refine'_2.intro X : Type u_1 𝕜 : Type u_2 inst✝³ : TopologicalSpace X inst✝² : CompactSpace X inst✝¹ : T2Space X inst✝ : IsROrC 𝕜 φ : ↑(characterSpace 𝕜 C(X, 𝕜)) x : X hx : idealOfSet 𝕜 {x}ᶜ = RingHom.ker φ f : C(X, 𝕜) ⊢ f ∈ RingHom.ker (↑(continuousMapEval X 𝕜) x) ↔ f ∈ RingHom.ker φ ** simpa only [RingHom.mem_ker, continuousMapEval_apply_apply, mem_idealOfSet_compl_singleton,
RingHom.mem_ker] using SetLike.ext_iff.mp hx f ** Qed
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CompHaus.isIso_of_bijective ** X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f ⊢ IsIso f ** let E := Equiv.ofBijective _ bij ** X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij ⊢ IsIso f ** have hE : Continuous E.symm := by
rw [continuous_iff_isClosed]
intro S hS
rw [← E.image_eq_preimage]
exact isClosedMap f S hS ** X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij hE : Continuous ↑E.symm ⊢ IsIso f ** refine' ⟨⟨⟨E.symm, hE⟩, _, _⟩⟩ ** X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij ⊢ Continuous ↑E.symm ** rw [continuous_iff_isClosed] ** X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij ⊢ ∀ (s : Set ((forget CompHaus).obj X)), IsClosed s → IsClosed (↑E.symm ⁻¹' s) ** intro S hS ** X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij S : Set ((forget CompHaus).obj X) hS : IsClosed S ⊢ IsClosed (↑E.symm ⁻¹' S) ** rw [← E.image_eq_preimage] ** X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij S : Set ((forget CompHaus).obj X) hS : IsClosed S ⊢ IsClosed (↑E '' S) ** exact isClosedMap f S hS ** case refine'_1 X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij hE : Continuous ↑E.symm ⊢ f ≫ ContinuousMap.mk ↑E.symm = 𝟙 X ** ext x ** case refine'_1.w X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij hE : Continuous ↑E.symm x : (forget CompHaus).obj X ⊢ ↑(f ≫ ContinuousMap.mk ↑E.symm) x = ↑(𝟙 X) x ** apply E.symm_apply_apply ** case refine'_2 X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij hE : Continuous ↑E.symm ⊢ ContinuousMap.mk ↑E.symm ≫ f = 𝟙 Y ** ext x ** case refine'_2.w X✝ : Type u_1 inst✝² : TopologicalSpace X✝ inst✝¹ : CompactSpace X✝ inst✝ : T2Space X✝ X Y : CompHaus f : X ⟶ Y bij : Function.Bijective ↑f E : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective (↑f) bij hE : Continuous ↑E.symm x : (forget CompHaus).obj Y ⊢ ↑(ContinuousMap.mk ↑E.symm ≫ f) x = ↑(𝟙 Y) x ** apply E.apply_symm_apply ** Qed
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CompHaus.epi_iff_surjective ** X Y : CompHaus f : X ⟶ Y ⊢ Epi f ↔ Function.Surjective ↑f ** constructor ** case mp X Y : CompHaus f : X ⟶ Y ⊢ Epi f → Function.Surjective ↑f ** dsimp [Function.Surjective] ** case mp X Y : CompHaus f : X ⟶ Y ⊢ Epi f → ∀ (b : (forget CompHaus).obj Y), ∃ a, ↑f a = b ** contrapose! ** case mp X Y : CompHaus f : X ⟶ Y ⊢ (∃ b, ∀ (a : (forget CompHaus).obj X), ↑f a ≠ b) → ¬Epi f ** rintro ⟨y, hy⟩ hf ** case mp.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f ⊢ False ** let C := Set.range f ** case mp.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f ⊢ False ** have hC : IsClosed C := (isCompact_range f.continuous).isClosed ** case mp.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C ⊢ False ** let D := ({y} : Set Y) ** case mp.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} ⊢ False ** have hD : IsClosed D := isClosed_singleton ** case mp.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D ⊢ False ** have hCD : Disjoint C D := by
rw [Set.disjoint_singleton_right]
rintro ⟨y', hy'⟩
exact hy y' hy' ** case mp.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D ⊢ False ** obtain ⟨φ, hφ0, hφ1, hφ01⟩ := exists_continuous_zero_one_of_closed hC hD hCD ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 ⊢ False ** haveI : CompactSpace (ULift.{u} <| Set.Icc (0 : ℝ) 1) := Homeomorph.ulift.symm.compactSpace ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) ⊢ False ** haveI : T2Space (ULift.{u} <| Set.Icc (0 : ℝ) 1) := Homeomorph.ulift.symm.t2Space ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) ⊢ False ** let Z := of (ULift.{u} <| Set.Icc (0 : ℝ) 1) ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) ⊢ False ** let g : Y ⟶ Z :=
⟨fun y' => ⟨⟨φ y', hφ01 y'⟩⟩,
continuous_uLift_up.comp (φ.continuous.subtype_mk fun y' => hφ01 y')⟩ ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } ⊢ False ** let h : Y ⟶ Z := ⟨fun _ => ⟨⟨0, Set.left_mem_Icc.mpr zero_le_one⟩⟩, continuous_const⟩ ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } ⊢ False ** have H : h = g := by
rw [← cancel_epi f]
ext x
apply ULift.ext
apply Subtype.ext
dsimp
change 0 = φ (f x)
simp only [hφ0 (Set.mem_range_self x), Pi.zero_apply] ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } H : h = g ⊢ False ** apply_fun fun e => (e y).down.1 at H ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } H : ↑(↑h y).down = ↑(↑g y).down ⊢ False ** dsimp at H ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } H : ↑(↑(ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } }) y).down = ↑(↑(ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } }) y).down ⊢ False ** change 0 = φ y at H ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } H : 0 = ↑φ y ⊢ False ** simp only [hφ1 (Set.mem_singleton y), Pi.one_apply] at H ** case mp.intro.intro.intro.intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } H : 0 = 1 ⊢ False ** exact zero_ne_one H ** X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D ⊢ Disjoint C D ** rw [Set.disjoint_singleton_right] ** X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D ⊢ ¬y ∈ C ** rintro ⟨y', hy'⟩ ** case intro X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D y' : (forget CompHaus).obj X hy' : ↑f y' = y ⊢ False ** exact hy y' hy' ** X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } ⊢ h = g ** rw [← cancel_epi f] ** X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } ⊢ f ≫ h = f ≫ g ** ext x ** case w X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } x : (forget CompHaus).obj X ⊢ ↑(f ≫ h) x = ↑(f ≫ g) x ** apply ULift.ext ** case w.h X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } x : (forget CompHaus).obj X ⊢ (↑(f ≫ h) x).down = (↑(f ≫ g) x).down ** apply Subtype.ext ** case w.h.a X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } x : (forget CompHaus).obj X ⊢ ↑(↑(f ≫ h) x).down = ↑(↑(f ≫ g) x).down ** dsimp ** case w.h.a X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } x : (forget CompHaus).obj X ⊢ ↑(↑(f ≫ ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } }) x).down = ↑(↑(f ≫ ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } }) x).down ** change 0 = φ (f x) ** case w.h.a X Y : CompHaus f : X ⟶ Y y : (forget CompHaus).obj Y hy : ∀ (a : (forget CompHaus).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget CompHaus).obj Y) := Set.range ↑f hC : IsClosed C D : Set ↑Y.toTop := {y} hD : IsClosed D hCD : Disjoint C D φ : C((forget CompHaus).obj Y, ℝ) hφ0 : Set.EqOn (↑φ) 0 C hφ1 : Set.EqOn (↑φ) 1 D hφ01 : ∀ (x : (forget CompHaus).obj Y), ↑φ x ∈ Set.Icc 0 1 this✝ : CompactSpace (ULift.{u, 0} ↑(Set.Icc 0 1)) this : T2Space (ULift.{u, 0} ↑(Set.Icc 0 1)) Z : CompHaus := of (ULift.{u, 0} ↑(Set.Icc 0 1)) g : Y ⟶ Z := ContinuousMap.mk fun y' => { down := { val := ↑φ y', property := (_ : ↑φ y' ∈ Set.Icc 0 1) } } h : Y ⟶ Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 ∈ Set.Icc 0 1) } } x : (forget CompHaus).obj X ⊢ 0 = ↑φ (↑f x) ** simp only [hφ0 (Set.mem_range_self x), Pi.zero_apply] ** case mpr X Y : CompHaus f : X ⟶ Y ⊢ Function.Surjective ↑f → Epi f ** rw [← CategoryTheory.epi_iff_surjective] ** case mpr X Y : CompHaus f : X ⟶ Y ⊢ Epi ↑f → Epi f ** apply (forget CompHaus).epi_of_epi_map ** Qed
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CompHaus.mono_iff_injective ** X Y : CompHaus f : X ⟶ Y ⊢ Mono f ↔ Function.Injective ↑f ** constructor ** case mp X Y : CompHaus f : X ⟶ Y ⊢ Mono f → Function.Injective ↑f ** intro hf x₁ x₂ h ** case mp X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ ⊢ x₁ = x₂ ** let g₁ : of PUnit ⟶ X := ⟨fun _ => x₁, continuous_const⟩ ** case mp X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ g₁ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₁ ⊢ x₁ = x₂ ** let g₂ : of PUnit ⟶ X := ⟨fun _ => x₂, continuous_const⟩ ** case mp X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ g₁ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₁ g₂ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₂ ⊢ x₁ = x₂ ** have : g₁ ≫ f = g₂ ≫ f := by
ext
exact h ** case mp X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ g₁ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₁ g₂ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₂ this : g₁ ≫ f = g₂ ≫ f ⊢ x₁ = x₂ ** rw [cancel_mono] at this ** case mp X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ g₁ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₁ g₂ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₂ this : g₁ = g₂ ⊢ x₁ = x₂ ** apply_fun fun e => e PUnit.unit at this ** case mp X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ g₁ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₁ g₂ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₂ this : ↑g₁ PUnit.unit = ↑g₂ PUnit.unit ⊢ x₁ = x₂ ** exact this ** X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ g₁ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₁ g₂ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₂ ⊢ g₁ ≫ f = g₂ ≫ f ** ext ** case w X Y : CompHaus f : X ⟶ Y hf : Mono f x₁ x₂ : (forget CompHaus).obj X h : ↑f x₁ = ↑f x₂ g₁ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₁ g₂ : of PUnit.{u + 1} ⟶ X := ContinuousMap.mk fun x => x₂ x✝ : (forget CompHaus).obj (of PUnit.{u + 1}) ⊢ ↑(g₁ ≫ f) x✝ = ↑(g₂ ≫ f) x✝ ** exact h ** case mpr X Y : CompHaus f : X ⟶ Y ⊢ Function.Injective ↑f → Mono f ** rw [← CategoryTheory.mono_iff_injective] ** case mpr X Y : CompHaus f : X ⟶ Y ⊢ Mono ↑f → Mono f ** apply (forget CompHaus).mono_of_mono_map ** Qed
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Profinite.pullback_fst_eq ** X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst ** dsimp [pullbackIsoPullback] ** X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.fst f g = (Limits.IsLimit.conePointUniqueUpToIso (pullback.isLimit f g) (Limits.limit.isLimit (Limits.cospan f g))).hom ≫ Limits.pullback.fst ** simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π] ** Qed
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Profinite.pullback_snd_eq ** X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.snd f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.snd ** dsimp [pullbackIsoPullback] ** X Y B : Profinite f : X ⟶ B g : Y ⟶ B ⊢ pullback.snd f g = (Limits.IsLimit.conePointUniqueUpToIso (pullback.isLimit f g) (Limits.limit.isLimit (Limits.cospan f g))).hom ≫ Limits.pullback.snd ** simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π] ** Qed
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Profinite.Sigma.ι_comp_toFiniteCoproduct ** α : Type inst✝ : Fintype α X : α → Profinite a : α ⊢ Limits.Sigma.ι X a ≫ (coproductIsoCoproduct X).inv = finiteCoproduct.ι X a ** simp only [coproductIsoCoproduct, Limits.colimit.comp_coconePointUniqueUpToIso_inv,
finiteCoproduct.cocone_pt, finiteCoproduct.cocone_ι, Discrete.natTrans_app] ** Qed
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Compactum.str_incl ** X : Compactum x : X.A ⊢ str X (incl X x) = x ** change ((β ).η.app _ ≫ X.a) _ = _ ** X : Compactum x : X.A ⊢ ((Monad.η β).app X.A ≫ X.a) x = x ** rw [Monad.Algebra.unit] ** X : Compactum x : X.A ⊢ 𝟙 X.A x = x ** rfl ** Qed
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Compactum.str_hom_commute ** X Y : Compactum f : X ⟶ Y xs : Ultrafilter X.A ⊢ Monad.Algebra.Hom.f f (str X xs) = str Y (Ultrafilter.map f.f xs) ** change (X.a ≫ f.f) _ = _ ** X Y : Compactum f : X ⟶ Y xs : Ultrafilter X.A ⊢ (X.a ≫ f.f) xs = str Y (Ultrafilter.map f.f xs) ** rw [← f.h] ** X Y : Compactum f : X ⟶ Y xs : Ultrafilter X.A ⊢ (β.toFunctor.map f.f ≫ Y.a) xs = str Y (Ultrafilter.map f.f xs) ** rfl ** Qed
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Compactum.join_distrib ** X : Compactum uux : Ultrafilter (Ultrafilter X.A) ⊢ str X (join X uux) = str X (Ultrafilter.map (str X) uux) ** change ((β ).μ.app _ ≫ X.a) _ = _ ** X : Compactum uux : Ultrafilter (Ultrafilter X.A) ⊢ ((Monad.μ β).app X.A ≫ X.a) uux = str X (Ultrafilter.map (str X) uux) ** rw [Monad.Algebra.assoc] ** X : Compactum uux : Ultrafilter (Ultrafilter X.A) ⊢ (β.toFunctor.map X.a ≫ X.a) uux = str X (Ultrafilter.map (str X) uux) ** rfl ** Qed
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Compactum.isClosed_iff ** X : Compactum S : Set X.A ⊢ IsClosed S ↔ ∀ (F : Ultrafilter X.A), S ∈ F → str X F ∈ S ** rw [← isOpen_compl_iff] ** X : Compactum S : Set X.A ⊢ IsOpen Sᶜ ↔ ∀ (F : Ultrafilter X.A), S ∈ F → str X F ∈ S ** constructor ** case mp X : Compactum S : Set X.A ⊢ IsOpen Sᶜ → ∀ (F : Ultrafilter X.A), S ∈ F → str X F ∈ S ** intro cond F h ** case mp X : Compactum S : Set X.A cond : IsOpen Sᶜ F : Ultrafilter X.A h : S ∈ F ⊢ str X F ∈ S ** by_contra c ** case mp X : Compactum S : Set X.A cond : IsOpen Sᶜ F : Ultrafilter X.A h : S ∈ F c : ¬str X F ∈ S ⊢ False ** specialize cond F c ** case mp X : Compactum S : Set X.A F : Ultrafilter X.A h : S ∈ F c : ¬str X F ∈ S cond : Sᶜ ∈ F ⊢ False ** rw [compl_mem_iff_not_mem] at cond ** case mp X : Compactum S : Set X.A F : Ultrafilter X.A h : S ∈ F c : ¬str X F ∈ S cond : ¬S ∈ F ⊢ False ** contradiction ** case mpr X : Compactum S : Set X.A ⊢ (∀ (F : Ultrafilter X.A), S ∈ F → str X F ∈ S) → IsOpen Sᶜ ** intro h1 F h2 ** case mpr X : Compactum S : Set X.A h1 : ∀ (F : Ultrafilter X.A), S ∈ F → str X F ∈ S F : Ultrafilter X.A h2 : str X F ∈ Sᶜ ⊢ Sᶜ ∈ F ** specialize h1 F ** case mpr X : Compactum S : Set X.A F : Ultrafilter X.A h2 : str X F ∈ Sᶜ h1 : S ∈ F → str X F ∈ S ⊢ Sᶜ ∈ F ** cases' F.mem_or_compl_mem S with h h ** case mpr.inl X : Compactum S : Set X.A F : Ultrafilter X.A h2 : str X F ∈ Sᶜ h1 : S ∈ F → str X F ∈ S h : S ∈ F ⊢ Sᶜ ∈ F case mpr.inr X : Compactum S : Set X.A F : Ultrafilter X.A h2 : str X F ∈ Sᶜ h1 : S ∈ F → str X F ∈ S h : Sᶜ ∈ F ⊢ Sᶜ ∈ F ** exacts [absurd (h1 h) h2, h] ** Qed
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Compactum.basic_inter ** X : Compactum A B : Set X.A ⊢ Compactum.basic (A ∩ B) = Compactum.basic A ∩ Compactum.basic B ** ext G ** case h X : Compactum A B : Set X.A G : Ultrafilter X.A ⊢ G ∈ Compactum.basic (A ∩ B) ↔ G ∈ Compactum.basic A ∩ Compactum.basic B ** constructor ** case h.mp X : Compactum A B : Set X.A G : Ultrafilter X.A ⊢ G ∈ Compactum.basic (A ∩ B) → G ∈ Compactum.basic A ∩ Compactum.basic B ** intro hG ** case h.mp X : Compactum A B : Set X.A G : Ultrafilter X.A hG : G ∈ Compactum.basic (A ∩ B) ⊢ G ∈ Compactum.basic A ∩ Compactum.basic B ** constructor <;> filter_upwards [hG] with _ ** case h X : Compactum A B : Set X.A G : Ultrafilter X.A hG : G ∈ Compactum.basic (A ∩ B) a✝ : X.A ⊢ a✝ ∈ A ∩ B → a✝ ∈ A case h X : Compactum A B : Set X.A G : Ultrafilter X.A hG : G ∈ Compactum.basic (A ∩ B) a✝ : X.A ⊢ a✝ ∈ A ∩ B → a✝ ∈ B ** exacts [And.left, And.right] ** case h.mpr X : Compactum A B : Set X.A G : Ultrafilter X.A ⊢ G ∈ Compactum.basic A ∩ Compactum.basic B → G ∈ Compactum.basic (A ∩ B) ** rintro ⟨h1, h2⟩ ** case h.mpr.intro X : Compactum A B : Set X.A G : Ultrafilter X.A h1 : G ∈ Compactum.basic A h2 : G ∈ Compactum.basic B ⊢ G ∈ Compactum.basic (A ∩ B) ** exact inter_mem h1 h2 ** Qed
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Compactum.subset_cl ** X : Compactum A : Set X.A a : X.A ha : a ∈ A ⊢ str X (incl X a) = a ** simp ** Qed
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Compactum.cl_cl ** X : Compactum A : Set X.A ⊢ Compactum.cl (Compactum.cl A) ⊆ Compactum.cl A ** rintro _ ⟨F, hF, rfl⟩ ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) ⊢ str X F ∈ Compactum.cl A ** let fsu := Finset (Set (Ultrafilter X)) ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ⊢ str X F ∈ Compactum.cl A ** let ssu := Set (Set (Ultrafilter X)) ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ⊢ str X F ∈ Compactum.cl A ** let ι : fsu → ssu := fun x ↦ ↑x ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x ⊢ str X F ∈ Compactum.cl A ** let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z } ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} ⊢ str X F ∈ Compactum.cl A ** let AA := { G : Ultrafilter X | A ∈ G } ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} ⊢ str X F ∈ Compactum.cl A ** let C1 := insert AA C0 ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 ⊢ str X F ∈ Compactum.cl A ** let C2 := finiteInterClosure C1 ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 ⊢ str X F ∈ Compactum.cl A ** have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
use Q ∩ R
simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
exact inter_sets _ hQ hR ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 ⊢ str X F ∈ Compactum.cl A ** have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
rintro B ⟨Q, hQ, rfl⟩
obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
use X.incl q
simpa ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B ⊢ str X F ∈ Compactum.cl A ** have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
rintro B ⟨Q, hQ, rfl⟩
have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
rcases this with ⟨q, hq1, P, hq2, hq3⟩
refine' ⟨P, hq2, _⟩
rw [← hq3] at hq1
simpa ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) ⊢ str X F ∈ Compactum.cl A ** suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
use X.join G
have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
rw [join_distrib, this]
exact ⟨h1 (Or.inl rfl), rfl⟩ ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) ⊢ ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) ** have claim4 := finiteInterClosure_finiteInter C1 ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) ⊢ ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) ** have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩ ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 claim6 : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → Set.Nonempty P ⊢ ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) ** intro T hT ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 claim6 : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → Set.Nonempty P T : fsu hT : ι T ⊆ C1 ⊢ Set.Nonempty (⋂₀ ι T) ** suffices ⋂₀ ι T ∈ C2 by exact claim6 _ this ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 claim6 : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → Set.Nonempty P T : fsu hT : ι T ⊆ C1 ⊢ ⋂₀ ι T ∈ C2 ** apply claim4.finiteInter_mem T ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 claim6 : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → Set.Nonempty P T : fsu hT : ι T ⊆ C1 ⊢ ↑T ⊆ finiteInterClosure C1 ** intro t ht ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 claim6 : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → Set.Nonempty P T : fsu hT : ι T ⊆ C1 t : Set (Ultrafilter X.A) ht : t ∈ ↑T ⊢ t ∈ finiteInterClosure C1 ** refine' finiteInterClosure.basic (@hT t ht) ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 ⊢ ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 ** rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩ ** case intro.intro.intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 Q : Set X.A hQ : Q ∈ F R : Set X.A hR : R ∈ F ⊢ str X ⁻¹' Q ∩ str X ⁻¹' R ∈ C0 ** use Q ∩ R ** case h X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 Q : Set X.A hQ : Q ∈ F R : Set X.A hR : R ∈ F ⊢ Q ∩ R ∈ F ∧ str X ⁻¹' (Q ∩ R) = str X ⁻¹' Q ∩ str X ⁻¹' R ** simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter] ** case h X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 Q : Set X.A hQ : Q ∈ F R : Set X.A hR : R ∈ F ⊢ Q ∩ R ∈ F ** exact inter_sets _ hQ hR ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 ⊢ ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B ** rintro B ⟨Q, hQ, rfl⟩ ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 Q : Set X.A hQ : Q ∈ F ⊢ Set.Nonempty (str X ⁻¹' Q) ** obtain ⟨q⟩ := Filter.nonempty_of_mem hQ ** case intro.intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 Q : Set X.A hQ : Q ∈ F q : X.A h✝ : q ∈ Q ⊢ Set.Nonempty (str X ⁻¹' Q) ** use X.incl q ** case h X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 Q : Set X.A hQ : Q ∈ F q : X.A h✝ : q ∈ Q ⊢ incl X q ∈ str X ⁻¹' Q ** simpa ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B ⊢ ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) ** rintro B ⟨Q, hQ, rfl⟩ ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B Q : Set X.A hQ : Q ∈ F ⊢ Set.Nonempty (AA ∩ str X ⁻¹' Q) ** have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF) ** case intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B Q : Set X.A hQ : Q ∈ F this : Set.Nonempty (Q ∩ Compactum.cl A) ⊢ Set.Nonempty (AA ∩ str X ⁻¹' Q) ** rcases this with ⟨q, hq1, P, hq2, hq3⟩ ** case intro.intro.intro.intro.intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B Q : Set X.A hQ : Q ∈ F q : X.A hq1 : q ∈ Q P : Ultrafilter X.A hq2 : P ∈ Compactum.basic A hq3 : str X P = q ⊢ Set.Nonempty (AA ∩ str X ⁻¹' Q) ** refine' ⟨P, hq2, _⟩ ** case intro.intro.intro.intro.intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B Q : Set X.A hQ : Q ∈ F q : X.A hq1 : q ∈ Q P : Ultrafilter X.A hq2 : P ∈ Compactum.basic A hq3 : str X P = q ⊢ P ∈ str X ⁻¹' Q ** rw [← hq3] at hq1 ** case intro.intro.intro.intro.intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B Q : Set X.A hQ : Q ∈ F q : X.A P : Ultrafilter X.A hq1 : str X P ∈ Q hq2 : P ∈ Compactum.basic A hq3 : str X P = q ⊢ P ∈ str X ⁻¹' Q ** simpa ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) this : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) ⊢ str X F ∈ Compactum.cl A ** obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this ** case intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) this : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) G : Ultrafilter (Ultrafilter X.A) h1 : C1 ⊆ G.sets ⊢ str X F ∈ Compactum.cl A ** use X.join G ** case h X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) this : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) G : Ultrafilter (Ultrafilter X.A) h1 : C1 ⊆ G.sets ⊢ join X G ∈ Compactum.basic A ∧ str X (join X G) = str X F ** have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩) ** case h X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) this✝ : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) G : Ultrafilter (Ultrafilter X.A) h1 : C1 ⊆ G.sets this : Ultrafilter.map (str X) G = F ⊢ join X G ∈ Compactum.basic A ∧ str X (join X G) = str X F ** rw [join_distrib, this] ** case h X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) this✝ : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T) G : Ultrafilter (Ultrafilter X.A) h1 : C1 ⊆ G.sets this : Ultrafilter.map (str X) G = F ⊢ join X G ∈ Compactum.basic A ∧ str X F = str X F ** exact ⟨h1 (Or.inl rfl), rfl⟩ ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 ⊢ ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → P ∈ C0 ∨ ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q ** intro P hP ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 P : Set (Ultrafilter X.A) hP : P ∈ C2 ⊢ P ∈ C0 ∨ ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q ** exact claim5.finiteInterClosure_insert _ hP ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 this : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → P ∈ C0 ∨ ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q ⊢ ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → Set.Nonempty P ** intro P hP ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 this : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → P ∈ C0 ∨ ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q P : Set (Ultrafilter X.A) hP : P ∈ C2 ⊢ Set.Nonempty P ** cases' this P hP with h h ** case inl X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 this : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → P ∈ C0 ∨ ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q P : Set (Ultrafilter X.A) hP : P ∈ C2 h : P ∈ C0 ⊢ Set.Nonempty P ** exact claim2 _ h ** case inr X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 this : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → P ∈ C0 ∨ ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q P : Set (Ultrafilter X.A) hP : P ∈ C2 h : ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q ⊢ Set.Nonempty P ** rcases h with ⟨Q, hQ, rfl⟩ ** case inr.intro.intro X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 this : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → P ∈ C0 ∨ ∃ Q, Q ∈ C0 ∧ P = AA ∩ Q Q : Set (Ultrafilter X.A) hQ : Q ∈ C0 hP : AA ∩ Q ∈ C2 ⊢ Set.Nonempty (AA ∩ Q) ** exact claim3 _ hQ ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : F ∈ Compactum.basic (Compactum.cl A) fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x C0 : ssu := {Z | ∃ B, B ∈ F ∧ str X ⁻¹' B = Z} AA : Set (Ultrafilter X.A) := {G | A ∈ G} C1 : ssu := insert AA C0 C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1 claim1 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → ∀ (C : Set (Ultrafilter X.A)), C ∈ C0 → B ∩ C ∈ C0 claim2 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty B claim3 : ∀ (B : Set (Ultrafilter X.A)), B ∈ C0 → Set.Nonempty (AA ∩ B) claim4 : FiniteInter (finiteInterClosure C1) claim5 : FiniteInter C0 claim6 : ∀ (P : Set (Ultrafilter X.A)), P ∈ C2 → Set.Nonempty P T : fsu hT : ι T ⊆ C1 this : ⋂₀ ι T ∈ C2 ⊢ Set.Nonempty (⋂₀ ι T) ** exact claim6 _ this ** Qed
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Compactum.isClosed_cl ** X : Compactum A : Set X.A ⊢ IsClosed (Compactum.cl A) ** rw [isClosed_iff] ** X : Compactum A : Set X.A ⊢ ∀ (F : Ultrafilter X.A), Compactum.cl A ∈ F → str X F ∈ Compactum.cl A ** intro F hF ** X : Compactum A : Set X.A F : Ultrafilter X.A hF : Compactum.cl A ∈ F ⊢ str X F ∈ Compactum.cl A ** exact cl_cl _ ⟨F, hF, rfl⟩ ** Qed
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Compactum.str_eq_of_le_nhds ** X : Compactum F : Ultrafilter X.A x : X.A ⊢ ↑F ≤ 𝓝 x → str X F = x ** let fsu := Finset (Set (Ultrafilter X)) ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ⊢ ↑F ≤ 𝓝 x → str X F = x ** let ssu := Set (Set (Ultrafilter X)) ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ⊢ ↑F ≤ 𝓝 x → str X F = x ** let ι : fsu → ssu := fun x ↦ ↑x ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x ⊢ ↑F ≤ 𝓝 x → str X F = x ** let T0 : ssu := { S | ∃ A ∈ F, S = basic A } ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} ⊢ ↑F ≤ 𝓝 x → str X F = x ** let AA := X.str ⁻¹' {x} ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} ⊢ ↑F ≤ 𝓝 x → str X F = x ** let T1 := insert AA T0 ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 ⊢ ↑F ≤ 𝓝 x → str X F = x ** let T2 := finiteInterClosure T1 ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 ⊢ ↑F ≤ 𝓝 x → str X F = x ** intro cond ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x ⊢ str X F = x ** have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
intro A hA h
by_contra H
rw [le_nhds_iff] at cond
specialize cond Aᶜ H hA.isOpen_compl
rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
contradiction ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A ⊢ str X F = x ** have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
intro A hA
exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A)) ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A ⊢ str X F = x ** have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 ⊢ str X F = x ** have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
rintro S ⟨S, hS, rfl⟩
rcases claim2 _ hS with ⟨G, hG, hG2⟩
exact ⟨G, hG2, hG⟩ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) ⊢ str X F = x ** have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
rintro S ⟨S, hS, rfl⟩
exact ⟨F, hS⟩ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S ⊢ str X F = x ** suffices ∀ F : fsu, ↑F ⊆ T1 → (⋂₀ ι F).Nonempty by
obtain ⟨G, h1⟩ := Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ this
have c1 : X.join G = F := Ultrafilter.coe_le_coe.1 fun P hP => h1 (Or.inr ⟨P, hP, rfl⟩)
have c2 : G.map X.str = X.incl x := by
refine' Ultrafilter.coe_le_coe.1 fun P hP => _
apply mem_of_superset (h1 (Or.inl rfl))
rintro x ⟨rfl⟩
exact hP
simp [← c1, c2] ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S ⊢ ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) ** intro T hT ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S T : fsu hT : ↑T ⊆ T1 ⊢ Set.Nonempty (⋂₀ ι T) ** refine' claim6 _ (finiteInter_mem (.finiteInterClosure_finiteInter _) _ _) ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S T : fsu hT : ↑T ⊆ T1 ⊢ ↑T ⊆ finiteInterClosure T1 ** intro t ht ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S T : fsu hT : ↑T ⊆ T1 t : Set (Ultrafilter X.A) ht : t ∈ ↑T ⊢ t ∈ finiteInterClosure T1 ** exact finiteInterClosure.basic (@hT t ht) ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x ⊢ ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A ** intro A hA h ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x A : Set X.A hA : IsClosed A h : A ∈ F ⊢ x ∈ A ** by_contra H ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x A : Set X.A hA : IsClosed A h : A ∈ F H : ¬x ∈ A ⊢ False ** rw [le_nhds_iff] at cond ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ∀ (s : Set X.A), x ∈ s → IsOpen s → s ∈ ↑F A : Set X.A hA : IsClosed A h : A ∈ F H : ¬x ∈ A ⊢ False ** specialize cond Aᶜ H hA.isOpen_compl ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 A : Set X.A hA : IsClosed A h : A ∈ F H : ¬x ∈ A cond : Aᶜ ∈ ↑F ⊢ False ** rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 A : Set X.A hA : IsClosed A h : A ∈ F H : ¬x ∈ A cond : ¬A ∈ F ⊢ False ** contradiction ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A ⊢ ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A ** intro A hA ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A A : Set X.A hA : A ∈ F ⊢ x ∈ Compactum.cl A ** exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A)) ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A ⊢ ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 ** rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩ ** case intro.intro.intro.intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A S1 : Set X.A hS1 : S1 ∈ F S2 : Set X.A hS2 : S2 ∈ F ⊢ Compactum.basic S1 ∩ Compactum.basic S2 ∈ T0 ** exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A S1 : Set X.A hS1 : S1 ∈ F S2 : Set X.A hS2 : S2 ∈ F ⊢ Compactum.basic S1 ∩ Compactum.basic S2 = Compactum.basic (S1 ∩ S2) ** simp [basic_inter] ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 ⊢ ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) ** rintro S ⟨S, hS, rfl⟩ ** case intro.intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 S : Set X.A hS : S ∈ F ⊢ Set.Nonempty (AA ∩ Compactum.basic S) ** rcases claim2 _ hS with ⟨G, hG, hG2⟩ ** case intro.intro.intro.intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 S : Set X.A hS : S ∈ F G : Ultrafilter X.A hG : G ∈ Compactum.basic S hG2 : str X G = x ⊢ Set.Nonempty (AA ∩ Compactum.basic S) ** exact ⟨G, hG2, hG⟩ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) ⊢ ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S ** rintro S ⟨S, hS, rfl⟩ ** case intro.intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) S : Set X.A hS : S ∈ F ⊢ Set.Nonempty (Compactum.basic S) ** exact ⟨F, hS⟩ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S ⊢ ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → S ∈ T0 ∨ ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q ** intro S hS ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ S ∈ T0 ∨ ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q ** apply finiteInterClosure_insert ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S this : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → S ∈ T0 ∨ ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q ⊢ ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S ** intro S hS ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S this : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → S ∈ T0 ∨ ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ Set.Nonempty S ** cases' this _ hS with h h ** case inl X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S this : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → S ∈ T0 ∨ ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q S : Set (Ultrafilter X.A) hS : S ∈ T2 h : S ∈ T0 ⊢ Set.Nonempty S ** exact claim5 S h ** case inr X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S this : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → S ∈ T0 ∨ ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q S : Set (Ultrafilter X.A) hS : S ∈ T2 h : ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q ⊢ Set.Nonempty S ** rcases h with ⟨Q, hQ, rfl⟩ ** case inr.intro.intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S this : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → S ∈ T0 ∨ ∃ Q, Q ∈ T0 ∧ S = AA ∩ Q Q : Set (Ultrafilter X.A) hQ : Q ∈ T0 hS : AA ∩ Q ∈ T2 ⊢ Set.Nonempty (AA ∩ Q) ** exact claim4 Q hQ ** case cond X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ FiniteInter T0 ** constructor ** case cond.univ_mem X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ Set.univ ∈ T0 ** use Set.univ ** case h X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ Set.univ ∈ F ∧ Set.univ = Compactum.basic Set.univ ** refine' ⟨Filter.univ_sets _, _⟩ ** case h X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ Set.univ = Compactum.basic Set.univ ** ext ** case h.h X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 x✝ : Ultrafilter X.A ⊢ x✝ ∈ Set.univ ↔ x✝ ∈ Compactum.basic Set.univ ** refine' ⟨_, by tauto⟩ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 x✝ : Ultrafilter X.A ⊢ x✝ ∈ Compactum.basic Set.univ → x✝ ∈ Set.univ ** tauto ** case h.h X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 x✝ : Ultrafilter X.A ⊢ x✝ ∈ Set.univ → x✝ ∈ Compactum.basic Set.univ ** intro ** case h.h X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 x✝ : Ultrafilter X.A a✝ : x✝ ∈ Set.univ ⊢ x✝ ∈ Compactum.basic Set.univ ** apply Filter.univ_sets ** case cond.inter_mem X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ ∀ ⦃s : Set (Ultrafilter X.A)⦄, s ∈ T0 → ∀ ⦃t : Set (Ultrafilter X.A)⦄, t ∈ T0 → s ∩ t ∈ T0 ** exact claim3 ** case H X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S S : Set (Ultrafilter X.A) hS : S ∈ T2 ⊢ S ∈ finiteInterClosure (insert AA T0) ** exact hS ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) ⊢ str X F = x ** obtain ⟨G, h1⟩ := Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ this ** case intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) G : Ultrafilter (Ultrafilter X.A) h1 : T1 ⊆ G.sets ⊢ str X F = x ** have c1 : X.join G = F := Ultrafilter.coe_le_coe.1 fun P hP => h1 (Or.inr ⟨P, hP, rfl⟩) ** case intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) G : Ultrafilter (Ultrafilter X.A) h1 : T1 ⊆ G.sets c1 : join X G = F ⊢ str X F = x ** have c2 : G.map X.str = X.incl x := by
refine' Ultrafilter.coe_le_coe.1 fun P hP => _
apply mem_of_superset (h1 (Or.inl rfl))
rintro x ⟨rfl⟩
exact hP ** case intro X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) G : Ultrafilter (Ultrafilter X.A) h1 : T1 ⊆ G.sets c1 : join X G = F c2 : Ultrafilter.map (str X) G = incl X x ⊢ str X F = x ** simp [← c1, c2] ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) G : Ultrafilter (Ultrafilter X.A) h1 : T1 ⊆ G.sets c1 : join X G = F ⊢ Ultrafilter.map (str X) G = incl X x ** refine' Ultrafilter.coe_le_coe.1 fun P hP => _ ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) G : Ultrafilter (Ultrafilter X.A) h1 : T1 ⊆ G.sets c1 : join X G = F P : Set X.A hP : P ∈ ↑(incl X x) ⊢ P ∈ ↑(Ultrafilter.map (str X) G) ** apply mem_of_superset (h1 (Or.inl rfl)) ** X : Compactum F : Ultrafilter X.A x : X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} AA : Set (Ultrafilter X.A) := str X ⁻¹' {x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 x claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → x ∈ Compactum.cl A claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) G : Ultrafilter (Ultrafilter X.A) h1 : T1 ⊆ G.sets c1 : join X G = F P : Set X.A hP : P ∈ ↑(incl X x) ⊢ AA ⊆ str X ⁻¹' P ** rintro x ⟨rfl⟩ ** case refl X : Compactum F : Ultrafilter X.A fsu : Type u_1 := Finset (Set (Ultrafilter X.A)) ssu : Type u_1 := Set (Set (Ultrafilter X.A)) ι : fsu → ssu := fun x => ↑x T0 : ssu := {S | ∃ A, A ∈ F ∧ S = Compactum.basic A} claim3 : ∀ (S1 : Set (Ultrafilter X.A)), S1 ∈ T0 → ∀ (S2 : Set (Ultrafilter X.A)), S2 ∈ T0 → S1 ∩ S2 ∈ T0 claim5 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty S G : Ultrafilter (Ultrafilter X.A) c1 : join X G = F P : Set X.A x : Ultrafilter X.A AA : Set (Ultrafilter X.A) := str X ⁻¹' {str X x} T1 : ssu := insert AA T0 T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1 cond : ↑F ≤ 𝓝 (str X x) claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → str X x ∈ A claim2 : ∀ (A : Set X.A), A ∈ F → str X x ∈ Compactum.cl A claim4 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T0 → Set.Nonempty (AA ∩ S) claim6 : ∀ (S : Set (Ultrafilter X.A)), S ∈ T2 → Set.Nonempty S this : ∀ (F : fsu), ↑F ⊆ T1 → Set.Nonempty (⋂₀ ι F) h1 : T1 ⊆ G.sets hP : P ∈ ↑(incl X (str X x)) ⊢ x ∈ str X ⁻¹' P ** exact hP ** Qed
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Compactum.le_nhds_of_str_eq ** X : Compactum F : Ultrafilter X.A x : X.A h : str X F = x s : Set X.A hx : x ∈ s hs : IsOpen s ⊢ str X F ∈ s ** rwa [h] ** Qed
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Compactum.lim_eq_str ** X : Compactum F : Ultrafilter X.A ⊢ Ultrafilter.lim F = str X F ** rw [Ultrafilter.lim_eq_iff_le_nhds, le_nhds_iff] ** X : Compactum F : Ultrafilter X.A ⊢ ∀ (s : Set X.A), str X F ∈ s → IsOpen s → s ∈ ↑F ** tauto ** Qed
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Compactum.cl_eq_closure ** X : Compactum A : Set X.A ⊢ Compactum.cl A = closure A ** ext ** case h X : Compactum A : Set X.A x✝ : X.A ⊢ x✝ ∈ Compactum.cl A ↔ x✝ ∈ closure A ** rw [mem_closure_iff_ultrafilter] ** case h X : Compactum A : Set X.A x✝ : X.A ⊢ x✝ ∈ Compactum.cl A ↔ ∃ u, A ∈ u ∧ ↑u ≤ 𝓝 x✝ ** constructor ** case h.mp X : Compactum A : Set X.A x✝ : X.A ⊢ x✝ ∈ Compactum.cl A → ∃ u, A ∈ u ∧ ↑u ≤ 𝓝 x✝ ** rintro ⟨F, h1, h2⟩ ** case h.mp.intro.intro X : Compactum A : Set X.A x✝ : X.A F : Ultrafilter X.A h1 : F ∈ Compactum.basic A h2 : str X F = x✝ ⊢ ∃ u, A ∈ u ∧ ↑u ≤ 𝓝 x✝ ** exact ⟨F, h1, le_nhds_of_str_eq _ _ h2⟩ ** case h.mpr X : Compactum A : Set X.A x✝ : X.A ⊢ (∃ u, A ∈ u ∧ ↑u ≤ 𝓝 x✝) → x✝ ∈ Compactum.cl A ** rintro ⟨F, h1, h2⟩ ** case h.mpr.intro.intro X : Compactum A : Set X.A x✝ : X.A F : Ultrafilter X.A h1 : A ∈ F h2 : ↑F ≤ 𝓝 x✝ ⊢ x✝ ∈ Compactum.cl A ** exact ⟨F, h1, str_eq_of_le_nhds _ _ h2⟩ ** Qed
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Compactum.continuous_of_hom ** X Y : Compactum f : X ⟶ Y ⊢ Continuous f.f ** rw [continuous_iff_ultrafilter] ** X Y : Compactum f : X ⟶ Y ⊢ ∀ (x : X.A) (g : Ultrafilter X.A), ↑g ≤ 𝓝 x → Tendsto f.f (↑g) (𝓝 (Monad.Algebra.Hom.f f x)) ** intro x g h ** X Y : Compactum f : X ⟶ Y x : X.A g : Ultrafilter X.A h : ↑g ≤ 𝓝 x ⊢ Tendsto f.f (↑g) (𝓝 (Monad.Algebra.Hom.f f x)) ** rw [Tendsto, ← coe_map] ** X Y : Compactum f : X ⟶ Y x : X.A g : Ultrafilter X.A h : ↑g ≤ 𝓝 x ⊢ ↑(Ultrafilter.map f.f g) ≤ 𝓝 (Monad.Algebra.Hom.f f x) ** apply le_nhds_of_str_eq ** case a X Y : Compactum f : X ⟶ Y x : X.A g : Ultrafilter X.A h : ↑g ≤ 𝓝 x ⊢ str Y (Ultrafilter.map f.f g) = Monad.Algebra.Hom.f f x ** rw [← str_hom_commute, str_eq_of_le_nhds _ x _] ** X Y : Compactum f : X ⟶ Y x : X.A g : Ultrafilter X.A h : ↑g ≤ 𝓝 x ⊢ ↑g ≤ 𝓝 x ** apply h ** Qed
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compactumToCompHaus.faithful ** ⊢ ∀ {X Y : Compactum}, Function.Injective compactumToCompHaus.map ** intro _ _ _ _ h ** X✝ Y✝ : Compactum a₁✝ a₂✝ : X✝ ⟶ Y✝ h : compactumToCompHaus.map a₁✝ = compactumToCompHaus.map a₂✝ ⊢ a₁✝ = a₂✝ ** apply Monad.Algebra.Hom.ext ** case f X✝ Y✝ : Compactum a₁✝ a₂✝ : X✝ ⟶ Y✝ h : compactumToCompHaus.map a₁✝ = compactumToCompHaus.map a₂✝ ⊢ a₁✝.f = a₂✝.f ** apply congrArg (fun f => f.toFun) h ** Qed
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dist_nonneg' ** α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Sort u_3 x y : α dist : α → α → ℝ dist_self : ∀ (x : α), dist x x = 0 dist_comm : ∀ (x y : α), dist x y = dist y x dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ⊢ dist x y + dist y x = 2 * dist x y ** rw [two_mul, dist_comm] ** Qed
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PseudoMetricSpace.ext ** α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 m m' : PseudoMetricSpace α h : toDist = toDist ⊢ m = m' ** cases' m with d _ _ _ ed hed U hU B hB ** case mk α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 m' : PseudoMetricSpace α d : Dist α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} h : toDist = toDist ⊢ mk dist_self✝ dist_comm✝ dist_triangle✝ ed hed U B = m' ** cases' m' with d' _ _ _ ed' hed' U' hU' B' hB' ** case mk.mk α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 d : Dist α dist_self✝¹ : ∀ (x : α), dist x x = 0 dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} d' : Dist α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed' : α → α → ℝ≥0∞ hed' : ∀ (x y : α), ed' x y = ENNReal.ofReal (dist x y) U' : UniformSpace α hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B' : Bornology α hB' : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} h : toDist = toDist ⊢ mk dist_self✝¹ dist_comm✝¹ dist_triangle✝¹ ed hed U B = mk dist_self✝ dist_comm✝ dist_triangle✝ ed' hed' U' B' ** obtain rfl : d = d' := h ** case mk.mk α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 d : Dist α dist_self✝¹ : ∀ (x : α), dist x x = 0 dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ed' : α → α → ℝ≥0∞ U' : UniformSpace α B' : Bornology α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z hed' : ∀ (x y : α), ed' x y = ENNReal.ofReal (dist x y) hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} hB' : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ⊢ mk dist_self✝¹ dist_comm✝¹ dist_triangle✝¹ ed hed U B = mk dist_self✝ dist_comm✝ dist_triangle✝ ed' hed' U' B' ** congr ** case mk.mk.e_edist α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 d : Dist α dist_self✝¹ : ∀ (x : α), dist x x = 0 dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ed' : α → α → ℝ≥0∞ U' : UniformSpace α B' : Bornology α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z hed' : ∀ (x y : α), ed' x y = ENNReal.ofReal (dist x y) hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} hB' : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ⊢ ed = ed' ** ext x y : 2 ** case mk.mk.e_edist.h.h α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 d : Dist α dist_self✝¹ : ∀ (x : α), dist x x = 0 dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ed' : α → α → ℝ≥0∞ U' : UniformSpace α B' : Bornology α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z hed' : ∀ (x y : α), ed' x y = ENNReal.ofReal (dist x y) hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} hB' : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} x y : α ⊢ ed x y = ed' x y ** rw [hed, hed'] ** case mk.mk.e_toUniformSpace α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 d : Dist α dist_self✝¹ : ∀ (x : α), dist x x = 0 dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ed' : α → α → ℝ≥0∞ U' : UniformSpace α B' : Bornology α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z hed' : ∀ (x y : α), ed' x y = ENNReal.ofReal (dist x y) hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} hB' : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ⊢ U = U' ** exact UniformSpace.ext (hU.trans hU'.symm) ** case mk.mk.e_toBornology α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 d : Dist α dist_self✝¹ : ∀ (x : α), dist x x = 0 dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ed' : α → α → ℝ≥0∞ U' : UniformSpace α B' : Bornology α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z hed' : ∀ (x y : α), ed' x y = ENNReal.ofReal (dist x y) hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} hB' : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ⊢ B = B' ** ext : 2 ** case mk.mk.e_toBornology.h_cobounded.a α✝ : Type u β : Type v X : Type u_1 ι : Type u_2 α : Type u_3 d : Dist α dist_self✝¹ : ∀ (x : α), dist x x = 0 dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z ed : α → α → ℝ≥0∞ hed : ∀ (x y : α), ed x y = ENNReal.ofReal (dist x y) U : UniformSpace α hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} B : Bornology α hB : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} ed' : α → α → ℝ≥0∞ U' : UniformSpace α B' : Bornology α dist_self✝ : ∀ (x : α), dist x x = 0 dist_comm✝ : ∀ (x y : α), dist x y = dist y x dist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z hed' : ∀ (x y : α), ed' x y = ENNReal.ofReal (dist x y) hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε} hB' : (cobounded α).sets = {s | ∃ C, ∀ (x : α), x ∈ sᶜ → ∀ (y : α), y ∈ sᶜ → dist x y ≤ C} s✝ : Set α ⊢ s✝ ∈ cobounded α ↔ s✝ ∈ cobounded α ** rw [← Filter.mem_sets, ← Filter.mem_sets, hB, hB'] ** Qed
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dist_triangle_left ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α ⊢ dist x y ≤ dist z x + dist z y ** rw [dist_comm z] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α ⊢ dist x y ≤ dist x z + dist z y ** apply dist_triangle ** Qed
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dist_triangle_right ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α ⊢ dist x y ≤ dist x z + dist y z ** rw [dist_comm y] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α ⊢ dist x y ≤ dist x z + dist z y ** apply dist_triangle ** Qed
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dist_triangle4_left ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x₁ y₁ x₂ y₂ : α ⊢ dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) ** rw [add_left_comm, dist_comm x₁, ← add_assoc] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x₁ y₁ x₂ y₂ : α ⊢ dist x₂ y₂ ≤ dist x₂ x₁ + dist x₁ y₁ + dist y₁ y₂ ** apply dist_triangle4 ** Qed
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dist_triangle4_right ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x₁ y₁ x₂ y₂ : α ⊢ dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ ** rw [add_right_comm, dist_comm y₁] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x₁ y₁ x₂ y₂ : α ⊢ dist x₁ y₁ ≤ dist x₁ x₂ + dist x₂ y₂ + dist y₂ y₁ ** apply dist_triangle4 ** Qed
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dist_le_Ico_sum_dist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α f : ℕ → α m n : ℕ h : m ≤ n ⊢ dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1)) ** induction n, h using Nat.le_induction with
| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]
| succ n hle ihn =>
calc
dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _
_ ≤ (∑ i in Finset.Ico m n, _) + _ := add_le_add ihn le_rfl
_ = ∑ i in Finset.Ico m (n + 1), _ := by
{ rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp } ** case base α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α f : ℕ → α m n : ℕ ⊢ dist (f m) (f m) ≤ ∑ i in Finset.Ico m m, dist (f i) (f (i + 1)) ** rw [Finset.Ico_self, Finset.sum_empty, dist_self] ** case succ α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α f : ℕ → α m n✝ n : ℕ hle : m ≤ n ihn : dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1)) ⊢ dist (f m) (f (n + 1)) ≤ ∑ i in Finset.Ico m (n + 1), dist (f i) (f (i + 1)) ** calc
dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _
_ ≤ (∑ i in Finset.Ico m n, _) + _ := add_le_add ihn le_rfl
_ = ∑ i in Finset.Ico m (n + 1), _ := by
{ rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp } ** Qed
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swap_dist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α ⊢ Function.swap dist = dist ** funext x y ** case h.h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α ⊢ Function.swap dist x y = dist x y ** exact dist_comm _ _ ** Qed
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edist_nndist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α ⊢ edist x y = ↑(nndist x y) ** rw [edist_dist, dist_nndist, ENNReal.ofReal_coe_nnreal] ** Qed
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nndist_edist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α ⊢ nndist x y = ENNReal.toNNReal (edist x y) ** simp [edist_nndist] ** Qed
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edist_lt_coe ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α c : ℝ≥0 ⊢ edist x y < ↑c ↔ nndist x y < c ** rw [edist_nndist, ENNReal.coe_lt_coe] ** Qed
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edist_le_coe ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α c : ℝ≥0 ⊢ edist x y ≤ ↑c ↔ nndist x y ≤ c ** rw [edist_nndist, ENNReal.coe_le_coe] ** Qed
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edist_lt_ofReal ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α r : ℝ ⊢ edist x y < ENNReal.ofReal r ↔ dist x y < r ** rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg] ** Qed
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edist_le_ofReal ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α r : ℝ hr : 0 ≤ r ⊢ edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r ** rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr] ** Qed
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nndist_dist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α ⊢ nndist x y = Real.toNNReal (dist x y) ** rw [dist_nndist, Real.toNNReal_coe] ** Qed
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dist_edist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y : α ⊢ dist x y = ENNReal.toReal (edist x y) ** rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg] ** Qed
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Metric.mem_ball' ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ y ∈ ball x ε ↔ dist x y < ε ** rw [dist_comm, mem_ball] ** Qed
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Metric.mem_ball_self ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : 0 < ε ⊢ x ∈ ball x ε ** rwa [mem_ball, dist_self] ** Qed
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Metric.ball_eq_empty ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ ball x ε = ∅ ↔ ε ≤ 0 ** rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt] ** Qed
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Metric.ball_zero ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ ball x 0 = ∅ ** rw [ball_eq_empty] ** Qed
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Metric.exists_lt_mem_ball_of_mem_ball ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : x ∈ ball y ε ⊢ ∃ ε', ε' < ε ∧ x ∈ ball y ε' ** simp only [mem_ball] at h ⊢ ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : dist x y < ε ⊢ ∃ ε', ε' < ε ∧ dist x y < ε' ** exact ⟨(dist x y + ε) / 2, by linarith, by linarith⟩ ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : dist x y < ε ⊢ (dist x y + ε) / 2 < ε ** linarith ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : dist x y < ε ⊢ dist x y < (dist x y + ε) / 2 ** linarith ** Qed
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Metric.ball_eq_ball' ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ y z : α δ ε✝ ε₁ ε₂ : ℝ s : Set α ε : ℝ x : α ⊢ UniformSpace.ball x {p | dist p.1 p.2 < ε} = ball x ε ** ext ** case h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝¹ y z : α δ ε✝ ε₁ ε₂ : ℝ s : Set α ε : ℝ x x✝ : α ⊢ x✝ ∈ UniformSpace.ball x {p | dist p.1 p.2 < ε} ↔ x✝ ∈ ball x ε ** simp [dist_comm, UniformSpace.ball] ** Qed
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Metric.mem_closedBall' ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ y ∈ closedBall x ε ↔ dist x y ≤ ε ** rw [dist_comm, mem_closedBall] ** Qed
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Metric.mem_sphere' ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ y ∈ sphere x ε ↔ dist x y = ε ** rw [dist_comm, mem_sphere] ** Qed
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Metric.ne_of_mem_sphere ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : y ∈ sphere x ε hε : ε ≠ 0 ⊢ ¬x ∈ sphere x ε ** simpa using hε.symm ** Qed
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Metric.sphere_isEmpty_of_subsingleton ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α inst✝ : Subsingleton α hε : ε ≠ 0 ⊢ IsEmpty ↑(sphere x ε) ** rw [sphere_eq_empty_of_subsingleton hε] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝¹ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α inst✝ : Subsingleton α hε : ε ≠ 0 ⊢ IsEmpty ↑∅ ** infer_instance ** Qed
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Metric.mem_closedBall_self ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : 0 ≤ ε ⊢ x ∈ closedBall x ε ** rwa [mem_closedBall, dist_self] ** Qed
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Metric.closedBall_eq_empty ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ closedBall x ε = ∅ ↔ ε < 0 ** rw [← not_nonempty_iff_eq_empty, nonempty_closedBall, not_le] ** Qed
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Metric.ball_disjoint_closedBall ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : δ + ε ≤ dist x y ⊢ ε + δ ≤ dist y x ** rwa [add_comm, dist_comm] ** Qed
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Metric.sphere_union_ball ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ sphere x ε ∪ ball x ε = closedBall x ε ** rw [union_comm, ball_union_sphere] ** Qed
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Metric.closedBall_diff_sphere ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ closedBall x ε \ sphere x ε = ball x ε ** rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot] ** Qed
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Metric.closedBall_diff_ball ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ closedBall x ε \ ball x ε = sphere x ε ** rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot] ** Qed
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Metric.mem_ball_comm ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ x ∈ ball y ε ↔ y ∈ ball x ε ** rw [mem_ball', mem_ball] ** Qed
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Metric.mem_closedBall_comm ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ x ∈ closedBall y ε ↔ y ∈ closedBall x ε ** rw [mem_closedBall', mem_closedBall] ** Qed
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Metric.mem_sphere_comm ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ x ∈ sphere y ε ↔ y ∈ sphere x ε ** rw [mem_sphere', mem_sphere] ** Qed
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Metric.closedBall_eq_bInter_ball ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ closedBall x ε = ⋂ δ, ⋂ (_ : δ > ε), ball x δ ** ext y ** case h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α y : α ⊢ y ∈ closedBall x ε ↔ y ∈ ⋂ δ, ⋂ (_ : δ > ε), ball x δ ** rw [mem_closedBall, ← forall_lt_iff_le', mem_iInter₂] ** case h α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α y : α ⊢ (∀ ⦃c : ℝ⦄, ε < c → dist y x < c) ↔ ∀ (i : ℝ), i > ε → y ∈ ball x i ** rfl ** Qed
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Metric.dist_lt_add_of_nonempty_ball_inter_closedBall ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : Set.Nonempty (ball x ε₁ ∩ closedBall y ε₂) ⊢ dist x y < ε₁ + ε₂ ** rw [inter_comm] at h ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : Set.Nonempty (closedBall y ε₂ ∩ ball x ε₁) ⊢ dist x y < ε₁ + ε₂ ** rw [add_comm, dist_comm] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : Set.Nonempty (closedBall y ε₂ ∩ ball x ε₁) ⊢ dist y x < ε₂ + ε₁ ** exact dist_lt_add_of_nonempty_closedBall_inter_ball h ** Qed
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Metric.iUnion_inter_closedBall_nat ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ y z : α δ ε ε₁ ε₂ : ℝ s✝ s : Set α x : α ⊢ ⋃ n, s ∩ closedBall x ↑n = s ** rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ] ** Qed
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Metric.ball_subset ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z✝ : α δ ε ε₁ ε₂ : ℝ s : Set α h : dist x y ≤ ε₂ - ε₁ z : α zx : z ∈ ball x ε₁ ⊢ z ∈ ball y ε₂ ** rw [← add_sub_cancel'_right ε₁ ε₂] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z✝ : α δ ε ε₁ ε₂ : ℝ s : Set α h : dist x y ≤ ε₂ - ε₁ z : α zx : z ∈ ball x ε₁ ⊢ z ∈ ball y (ε₁ + (ε₂ - ε₁)) ** exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h) ** Qed
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Metric.ball_half_subset ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α y : α h : y ∈ ball x (ε / 2) ⊢ dist y x ≤ ε - ε / 2 ** rw [sub_self_div_two] ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α y : α h : y ∈ ball x (ε / 2) ⊢ dist y x ≤ ε / 2 ** exact le_of_lt h ** Qed
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Metric.exists_ball_subset_ball ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α h : y ∈ ball x ε ⊢ dist y x ≤ ε - (ε - dist y x) ** rw [sub_sub_self] ** Qed
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Metric.forall_of_forall_mem_closedBall ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α p : α → Prop x : α H : ∃ᶠ (R : ℝ) in atTop, ∀ (y : α), y ∈ closedBall x R → p y y : α ⊢ p y ** obtain ⟨R, hR, h⟩ : ∃ R ≥ dist y x, ∀ z : α, z ∈ closedBall x R → p z :=
frequently_iff.1 H (Ici_mem_atTop (dist y x)) ** case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α p : α → Prop x : α H : ∃ᶠ (R : ℝ) in atTop, ∀ (y : α), y ∈ closedBall x R → p y y : α R : ℝ hR : R ≥ dist y x h : ∀ (z : α), z ∈ closedBall x R → p z ⊢ p y ** exact h _ hR ** Qed
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Metric.forall_of_forall_mem_ball ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α p : α → Prop x : α H : ∃ᶠ (R : ℝ) in atTop, ∀ (y : α), y ∈ ball x R → p y y : α ⊢ p y ** obtain ⟨R, hR, h⟩ : ∃ R > dist y x, ∀ z : α, z ∈ ball x R → p z :=
frequently_iff.1 H (Ioi_mem_atTop (dist y x)) ** case intro.intro α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x✝ y✝ z : α δ ε ε₁ ε₂ : ℝ s : Set α p : α → Prop x : α H : ∃ᶠ (R : ℝ) in atTop, ∀ (y : α), y ∈ ball x R → p y y : α R : ℝ hR : R > dist y x h : ∀ (z : α), z ∈ ball x R → p z ⊢ p y ** exact h _ hR ** Qed
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Metric.isBounded_iff ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ s : Set α ⊢ IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C ** rw [isBounded_def, ← Filter.mem_sets, @PseudoMetricSpace.cobounded_sets α, mem_setOf_eq,
compl_compl] ** Qed
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Metric.isBounded_iff_nndist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ s : Set α ⊢ IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → nndist x y ≤ C ** simp only [isBounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
NNReal.coe_mk, exists_prop] ** Qed
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Metric.uniformity_basis_dist ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α ⊢ HasBasis (𝓤 α) (fun ε => 0 < ε) fun ε => {p | dist p.1 p.2 < ε} ** exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _ ** Qed
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Metric.mk_uniformity_basis ** α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (i : β), p i → 0 < f i hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε ⊢ HasBasis (𝓤 α) p fun i => {p | dist p.1 p.2 < f i} ** refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩ ** α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (i : β), p i → 0 < f i hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε s : Set (α × α) ⊢ (∃ i, 0 < i ∧ {p | dist p.1 p.2 < i} ⊆ s) ↔ ∃ i, p i ∧ {p | dist p.1 p.2 < f i} ⊆ s ** constructor ** case mp α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (i : β), p i → 0 < f i hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε s : Set (α × α) ⊢ (∃ i, 0 < i ∧ {p | dist p.1 p.2 < i} ⊆ s) → ∃ i, p i ∧ {p | dist p.1 p.2 < f i} ⊆ s ** rintro ⟨ε, ε₀, hε⟩ ** case mp.intro.intro α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε✝ ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (i : β), p i → 0 < f i hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε s : Set (α × α) ε : ℝ ε₀ : 0 < ε hε : {p | dist p.1 p.2 < ε} ⊆ s ⊢ ∃ i, p i ∧ {p | dist p.1 p.2 < f i} ⊆ s ** rcases hf ε₀ with ⟨i, hi, H⟩ ** case mp.intro.intro.intro.intro α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε✝ ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (i : β), p i → 0 < f i hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε s : Set (α × α) ε : ℝ ε₀ : 0 < ε hε : {p | dist p.1 p.2 < ε} ⊆ s i : β hi : p i H : f i ≤ ε ⊢ ∃ i, p i ∧ {p | dist p.1 p.2 < f i} ⊆ s ** exact ⟨i, hi, fun x (hx : _ < _) => hε <| lt_of_lt_of_le hx H⟩ ** case mpr α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (i : β), p i → 0 < f i hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε s : Set (α × α) ⊢ (∃ i, p i ∧ {p | dist p.1 p.2 < f i} ⊆ s) → ∃ i, 0 < i ∧ {p | dist p.1 p.2 < i} ⊆ s ** exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩ ** Qed
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Metric.uniformity_basis_dist_inv_nat_pos ** α : Type u β : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε✝ ε₁ ε₂ : ℝ s : Set α ε : ℝ ε0 : 0 < ε n : ℕ hn : 1 / (↑n + 1) < ε ⊢ 1 / ↑(n + 1) ≤ ε ** exact_mod_cast hn.le ** Qed
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Metric.mk_uniformity_basis_le ** α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (x : β), p x → 0 < f x hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε ⊢ HasBasis (𝓤 α) p fun x => {p | dist p.1 p.2 ≤ f x} ** refine' ⟨fun s => uniformity_basis_dist.mem_iff.trans _⟩ ** α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (x : β), p x → 0 < f x hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε s : Set (α × α) ⊢ (∃ i, 0 < i ∧ {p | dist p.1 p.2 < i} ⊆ s) ↔ ∃ i, p i ∧ {p | dist p.1 p.2 ≤ f i} ⊆ s ** constructor ** case mp α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (x : β), p x → 0 < f x hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε s : Set (α × α) ⊢ (∃ i, 0 < i ∧ {p | dist p.1 p.2 < i} ⊆ s) → ∃ i, p i ∧ {p | dist p.1 p.2 ≤ f i} ⊆ s ** rintro ⟨ε, ε₀, hε⟩ ** case mp.intro.intro α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε✝ ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (x : β), p x → 0 < f x hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε s : Set (α × α) ε : ℝ ε₀ : 0 < ε hε : {p | dist p.1 p.2 < ε} ⊆ s ⊢ ∃ i, p i ∧ {p | dist p.1 p.2 ≤ f i} ⊆ s ** rcases exists_between ε₀ with ⟨ε', hε'⟩ ** case mp.intro.intro.intro α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε✝ ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (x : β), p x → 0 < f x hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε s : Set (α × α) ε : ℝ ε₀ : 0 < ε hε : {p | dist p.1 p.2 < ε} ⊆ s ε' : ℝ hε' : 0 < ε' ∧ ε' < ε ⊢ ∃ i, p i ∧ {p | dist p.1 p.2 ≤ f i} ⊆ s ** rcases hf ε' hε'.1 with ⟨i, hi, H⟩ ** case mp.intro.intro.intro.intro.intro α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε✝ ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (x : β), p x → 0 < f x hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε s : Set (α × α) ε : ℝ ε₀ : 0 < ε hε : {p | dist p.1 p.2 < ε} ⊆ s ε' : ℝ hε' : 0 < ε' ∧ ε' < ε i : β hi : p i H : f i ≤ ε' ⊢ ∃ i, p i ∧ {p | dist p.1 p.2 ≤ f i} ⊆ s ** exact ⟨i, hi, fun x (hx : _ ≤ _) => hε <| lt_of_le_of_lt (le_trans hx H) hε'.2⟩ ** case mpr α : Type u β✝ : Type v X : Type u_1 ι : Type u_2 inst✝ : PseudoMetricSpace α x y z : α δ ε ε₁ ε₂ : ℝ s✝ : Set α β : Type u_3 p : β → Prop f : β → ℝ hf₀ : ∀ (x : β), p x → 0 < f x hf : ∀ (ε : ℝ), 0 < ε → ∃ x, p x ∧ f x ≤ ε s : Set (α × α) ⊢ (∃ i, p i ∧ {p | dist p.1 p.2 ≤ f i} ⊆ s) → ∃ i, 0 < i ∧ {p | dist p.1 p.2 < i} ⊆ s ** exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (mem_setOf.2 hx.le)⟩ ** Qed
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