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leandojo-lean4-formal-informal-strings-split

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Split (3)

train · 47.4k rows

formal stringlengths 41 427k	informal stringclasses 1 value
continuousAt_sign_of_ne_zero α : Type u_1 inst✝³ : Zero α inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α h : a ≠ 0 ⊢ ContinuousAt (↑SignType.sign) a rcases h.lt_or_lt with (h_neg \| h_pos) case inl α : Type u_1 inst✝³ : Zero α inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α h : a ≠ 0 h_neg : a < 0 ⊢ ContinuousAt (↑SignType.sign) a exact continuousAt_sign_of_neg h_neg case inr α : Type u_1 inst✝³ : Zero α inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α h : a ≠ 0 h_pos : 0 < a ⊢ ContinuousAt (↑SignType.sign) a exact continuousAt_sign_of_pos h_pos ** Qed
dense_irrational ⊢ Dense {x \| Irrational x} refine' Real.isTopologicalBasis_Ioo_rat.dense_iff.2 _ ⊢ ∀ (o : Set ℝ), o ∈ ⋃ a, ⋃ b, ⋃ (_ : a < b), {Ioo ↑a ↑b} → Set.Nonempty o → Set.Nonempty (o ∩ {x \| Irrational x}) simp only [gt_iff_lt, Rat.cast_lt, not_lt, ge_iff_le, Rat.cast_le, mem_iUnion, mem_singleton_iff, exists_prop, forall_exists_index, and_imp] ⊢ ∀ (o : Set ℝ) (x x_1 : ℚ), x < x_1 → o = Ioo ↑x ↑x_1 → Set.Nonempty o → Set.Nonempty (o ∩ {x \| Irrational x}) rintro _ a b hlt rfl _ a b : ℚ hlt : a < b a✝ : Set.Nonempty (Ioo ↑a ↑b) ⊢ Set.Nonempty (Ioo ↑a ↑b ∩ {x \| Irrational x}) rw [inter_comm] a b : ℚ hlt : a < b a✝ : Set.Nonempty (Ioo ↑a ↑b) ⊢ Set.Nonempty ({x \| Irrational x} ∩ Ioo ↑a ↑b) exact exists_irrational_btwn (Rat.cast_lt.2 hlt) ** Qed
Irrational.eventually_forall_le_dist_cast_div x : ℝ hx : Irrational x n : ℕ ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) := ((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.closedEmbedding_coe_real.isClosedMap).closed_range ** x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) ** have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by rintro ⟨m, rfl⟩ simp at hx ** x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with ⟨ε, ε0, hε⟩ case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) refine' (ge_mem_nhds ε0).mono fun δ hδ m => not_lt.1 fun hlt => _ case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ δ : ℝ hδ : δ ≤ ε m : ℤ hlt : dist x (↑m / ↑n) < δ ⊢ False rw [dist_comm] at hlt case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ δ : ℝ hδ : δ ≤ ε m : ℤ hlt : dist (↑m / ↑n) x < δ ⊢ False refine' hε (ball_subset_ball hδ hlt) ⟨m, _⟩ case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ δ : ℝ hδ : δ ≤ ε m : ℤ hlt : dist (↑m / ↑n) x < δ ⊢ (fun m => (↑n)⁻¹ * ↑m) m = ↑m / ↑n simp [div_eq_inv_mul] x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) ⊢ ¬x ∈ range fun m => (↑n)⁻¹ * ↑m rintro ⟨m, rfl⟩ case intro n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) m : ℤ hx : Irrational ((fun m => (↑n)⁻¹ * ↑m) m) ⊢ False simp at hx Qed
Irrational.eventually_forall_le_dist_cast_rat_of_den_le x : ℝ hx : Irrational x n : ℕ ε : ℝ H : ∀ (k : ℕ), k ≤ n → ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k) r : ℚ hr : r.den ≤ n ⊢ ε ≤ dist x ↑r simpa only [Rat.cast_def] using H r.den hr r.num ** Qed
ContinuousOpenMap.cancel_left F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β →CO γ f₁ f₂ : α →CO β hg : Injective ↑g h : comp g f₁ = comp g f₂ a : α ⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a) rw [← comp_apply, h, comp_apply] ** Qed
OnePoint.compl_image_coe X : Type u_1 s : Set X ⊢ (some '' s)ᶜ = some '' sᶜ ∪ {∞} rw [coe_injective.compl_image_eq, compl_range_coe] ** Qed
OnePoint.ne_infty_iff_exists X : Type u_1 x : OnePoint X ⊢ x ≠ ∞ ↔ ∃ y, ↑y = x induction x using OnePoint.rec <;> simp ** Qed
OnePoint.not_mem_range_coe_iff X : Type u_1 x : OnePoint X ⊢ ¬x ∈ range some ↔ x = ∞ rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff] ** Qed
OnePoint.coe_preimage_infty X : Type u_1 ⊢ some ⁻¹' {∞} = ∅ ext case h X : Type u_1 x✝ : X ⊢ x✝ ∈ some ⁻¹' {∞} ↔ x✝ ∈ ∅ simp ** Qed
OnePoint.isOpen_iff_of_mem' X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s ⊢ IsOpen s ↔ IsCompact (some ⁻¹' s)ᶜ ∧ IsOpen (some ⁻¹' s) simp [isOpen_def, h] ** Qed
OnePoint.isOpen_iff_of_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s ⊢ IsOpen s ↔ IsClosed (some ⁻¹' s)ᶜ ∧ IsCompact (some ⁻¹' s)ᶜ simp only [isOpen_iff_of_mem' h, isClosed_compl_iff, and_comm] ** Qed
OnePoint.isOpen_iff_of_not_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ¬∞ ∈ s ⊢ IsOpen s ↔ IsOpen (some ⁻¹' s) simp [isOpen_def, h] ** Qed
OnePoint.isClosed_iff_of_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s ⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) have : ∞ ∉ sᶜ := fun H => H h X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s this : ¬∞ ∈ sᶜ ⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) rw [← isOpen_compl_iff, isOpen_iff_of_not_mem this, ← isOpen_compl_iff, preimage_compl] ** Qed
OnePoint.isClosed_iff_of_not_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ¬∞ ∈ s ⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) ∧ IsCompact (some ⁻¹' s) rw [← isOpen_compl_iff, isOpen_iff_of_mem (mem_compl h), ← preimage_compl, compl_compl] ** Qed
OnePoint.isOpen_image_coe X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ IsOpen (some '' s) ↔ IsOpen s rw [isOpen_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective] ** Qed
OnePoint.isOpen_compl_image_coe X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ IsOpen (some '' s)ᶜ ↔ IsClosed s ∧ IsCompact s rw [isOpen_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective] X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ ∞ ∈ (some '' s)ᶜ exact infty_not_mem_image_coe ** Qed
OnePoint.isClosed_image_coe X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ IsClosed (some '' s) ↔ IsClosed s ∧ IsCompact s rw [← isOpen_compl_iff, isOpen_compl_image_coe] ** Qed
OnePoint.isClosed_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ IsClosed {∞} rw [← compl_range_coe, isClosed_compl_iff] X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ IsOpen (range some) exact isOpen_range_coe ** Qed
OnePoint.nhdsWithin_compl_infty_eq X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ 𝓝[{∞}ᶜ] ∞ = map some (coclosedCompact X) refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _ case refine'_1 X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ ∀ (i : Set (OnePoint X)), ∞ ∈ i ∧ IsOpen i → ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ i ∩ {∞}ᶜ rintro s ⟨hs, hso⟩ case refine'_1.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t : Set X s : Set (OnePoint X) hs : ∞ ∈ s hso : IsOpen s ⊢ ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ s ∩ {∞}ᶜ refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩ case refine'_1.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t : Set X s : Set (OnePoint X) hs : ∞ ∈ s hso : IsOpen s ⊢ some '' (some ⁻¹' s)ᶜᶜ ⊆ s ∩ {∞}ᶜ simp [Subset.rfl] case refine'_2 X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ ∀ (i' : Set X), IsClosed i' ∧ IsCompact i' → ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' i'ᶜ rintro s ⟨h₁, h₂⟩ case refine'_2.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X h₁ : IsClosed s h₂ : IsCompact s ⊢ ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' sᶜ refine' ⟨_, ⟨mem_compl infty_not_mem_image_coe, isOpen_compl_image_coe.2 ⟨h₁, h₂⟩⟩, _⟩ case refine'_2.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X h₁ : IsClosed s h₂ : IsCompact s ⊢ (some '' s)ᶜ ∩ {∞}ᶜ ⊆ some '' sᶜ simp [compl_image_coe, ← diff_eq, subset_preimage_image] ** Qed
OnePoint.nhds_infty_eq X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ 𝓝 ∞ = map some (coclosedCompact X) ⊔ pure ∞ rw [← nhdsWithin_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure] ** Qed
OnePoint.hasBasis_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ HasBasis (𝓝 ∞) (fun s => IsClosed s ∧ IsCompact s) fun s => some '' sᶜ ∪ {∞} rw [nhds_infty_eq] X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ HasBasis (map some (coclosedCompact X) ⊔ pure ∞) (fun s => IsClosed s ∧ IsCompact s) fun s => some '' sᶜ ∪ {∞} exact (hasBasis_coclosedCompact.map _).sup_pure _ ** Qed
OnePoint.comap_coe_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ comap some (𝓝 ∞) = coclosedCompact X simp [nhds_infty_eq, comap_sup, comap_map coe_injective] ** Qed
OnePoint.le_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X f : Filter (OnePoint X) ⊢ f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed s → IsCompact s → some '' sᶜ ∪ {∞} ∈ f simp only [hasBasis_nhds_infty.ge_iff, and_imp] ** Qed
OnePoint.ultrafilter_le_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X f : Ultrafilter (OnePoint X) ⊢ ↑f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed s → IsCompact s → ¬some '' s ∈ f simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] ** Qed
OnePoint.tendsto_nhds_infty' X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X α : Type u_2 f : OnePoint X → α l : Filter α ⊢ Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l simp [nhds_infty_eq, and_comm] ** Qed
OnePoint.tendsto_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X α : Type u_2 f : OnePoint X → α l : Filter α ⊢ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l ↔ ∀ (s : Set α), s ∈ l → f ∞ ∈ s ∧ ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s simp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and, and_assoc, exists_prop] ** Qed
OnePoint.continuousAt_infty X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X Y : Type u_2 inst✝ : TopologicalSpace Y f : OnePoint X → Y ⊢ Tendsto (f ∘ some) (coclosedCompact X) (𝓝 (f ∞)) ↔ ∀ (s : Set Y), s ∈ 𝓝 (f ∞) → ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s simp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc] ** Qed
OnePoint.continuousAt_coe X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X Y : Type u_2 inst✝ : TopologicalSpace Y f : OnePoint X → Y x : X ⊢ ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt] X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X Y : Type u_2 inst✝ : TopologicalSpace Y f : OnePoint X → Y x : X ⊢ Tendsto (f ∘ some) (𝓝 x) (𝓝 (f ↑x)) ↔ Tendsto (f ∘ some) (𝓝 x) (𝓝 ((f ∘ some) x)) rfl ** Qed
OnePoint.denseRange_coe X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝ : NoncompactSpace X ⊢ DenseRange some rw [DenseRange, ← compl_infty] X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝ : NoncompactSpace X ⊢ Dense {∞}ᶜ exact dense_compl_singleton _ ** Qed
OnePoint.inseparable_iff X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X x y : OnePoint X ⊢ Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x', x = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y' induction x using OnePoint.rec <;> induction y using OnePoint.rec <;> simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl] ** Qed
OnePoint.not_continuous_cofiniteTopology_of_symm X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X ⊢ ¬Continuous ↑CofiniteTopology.of.symm inhabit X X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X inhabited_h : Inhabited X ⊢ ¬Continuous ↑CofiniteTopology.of.symm simp only [continuous_iff_continuousAt, ContinuousAt, not_forall] X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X inhabited_h : Inhabited X ⊢ ∃ x, ¬Tendsto (↑CofiniteTopology.of.symm) (𝓝 x) (𝓝 (↑CofiniteTopology.of.symm x)) use CofiniteTopology.of ↑(default : X) case h X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X inhabited_h : Inhabited X ⊢ ¬Tendsto (↑CofiniteTopology.of.symm) (𝓝 (↑CofiniteTopology.of ↑default)) (𝓝 (↑CofiniteTopology.of.symm (↑CofiniteTopology.of ↑default))) simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using (finite_singleton ((default : X) : OnePoint X)).infinite_compl ** Qed
Complex.subfield_eq_of_closed K : Subfield ℂ hc : IsClosed ↑K ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ suffices range (ofReal' : ℝ → ℂ) ⊆ K by rw [range_subset_iff, ← coe_algebraMap] at this have := (Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top (Subfield.toIntermediateField K this).toSubalgebra simp_rw [← SetLike.coe_set_eq] at this ⊢ convert this using 2 simp only [RingHom.coe_fieldRange, Algebra.coe_bot, coe_algebraMap] rfl K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ ↑K suffices range (ofReal' : ℝ → ℂ) ⊆ closure (Set.range ((ofReal' : ℝ → ℂ) ∘ ((↑) : ℚ → ℝ))) by refine' subset_trans this _ rw [← IsClosed.closure_eq hc] apply closure_mono rintro _ ⟨_, rfl⟩ simp only [Function.comp_apply, ofReal_rat_cast, SetLike.mem_coe, SubfieldClass.coe_rat_mem] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) nth_rw 1 [range_comp] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ closure (ofReal' '' range Rat.cast) refine' subset_trans _ (image_closure_subset_closure_image continuous_ofReal) K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ ofReal' '' closure (range Rat.cast) rw [DenseRange.closure_range Rat.denseEmbedding_coe_real.dense] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ ofReal' '' univ simp only [image_univ] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ range ofReal' rfl K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ ↑K ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ rw [range_subset_iff, ← coe_algebraMap] at this K : Subfield ℂ hc : IsClosed ↑K this : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ have := (Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top (Subfield.toIntermediateField K this).toSubalgebra K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : (Subfield.toIntermediateField K this✝).toSubalgebra = ⊥ ∨ (Subfield.toIntermediateField K this✝).toSubalgebra = ⊤ ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ simp_rw [← SetLike.coe_set_eq] at this ⊢ K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊥ ∨ ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊤ ⊢ ↑K = ↑(RingHom.fieldRange ofReal) ∨ ↑K = ↑⊤ convert this using 2 case h.e'_1.h.e'_3 K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊥ ∨ ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊤ ⊢ ↑(RingHom.fieldRange ofReal) = ↑⊥ simp only [RingHom.coe_fieldRange, Algebra.coe_bot, coe_algebraMap] case h.e'_1.h.e'_3 K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊥ ∨ ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊤ ⊢ range ↑ofReal = range ofReal' rfl K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ range ofReal' ⊆ ↑K refine' subset_trans this _ K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ closure (range (ofReal' ∘ Rat.cast)) ⊆ ↑K rw [← IsClosed.closure_eq hc] K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ closure (range (ofReal' ∘ Rat.cast)) ⊆ closure ↑K apply closure_mono case h K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ range (ofReal' ∘ Rat.cast) ⊆ ↑K rintro _ ⟨_, rfl⟩ case h.intro K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) w✝ : ℚ ⊢ (ofReal' ∘ Rat.cast) w✝ ∈ ↑K simp only [Function.comp_apply, ofReal_rat_cast, SetLike.mem_coe, SubfieldClass.coe_rat_mem] ** Qed
Complex.uniformContinuous_ringHom_eq_id_or_conj ** K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) letI : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) letI : TopologicalRing K.topologicalClosure := Subring.instTopologicalRing K.topologicalClosure.toSubring K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) set ι : K → K.topologicalClosure := ⇑(Subfield.inclusion K.le_topologicalClosure) K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) have ui : UniformInducing ι := ⟨by erw [uniformity_subtype, uniformity_subtype, Filter.comap_comap] congr ⟩ K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) let di := ui.denseInducing (?_ : DenseRange ι) K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ⊢ Filter.comap (fun x => (ι x.1, ι x.2)) (uniformity { x // x ∈ Subfield.topologicalClosure K }) = uniformity { x // x ∈ K } erw [uniformity_subtype, uniformity_subtype, Filter.comap_comap] K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ⊢ Filter.comap ((fun q => (↑q.1, ↑q.2)) ∘ fun x => (ι x.1, ι x.2)) (uniformity ℂ) = Filter.comap (fun q => (↑q.1, ↑q.2)) (uniformity ℂ) congr case refine_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) let extψ := DenseInducing.extendRingHom ui di.dense hc case refine_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) haveI hψ := (uniformContinuous_uniformly_extend ui di.dense hc).continuous case refine_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) cases' Complex.subfield_eq_of_closed (Subfield.isClosed_topologicalClosure K) with h h case refine_2.inl K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) left case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal ⊢ ψ.toFun = ↑(Subfield.subtype K) let j := RingEquiv.subfieldCongr h case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ⊢ ψ.toFun = ↑(Subfield.subtype K) let ψ₁ := RingHom.comp extψ (RingHom.comp j.symm.toRingHom ofReal.rangeRestrict) case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) ⊢ ψ.toFun = ↑(Subfield.subtype K) have hψ₁ : Continuous ψ₁ := by simpa only [RingHom.coe_comp] using hψ.comp ((continuous_algebraMap ℝ ℂ).subtype_mk _) case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ ⊢ ψ.toFun = ↑(Subfield.subtype K) ext1 x case refine_2.inl.h.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ofReal.rangeRestrict r = j (ι x) K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } ⊢ ∃ r, ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) obtain ⟨r, hr⟩ := SetLike.coe_mem (j (ι x)) case intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑ofReal r = ↑(↑j (ι x)) ⊢ ∃ r, ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) exact ⟨r, Subtype.ext hr⟩ K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) ⊢ Continuous ↑ψ₁ simpa only [RingHom.coe_comp] using hψ.comp ((continuous_algebraMap ℝ ℂ).subtype_mk _) case refine_2.inl.h.h.intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x have := RingHom.congr_fun (ringHom_eq_ofReal_of_continuous hψ₁) r case refine_2.inl.h.h.intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑ψ₁ r = ↑ofReal r ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x erw [RingHom.comp_apply, RingHom.comp_apply, hr, RingEquiv.toRingHom_eq_coe] at this case refine_2.inl.h.h.intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x convert this using 1 case h.e'_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ OneHom.toFun (↑↑ψ) x = ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) exact (DenseInducing.extend_eq di hc.continuous _).symm case h.e'_3 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ ↑(Subfield.subtype K) x = ↑ofReal r rw [← ofReal.coe_rangeRestrict, hr] case h.e'_3 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ ↑(Subfield.subtype K) x = ↑(↑j (ι x)) rfl case refine_2.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) let ψ₁ := RingHom.comp extψ (RingHom.comp (RingEquiv.subfieldCongr h).symm.toRingHom (@Subfield.topEquiv ℂ _).symm.toRingHom) case refine_2.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) have hψ₁ : Continuous ψ₁ := by simpa only [RingHom.coe_comp] using hψ.comp (continuous_id.subtype_mk _) case refine_2.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) cases' ringHom_eq_id_or_conj_of_continuous hψ₁ with h h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) ⊢ Continuous ↑ψ₁ simpa only [RingHom.coe_comp] using hψ.comp (continuous_id.subtype_mk _) case refine_2.inr.inl K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) left case refine_2.inr.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ ⊢ ψ.toFun = ↑(Subfield.subtype K) ext1 z case refine_2.inr.inl.h.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = ↑(Subfield.subtype K) z convert RingHom.congr_fun h z using 1 case h.e'_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = ↑ψ₁ ↑z exact (DenseInducing.extend_eq di hc.continuous z).symm case refine_2.inr.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) right case refine_2.inr.inr.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ ⊢ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) ext1 z case refine_2.inr.inr.h.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = (↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K)) z convert RingHom.congr_fun h z using 1 case h.e'_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = ↑ψ₁ ↑z exact (DenseInducing.extend_eq di hc.continuous z).symm case refine_1 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι ⊢ DenseRange ι let j : { x // x ∈ closure (id '' { x \| (K : Set ℂ) x }) } → (K.topologicalClosure : Set ℂ) := fun x => ⟨x, by convert x.prop simp only [id.def, Set.image_id'] rfl ⟩ case refine_1 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } ⊢ DenseRange ι convert DenseRange.comp (Function.Surjective.denseRange _) (DenseEmbedding.subtype denseEmbedding_id (K : Set ℂ)).dense (by continuity : Continuous j) case refine_1 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } ⊢ Function.Surjective j rintro ⟨y, hy⟩ case refine_1.mk K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ ∃ a, j a = { val := y, property := hy } use ⟨y, by convert hy simp only [id.def, Set.image_id'] rfl ⟩ K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι x : { x // x ∈ closure (id '' {x \| ↑K x}) } ⊢ ↑x ∈ ↑(Subfield.topologicalClosure K) convert x.prop case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι x : { x // x ∈ closure (id '' {x \| ↑K x}) } ⊢ ↑(Subfield.topologicalClosure K) = closure (id '' {x \| ↑K x}) simp only [id.def, Set.image_id'] case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι x : { x // x ∈ closure (id '' {x \| ↑K x}) } ⊢ ↑(Subfield.topologicalClosure K) = closure {x \| ↑K x} rfl K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } ⊢ Continuous j continuity K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ y ∈ closure (id '' {x \| ↑K x}) convert hy case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ closure (id '' {x \| ↑K x}) = ↑(Subfield.topologicalClosure K) simp only [id.def, Set.image_id'] case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ closure {x \| ↑K x} = ↑(Subfield.topologicalClosure K) rfl Qed
continuous_right_toIcoMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 ⊢ ContinuousWithinAt (toIcoMod hp a) (Ici x) x intro s h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 h : s ∈ 𝓝 (toIcoMod hp a x) ⊢ s ∈ Filter.map (toIcoMod hp a) (𝓝[Ici x] x) rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 h : s ∈ 𝓝 (toIcoMod hp a x) ⊢ ∃ u, u ∈ 𝓝 x ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s haveI : Nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩ 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 h : s ∈ 𝓝 (toIcoMod hp a x) this : Nontrivial 𝕜 ⊢ ∃ u, u ∈ 𝓝 x ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s simp_rw [mem_nhds_iff_exists_Ioo_subset] at h ⊢ 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 h : ∃ l u, toIcoMod hp a x ∈ Ioo l u ∧ Ioo l u ⊆ s ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s obtain ⟨l, u, hxI, hIs⟩ := h case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s let d := toIcoDiv hp a x • p case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s have hd := toIcoMod_mem_Ico hp a x case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s simp_rw [subset_def, mem_inter_iff] case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ ∀ (x : 𝕜), x ∈ Ioo l u_1 → x ∈ u) ∧ ∀ (x_1 : 𝕜), x_1 ∈ u ∧ x_1 ∈ Ici x → x_1 ∈ toIcoMod hp a ⁻¹' s refine' ⟨_, ⟨l + d, min (a + p) u + d, _, fun x => id⟩, fun y => _⟩ <;> simp_rw [← sub_mem_Ioo_iff_left, mem_Ioo, lt_min_iff] case intro.intro.intro.refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) ⊢ l < x - toIcoDiv hp a x • p ∧ x - toIcoDiv hp a x • p < a + p ∧ x - toIcoDiv hp a x • p < u exact ⟨hxI.1, hd.2, hxI.2⟩ case intro.intro.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 ⊢ (l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u) ∧ y ∈ Ici x → y ∈ toIcoMod hp a ⁻¹' s rintro ⟨h, h'⟩ case intro.intro.intro.refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ y ∈ toIcoMod hp a ⁻¹' s apply hIs case intro.intro.intro.refine'_2.intro.a 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ toIcoMod hp a y ∈ Ioo l u rw [← toIcoMod_sub_zsmul, (toIcoMod_eq_self _).2] case intro.intro.intro.refine'_2.intro.a 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ y - ?intro.intro.intro.refine'_2.intro.a.m • p ∈ Ioo l u case intro.intro.intro.refine'_2.intro.a 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ y - ?intro.intro.intro.refine'_2.intro.a.m • p ∈ Ico a (a + p) case intro.intro.intro.refine'_2.intro.a.m 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ ℤ case intro.intro.intro.refine'_2.intro.a.m 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ ℤ exacts [⟨h.1, h.2.2⟩, ⟨hd.1.trans (sub_le_sub_right h' _), h.2.1⟩] ** Qed
continuous_left_toIocMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 ⊢ ContinuousWithinAt (toIocMod hp a) (Iic x) x rw [(funext fun y => Eq.trans (by rw [neg_neg]) <\| toIocMod_neg _ _ _ : toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 ⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 this : ContinuousNeg 𝕜 ⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x exact (continuous_sub_left _).continuousAt.comp_continuousWithinAt <\| (continuous_right_toIcoMod _ _ _).comp continuous_neg.continuousWithinAt fun y => neg_le_neg 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x y : 𝕜 ⊢ toIocMod hp a y = toIocMod hp a (- -y) rw [neg_neg] ** Qed
toIcoMod_eventuallyEq_toIocMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 hx : ↑x ≠ ↑a ⊢ IsOpen {x \| (fun x => toIcoMod hp a x = toIocMod hp a x) x} rw [Ico_eq_locus_Ioc_eq_iUnion_Ioo] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 hx : ↑x ≠ ↑a ⊢ IsOpen (⋃ z, Ioo (a + z • p) (a + p + z • p)) exact isOpen_iUnion fun i => isOpen_Ioo ** Qed
AddCircle.coe_eq_zero_iff 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p x : 𝕜 ⊢ ↑x = 0 ↔ ∃ n, n • p = x simp [AddSubgroup.mem_zmultiples_iff] ** Qed
AddCircle.coe_eq_zero_of_pos_iff 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p x : 𝕜 hx : 0 < x ⊢ ↑x = 0 ↔ ∃ n, n • p = x rw [coe_eq_zero_iff] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p x : 𝕜 hx : 0 < x ⊢ (∃ n, n • p = x) ↔ ∃ n, n • p = x constructor <;> rintro ⟨n, rfl⟩ case mp.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n • p ⊢ ∃ n_1, n_1 • p = n • p replace hx : 0 < n case mp.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n ⊢ ∃ n_1, n_1 • p = n • p exact ⟨n.toNat, by rw [← coe_nat_zsmul, Int.toNat_of_nonneg hx.le]⟩ case hx 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n • p ⊢ 0 < n contrapose! hx case hx 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : n ≤ 0 ⊢ n • p ≤ 0 simpa only [← neg_nonneg, ← zsmul_neg, zsmul_neg'] using zsmul_nonneg hp.le (neg_nonneg.2 hx) 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n ⊢ Int.toNat n • p = n • p rw [← coe_nat_zsmul, Int.toNat_of_nonneg hx.le] case mpr.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℕ hx : 0 < n • p ⊢ ∃ n_1, n_1 • p = n • p exact ⟨(n : ℤ), by simp⟩ 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℕ hx : 0 < n • p ⊢ ↑n • p = n • p simp ** Qed
AddCircle.coe_add_period 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p x : 𝕜 ⊢ ↑(x + p) = ↑x rw [coe_add, ← eq_sub_iff_add_eq', sub_self, coe_period] ** Qed
AddCircle.coe_eq_coe_iff_of_mem_Ico 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) ⊢ ↑x = ↑y ↔ x = y refine' ⟨fun h => _, by tauto⟩ 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑x = ↑y ⊢ x = y suffices (⟨x, hx⟩ : Ico a (a + p)) = ⟨y, hy⟩ by exact Subtype.mk.inj this 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑x = ↑y ⊢ { val := x, property := hx } = { val := y, property := hy } apply_fun equivIco p a at h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y ⊢ { val := x, property := hx } = { val := y, property := hy } rw [← (equivIco p a).right_inv ⟨x, hx⟩, ← (equivIco p a).right_inv ⟨y, hy⟩] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y ⊢ Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := x, property := hx }) = Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := y, property := hy }) exact h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) ⊢ x = y → ↑x = ↑y tauto 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑x = ↑y this : { val := x, property := hx } = { val := y, property := hy } ⊢ x = y exact Subtype.mk.inj this ** Qed
AddCircle.liftIco_coe_apply 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) ⊢ liftIco p a f ↑x = f x have : (equivIco p a) x = ⟨x, hx⟩ := by rw [Equiv.apply_eq_iff_eq_symm_apply] rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) this : ↑(equivIco p a) ↑x = { val := x, property := hx } ⊢ liftIco p a f ↑x = f x rw [liftIco, comp_apply, this] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) this : ↑(equivIco p a) ↑x = { val := x, property := hx } ⊢ restrict (Ico a (a + p)) f { val := x, property := hx } = f x rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) ⊢ ↑(equivIco p a) ↑x = { val := x, property := hx } rw [Equiv.apply_eq_iff_eq_symm_apply] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) ⊢ ↑x = ↑(equivIco p a).symm { val := x, property := hx } rfl ** Qed
AddCircle.liftIoc_coe_apply 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) ⊢ liftIoc p a f ↑x = f x have : (equivIoc p a) x = ⟨x, hx⟩ := by rw [Equiv.apply_eq_iff_eq_symm_apply] rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) this : ↑(equivIoc p a) ↑x = { val := x, property := hx } ⊢ liftIoc p a f ↑x = f x rw [liftIoc, comp_apply, this] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) this : ↑(equivIoc p a) ↑x = { val := x, property := hx } ⊢ restrict (Ioc a (a + p)) f { val := x, property := hx } = f x rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) ⊢ ↑(equivIoc p a) ↑x = { val := x, property := hx } rw [Equiv.apply_eq_iff_eq_symm_apply] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) ⊢ ↑x = ↑(equivIoc p a).symm { val := x, property := hx } rfl ** Qed
AddCircle.continuousAt_equivIco 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p hx : x ≠ ↑a ⊢ ContinuousAt (↑(equivIco p a)) x induction x using QuotientAddGroup.induction_on' case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ ContinuousAt ↑(equivIco p a) ↑z✝ rw [ContinuousAt, Filter.Tendsto, QuotientAddGroup.nhds_eq, Filter.map_map] case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ Filter.map (↑(equivIco p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 (↑(equivIco p a) ↑z✝) exact (continuousAt_toIcoMod hp.out a hx).codRestrict _ ** Qed
AddCircle.continuousAt_equivIoc 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p hx : x ≠ ↑a ⊢ ContinuousAt (↑(equivIoc p a)) x induction x using QuotientAddGroup.induction_on' case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ ContinuousAt ↑(equivIoc p a) ↑z✝ rw [ContinuousAt, Filter.Tendsto, QuotientAddGroup.nhds_eq, Filter.map_map] case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ Filter.map (↑(equivIoc p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 (↑(equivIoc p a) ↑z✝) exact (continuousAt_toIocMod hp.out a hx).codRestrict _ ** Qed
AddCircle.coe_image_Ico_eq 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ QuotientAddGroup.mk '' Ico a (a + p) = univ rw [image_eq_range] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ (range fun x => ↑↑x) = univ exact (equivIco p a).symm.range_eq_univ ** Qed
AddCircle.coe_image_Ioc_eq 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ QuotientAddGroup.mk '' Ioc a (a + p) = univ rw [image_eq_range] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ (range fun x => ↑↑x) = univ exact (equivIoc p a).symm.range_eq_univ ** Qed
AddCircle.addOrderOf_period_div 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < n ⊢ addOrderOf ↑(p / ↑n) = n rw [addOrderOf_eq_iff h] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < n ⊢ n • ↑(p / ↑n) = 0 ∧ ∀ (m : ℕ), m < n → 0 < m → m • ↑(p / ↑n) ≠ 0 replace h : 0 < (n : 𝕜) := Nat.cast_pos.2 h 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n ⊢ n • ↑(p / ↑n) = 0 ∧ ∀ (m : ℕ), m < n → 0 < m → m • ↑(p / ↑n) ≠ 0 refine' ⟨_, fun m hn h0 => _⟩ <;> simp only [Ne, ← coe_nsmul, nsmul_eq_mul] ** case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m ⊢ ¬↑(↑m * (p / ↑n)) = 0 rw [coe_eq_zero_of_pos_iff p hp.out (mul_pos (Nat.cast_pos.2 h0) <\| div_pos hp.out h)] case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m ⊢ ¬∃ n_1, n_1 • p = ↑m * (p / ↑n) rintro ⟨k, hk⟩ case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m k : ℕ hk : k • p = ↑m * (p / ↑n) ⊢ False rw [mul_div, eq_div_iff h.ne', nsmul_eq_mul, mul_right_comm, ← Nat.cast_mul, (mul_left_injective₀ hp.out.ne').eq_iff, Nat.cast_inj, mul_comm] at hk case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m k : ℕ hk : n * k = m ⊢ False exact (Nat.le_of_dvd h0 ⟨_, hk.symm⟩).not_lt hn case refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n ⊢ ↑(↑n * (p / ↑n)) = 0 rw [mul_div_cancel' _ h.ne', coe_period] Qed
AddCircle.gcd_mul_addOrderOf_div_eq ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ Nat.gcd m n * addOrderOf ↑(↑m / ↑n * p) = n rw [mul_comm_div, ← nsmul_eq_mul, coe_nsmul, addOrderOf_nsmul''] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ Nat.gcd m n * (addOrderOf ↑(p / ↑n) / Nat.gcd (addOrderOf ↑(p / ↑n)) m) = n rw [addOrderOf_period_div hn, Nat.gcd_comm, Nat.mul_div_cancel'] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ Nat.gcd n m ∣ n exact n.gcd_dvd_left m case h 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ IsOfFinAddOrder ↑(p / ↑n) rw [← addOrderOf_pos_iff, addOrderOf_period_div hn] case h 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ 0 < n exact hn Qed
AddCircle.addOrderOf_div_of_gcd_eq_one ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) m n : ℕ hn : 0 < n h : Nat.gcd m n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = n convert gcd_mul_addOrderOf_div_eq p m hn case h.e'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) m n : ℕ hn : 0 < n h : Nat.gcd m n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = Nat.gcd m n * addOrderOf ↑(↑m / ↑n * p) rw [h, one_mul] Qed
AddCircle.addOrderOf_div_of_gcd_eq_one' ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) m : ℤ n : ℕ hn : 0 < n h : Nat.gcd (Int.natAbs m) n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = n induction m case ofNat 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd (Int.natAbs (Int.ofNat a✝)) n = 1 ⊢ addOrderOf ↑(↑(Int.ofNat a✝) / ↑n * p) = n simp only [Int.ofNat_eq_coe, Int.cast_ofNat, Int.natAbs_ofNat] at h ⊢ case ofNat 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd a✝ n = 1 ⊢ addOrderOf ↑(↑a✝ / ↑n * p) = n exact addOrderOf_div_of_gcd_eq_one hn h case negSucc 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd (Int.natAbs (Int.negSucc a✝)) n = 1 ⊢ addOrderOf ↑(↑(Int.negSucc a✝) / ↑n * p) = n simp only [Int.cast_negSucc, neg_div, neg_mul, coe_neg, addOrderOf_neg] case negSucc 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd (Int.natAbs (Int.negSucc a✝)) n = 1 ⊢ addOrderOf ↑(↑(a✝ + 1) / ↑n * p) = n exact addOrderOf_div_of_gcd_eq_one hn h Qed
AddCircle.addOrderOf_coe_rat ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ ⊢ addOrderOf ↑(↑q * p) = q.den have : (↑(q.den : ℤ) : 𝕜) ≠ 0 := by norm_cast exact q.pos.ne.symm 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ this : ↑↑q.den ≠ 0 ⊢ addOrderOf ↑(↑q * p) = q.den rw [← @Rat.num_den q, Rat.cast_mk_of_ne_zero _ _ this, Int.cast_ofNat, Rat.num_den, addOrderOf_div_of_gcd_eq_one' q.pos q.reduced] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ ⊢ ↑↑q.den ≠ 0 norm_cast 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ ⊢ ¬q.den = 0 exact q.pos.ne.symm Qed
AddCircle.addOrderOf_eq_pos_iff ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n ⊢ addOrderOf u = n ↔ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = u refine' ⟨QuotientAddGroup.induction_on' u fun k hk => _, _⟩ case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k have h0 := addOrderOf_nsmul_eq_zero (k : AddCircle p) case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n h0 : addOrderOf ↑k • ↑k = 0 ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k rw [hk, ← coe_nsmul, coe_eq_zero_iff] at h0 case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n h0 : ∃ n_1, n_1 • p = n • k ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k obtain ⟨a, ha⟩ := h0 case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : a • p = n • k ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k have h0 : (_ : 𝕜) ≠ 0 := Nat.cast_ne_zero.2 h.ne' case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : a • p = n • k h0 : ↑n ≠ 0 ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k rw [nsmul_eq_mul, mul_comm, ← div_eq_iff h0, ← a.ediv_add_emod' n, add_smul, add_div, zsmul_eq_mul, Int.cast_mul, Int.cast_ofNat, mul_assoc, ← mul_div, mul_comm _ p, mul_div_cancel p h0] at ha case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k have han : _ = a % n := Int.toNat_of_nonneg (Int.emod_nonneg _ <\| by exact_mod_cast h.ne') case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k ** have he : (↑(↑((a % n).toNat) / ↑n * p) : AddCircle p) = k ** case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k refine' ⟨(a % n).toNat, _, _, he⟩ case refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n ⊢ (∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = u) → addOrderOf u = n rintro ⟨m, _, h₁, rfl⟩ case refine'_1.intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < n m : ℕ left✝ : m < n h₁ : Nat.gcd m n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = n exact addOrderOf_div_of_gcd_eq_one h h₁ 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 ⊢ ↑n ≠ 0 exact_mod_cast h.ne' case he 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n ⊢ ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k convert congr_arg (QuotientAddGroup.mk : 𝕜 → (AddCircle p)) ha using 1 case h.e'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n ⊢ ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑(↑(a / ↑n) * p + (a % ↑n) • p / ↑n) rw [coe_add, ← Int.cast_ofNat, han, zsmul_eq_mul, mul_div_right_comm, eq_comm, add_left_eq_self, ←zsmul_eq_mul, coe_zsmul, coe_period, smul_zero] case refine'_2.intro.refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ Int.toNat (a % ↑n) < n rw [← Int.ofNat_lt, han] case refine'_2.intro.refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ a % ↑n < ↑n exact Int.emod_lt_of_pos _ (Int.ofNat_lt.2 h) case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ Nat.gcd (Int.toNat (a % ↑n)) n = 1 have := (gcd_mul_addOrderOf_div_eq p (Int.toNat (a % ↑n)) h).trans ((congr_arg addOrderOf he).trans hk).symm case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k this : Nat.gcd (Int.toNat (a % ↑n)) n * addOrderOf ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = addOrderOf ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) ⊢ Nat.gcd (Int.toNat (a % ↑n)) n = 1 rw [he, Nat.mul_left_eq_self_iff] at this case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k this : Nat.gcd (Int.toNat (a % ↑n)) n = 1 ⊢ Nat.gcd (Int.toNat (a % ↑n)) n = 1 exact this case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k this : Nat.gcd (Int.toNat (a % ↑n)) n * addOrderOf ↑k = addOrderOf ↑k ⊢ 0 < addOrderOf ↑k rwa [hk] Qed
AddCircle.card_addOrderOf_eq_totient 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ ⊢ Nat.card { u // addOrderOf u = n } = Nat.totient n rcases n.eq_zero_or_pos with (rfl \| hn) case inl 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) ⊢ Nat.card { u // addOrderOf u = 0 } = Nat.totient 0 simp only [Nat.totient_zero, addOrderOf_eq_zero_iff] case inl 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 rcases em (∃ u : AddCircle p, ¬IsOfFinAddOrder u) with (⟨u, hu⟩ \| h) case inl.inl.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 have : Infinite { u : AddCircle p // ¬IsOfFinAddOrder u } := by erw [infinite_coe_iff] exact infinite_not_isOfFinAddOrder hu case inl.inl.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u this : Infinite { u // ¬IsOfFinAddOrder u } ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 exact Nat.card_eq_zero_of_infinite 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u ⊢ Infinite { u // ¬IsOfFinAddOrder u } erw [infinite_coe_iff] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u ⊢ Set.Infinite fun u => IsOfFinAddOrder u → False exact infinite_not_isOfFinAddOrder hu case inl.inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) h : ¬∃ u, ¬IsOfFinAddOrder u ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 have : IsEmpty { u : AddCircle p // ¬IsOfFinAddOrder u } := by simpa using h case inl.inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) h : ¬∃ u, ¬IsOfFinAddOrder u this : IsEmpty { u // ¬IsOfFinAddOrder u } ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 exact Nat.card_of_isEmpty 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) h : ¬∃ u, ¬IsOfFinAddOrder u ⊢ IsEmpty { u // ¬IsOfFinAddOrder u } simpa using h case inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : n > 0 ⊢ Nat.card { u // addOrderOf u = n } = Nat.totient n rw [← coe_setOf, Nat.card_congr (setAddOrderOfEquiv p hn), n.totient_eq_card_lt_and_coprime] case inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : n > 0 ⊢ Nat.card ↑{m \| m < n ∧ Nat.gcd m n = 1} = Nat.card ↑{m \| m < n ∧ Nat.Coprime n m} simp only [Nat.gcd_comm] ** Qed
AddCircle.finite_setOf_add_order_eq 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n ⊢ Nat.card ↑{u \| addOrderOf u = n} ≠ 0 simpa only [coe_setOf, card_addOrderOf_eq_totient p] using (Nat.totient_pos hn).ne' ** Qed
AddCircle.equivIccQuot_comp_mk_eq_toIocMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 ⊢ ↑(equivIccQuot p a) ∘ Quotient.mk'' = fun x => Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } rw [equivIccQuot_comp_mk_eq_toIcoMod] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 ⊢ (fun x => Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) }) = fun x => Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } funext x case h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 ⊢ Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) } = Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } by_cases a ≡ x [PMOD p] case pos 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 h : a ≡ x [PMOD p] ⊢ Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) } = Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } simp_rw [(modEq_iff_toIcoMod_eq_left hp.out).1 h, (modEq_iff_toIocMod_eq_right hp.out).1 h] case pos 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 h : a ≡ x [PMOD p] ⊢ Quot.mk (EndpointIdent p a) { val := a, property := (_ : (fun x => x ∈ Icc a (a + p)) a) } = Quot.mk (EndpointIdent p a) { val := a + p, property := (_ : (fun x => x ∈ Icc a (a + p)) (a + p)) } exact Quot.sound EndpointIdent.mk case neg 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 h : ¬a ≡ x [PMOD p] ⊢ Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) } = Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } simp_rw [(not_modEq_iff_toIcoMod_eq_toIocMod hp.out).1 h] ** Qed
AddCircle.liftIco_eq_lift_Icc 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 f : 𝕜 → B h : f a = f (a + p) ⊢ ∀ (a_1 b : ↑(Icc a (a + p))), EndpointIdent p a a_1 b → restrict (Icc a (a + p)) f a_1 = restrict (Icc a (a + p)) f b rintro _ _ ⟨_⟩ case mk 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 f : 𝕜 → B h : f a = f (a + p) ⊢ restrict (Icc a (a + p)) f { val := a, property := (_ : a ∈ Icc a (a + p)) } = restrict (Icc a (a + p)) f { val := a + p, property := (_ : a + p ∈ Icc a (a + p)) } exact h ** Qed
AddCircle.liftIco_continuous 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f a = f (a + p) hc : ContinuousOn f (Icc a (a + p)) ⊢ Continuous (liftIco p a f) rw [liftIco_eq_lift_Icc hf] 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f a = f (a + p) hc : ContinuousOn f (Icc a (a + p)) ⊢ Continuous (Quot.lift (restrict (Icc a (a + p)) f) (_ : ∀ (a_1 b : ↑(Icc a (a + p))), EndpointIdent p a a_1 b → restrict (Icc a (a + p)) f a_1 = restrict (Icc a (a + p)) f b) ∘ ↑(equivIccQuot p a)) refine' Continuous.comp _ (homeoIccQuot p a).continuous_toFun 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f a = f (a + p) hc : ContinuousOn f (Icc a (a + p)) ⊢ Continuous (Quot.lift (restrict (Icc a (a + p)) f) (_ : ∀ (a_1 b : ↑(Icc a (a + p))), EndpointIdent p a a_1 b → restrict (Icc a (a + p)) f a_1 = restrict (Icc a (a + p)) f b)) exact continuous_coinduced_dom.mpr (continuousOn_iff_continuous_restrict.mp hc) ** Qed
AddCircle.liftIco_zero_coe_apply 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico 0 p ⊢ x ∈ Ico 0 (0 + p) rwa [zero_add] ** Qed
AddCircle.liftIco_zero_continuous 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f 0 = f p hc : ContinuousOn f (Icc 0 p) ⊢ f 0 = f (0 + p) rwa [zero_add] 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f 0 = f p hc : ContinuousOn f (Icc 0 p) ⊢ ContinuousOn f (Icc 0 (0 + p)) rwa [zero_add] ** Qed
ContinuousMap.continuous_isUnit_unit X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) ⊢ Continuous fun x => IsUnit.unit (_ : IsUnit (↑f x)) refine' continuous_induced_rng.2 (Continuous.prod_mk f.continuous (MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => _))) X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X ⊢ ContinuousAt (fun x => ↑((fun x => IsUnit.unit (_ : IsUnit (↑f x))) x)⁻¹) x have := NormedRing.inverse_continuousAt (h x).unit X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X this : ContinuousAt Ring.inverse ↑(IsUnit.unit (_ : IsUnit (↑f x))) ⊢ ContinuousAt (fun x => ↑((fun x => IsUnit.unit (_ : IsUnit (↑f x))) x)⁻¹) x simp only X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X this : ContinuousAt Ring.inverse ↑(IsUnit.unit (_ : IsUnit (↑f x))) ⊢ ContinuousAt (fun x => ↑(IsUnit.unit (_ : IsUnit (↑f x)))⁻¹) x simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢ X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X this : ContinuousAt Ring.inverse (↑f x) ⊢ ContinuousAt (fun x => Ring.inverse (↑f x)) x exact this.comp (f.continuousAt x) ** Qed
ContinuousMap.isUnit_iff_forall_isUnit X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) ⊢ ↑(↑unitsLift (unitsOfForallIsUnit h)) = f ext case h X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) a✝ : X ⊢ ↑↑(↑unitsLift (unitsOfForallIsUnit h)) a✝ = ↑f a✝ rfl ** Qed
ContinuousMap.isUnit_iff_forall_ne_zero X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedField 𝕜 inst✝ : CompleteSpace 𝕜 f : C(X, 𝕜) ⊢ IsUnit f ↔ ∀ (x : X), ↑f x ≠ 0 simp_rw [f.isUnit_iff_forall_isUnit, isUnit_iff_ne_zero] ** Qed
ContinuousMap.spectrum_eq_range X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedField 𝕜 inst✝ : CompleteSpace 𝕜 f : C(X, 𝕜) ⊢ spectrum 𝕜 f = Set.range ↑f ext x case h X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedField 𝕜 inst✝ : CompleteSpace 𝕜 f : C(X, 𝕜) x : 𝕜 ⊢ x ∈ spectrum 𝕜 f ↔ x ∈ Set.range ↑f simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, coe_sub, Pi.sub_apply, algebraMap_apply, Algebra.id.smul_eq_mul, mul_one, Classical.not_not, Set.mem_range, sub_eq_zero, @eq_comm _ x _] ** Qed
edist_div_right M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : DivInvMonoid M inst✝¹ : PseudoEMetricSpace M inst✝ : IsometricSMul Mᵐᵒᵖ M a b c : M ⊢ edist (a / c) (b / c) = edist a b simp only [div_eq_mul_inv, edist_mul_right] ** Qed
edist_inv_inv M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : PseudoEMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G ⊢ edist a⁻¹ b⁻¹ = edist a b rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_right_inv, one_mul, inv_mul_cancel_right, edist_comm] ** Qed
edist_inv M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : PseudoEMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G x y : G ⊢ edist x⁻¹ y = edist x y⁻¹ rw [← edist_inv_inv, inv_inv] ** Qed
edist_div_left M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : PseudoEMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b c : G ⊢ edist (a / b) (a / c) = edist b c rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv] ** Qed
EMetric.preimage_smul_ball M : Type u G : Type v X : Type w inst✝³ : PseudoEMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ≥0∞ ⊢ (fun x x_1 => x • x_1) c ⁻¹' ball x r = ball (c⁻¹ • x) r rw [preimage_smul, smul_ball] ** Qed
EMetric.preimage_smul_closedBall M : Type u G : Type v X : Type w inst✝³ : PseudoEMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ≥0∞ ⊢ (fun x x_1 => x • x_1) c ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r rw [preimage_smul, smul_closedBall] ** Qed
EMetric.preimage_mul_right_ball ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b * a⁻¹) r exact preimage_smul_ball (MulOpposite.op a) b r Qed
EMetric.preimage_mul_right_closedBall ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b * a⁻¹) r exact preimage_smul_closedBall (MulOpposite.op a) b r Qed
dist_div_right M : Type u G : Type v X : Type w inst✝² : DivInvMonoid M inst✝¹ : PseudoMetricSpace M inst✝ : IsometricSMul Mᵐᵒᵖ M a b c : M ⊢ dist (a / c) (b / c) = dist a b simp only [div_eq_mul_inv, dist_mul_right] ** Qed
nndist_div_right M : Type u G : Type v X : Type w inst✝² : DivInvMonoid M inst✝¹ : PseudoMetricSpace M inst✝ : IsometricSMul Mᵐᵒᵖ M a b c : M ⊢ nndist (a / c) (b / c) = nndist a b simp only [div_eq_mul_inv, nndist_mul_right] ** Qed
dist_div_left M : Type u G : Type v X : Type w inst✝³ : Group G inst✝² : PseudoMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b c : G ⊢ dist (a / b) (a / c) = dist b c simp [div_eq_mul_inv] ** Qed
nndist_div_left M : Type u G : Type v X : Type w inst✝³ : Group G inst✝² : PseudoMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b c : G ⊢ nndist (a / b) (a / c) = nndist b c simp [div_eq_mul_inv] ** Qed
Metric.preimage_smul_ball M : Type u G : Type v X : Type w inst✝³ : PseudoMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ ⊢ (fun x x_1 => x • x_1) c ⁻¹' ball x r = ball (c⁻¹ • x) r rw [preimage_smul, smul_ball] ** Qed
Metric.preimage_smul_closedBall M : Type u G : Type v X : Type w inst✝³ : PseudoMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ ⊢ (fun x x_1 => x • x_1) c ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r rw [preimage_smul, smul_closedBall] ** Qed
Metric.preimage_smul_sphere M : Type u G : Type v X : Type w inst✝³ : PseudoMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ ⊢ (fun x x_1 => x • x_1) c ⁻¹' sphere x r = sphere (c⁻¹ • x) r rw [preimage_smul, smul_sphere] ** Qed
Metric.preimage_mul_right_ball ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b * a⁻¹) r exact preimage_smul_ball (MulOpposite.op a) b r Qed
Metric.preimage_mul_right_closedBall ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b * a⁻¹) r exact preimage_smul_closedBall (MulOpposite.op a) b r Qed
ContinuousMap.sup'_coe α : Type u_1 β : Type u_2 γ : Type u_3 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : LinearOrder γ inst✝ : OrderClosedTopology γ ι : Type u_4 s : Finset ι H : Finset.Nonempty s f : ι → C(β, γ) ⊢ ↑(Finset.sup' s H f) = Finset.sup' s H fun a => ↑(f a) ext case h α : Type u_1 β : Type u_2 γ : Type u_3 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : LinearOrder γ inst✝ : OrderClosedTopology γ ι : Type u_4 s : Finset ι H : Finset.Nonempty s f : ι → C(β, γ) x✝ : β ⊢ ↑(Finset.sup' s H f) x✝ = Finset.sup' s H (fun a => ↑(f a)) x✝ simp [sup'_apply] ** Qed
exists_subset_iUnion_ball_radius_lt α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with ⟨v, hsv, hvc, hcv⟩ case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i have := fun i => exists_lt_subset_ball (hvc i) (hcv i) case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) this : ∀ (i : ι), ∃ r', r' < r i ∧ v i ⊆ ball (c i) r' ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i choose r' hlt hsub using this case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) r' : ι → ℝ hlt : ∀ (i : ι), r' i < r i hsub : ∀ (i : ι), v i ⊆ ball (c i) (r' i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i exact ⟨r', hsv.trans <\| iUnion_mono <\| hsub, hlt⟩ ** Qed
exists_subset_iUnion_ball_radius_pos_lt α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with ⟨v, hsv, hvc, hcv⟩ case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) have := fun i => exists_pos_lt_subset_ball (hr i) (hvc i) (hcv i) case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) this : ∀ (i : ι), ∃ r', r' ∈ Ioo 0 (r i) ∧ v i ⊆ ball (c i) r' ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) choose r' hlt hsub using this case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) r' : ι → ℝ hlt : ∀ (i : ι), r' i ∈ Ioo 0 (r i) hsub : ∀ (i : ι), v i ⊆ ball (c i) (r' i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) exact ⟨r', hsv.trans <\| iUnion_mono hsub, hlt⟩ ** Qed
exists_locallyFinite_subset_iUnion_ball_radius_lt α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) have : ∀ x ∈ s, (𝓝 x).HasBasis (fun r : ℝ => 0 < r ∧ r < R x) fun r => ball x r := fun x hx => nhds_basis_uniformity (uniformity_basis_dist_lt (hR x hx)) α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x this : ∀ (x : α), x ∈ s → Filter.HasBasis (𝓝 x) (fun r => 0 < r ∧ r < R x) fun r => ball x r ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs this with ⟨ι, c, r', hr', hsub', hfin⟩ case intro.intro.intro.intro.intro α : Type u ι✝ : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c✝ : ι✝ → α x : α r : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x this : ∀ (x : α), x ∈ s → Filter.HasBasis (𝓝 x) (fun r => 0 < r ∧ r < R x) fun r => ball x r ι : Type u c : ι → α r' : ι → ℝ hr' : ∀ (a : ι), c a ∈ s ∧ 0 < r' a ∧ r' a < R (c a) hsub' : s ⊆ ⋃ a, ball (c a) (r' a) hfin : LocallyFinite fun a => ball (c a) (r' a) ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) rcases exists_subset_iUnion_ball_radius_pos_lt (fun i => (hr' i).2.1) hs (fun x _ => hfin.point_finite x) hsub' with ⟨r, hsub, hlt⟩ case intro.intro.intro.intro.intro.intro.intro α : Type u ι✝ : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c✝ : ι✝ → α x : α r✝ : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x this : ∀ (x : α), x ∈ s → Filter.HasBasis (𝓝 x) (fun r => 0 < r ∧ r < R x) fun r => ball x r ι : Type u c : ι → α r' : ι → ℝ hr' : ∀ (a : ι), c a ∈ s ∧ 0 < r' a ∧ r' a < R (c a) hsub' : s ⊆ ⋃ a, ball (c a) (r' a) hfin : LocallyFinite fun a => ball (c a) (r' a) r : ι → ℝ hsub : s ⊆ ⋃ i, ball (c i) (r i) hlt : ∀ (i : ι), r i ∈ Ioo 0 (r' i) ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) exact ⟨ι, c, r, r', fun i => ⟨(hr' i).1, (hlt i).1, (hlt i).2, (hr' i).2.2⟩, hfin, hsub⟩ ** Qed
Path.ext X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 ⊢ ∀ {γ₁ γ₂ : Path x y}, ↑γ₁ = ↑γ₂ → γ₁ = γ₂ rintro ⟨⟨x, h11⟩, h12, h13⟩ ⟨⟨x, h21⟩, h22, h23⟩ rfl case mk.mk.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 x : ↑I → X h11 : Continuous x h12 : ContinuousMap.toFun (ContinuousMap.mk x) 0 = x✝ h13 : ContinuousMap.toFun (ContinuousMap.mk x) 1 = y h21 : Continuous ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 } h22 : ContinuousMap.toFun (ContinuousMap.mk ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 }) 0 = x✝ h23 : ContinuousMap.toFun (ContinuousMap.mk ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 }) 1 = y ⊢ { toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 } = { toContinuousMap := ContinuousMap.mk ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 }, source' := h22, target' := h23 } rfl ** Qed
Path.refl_range X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a : X ⊢ range ↑(refl a) = {a} simp [Path.refl, CoeFun.coe] ** Qed
Path.symm_symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y ⊢ symm (symm γ) = γ ext t case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y t : ↑I ⊢ ↑(symm (symm γ)) t = ↑γ t show γ (σ (σ t)) = γ t case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y t : ↑I ⊢ ↑γ (σ (σ t)) = ↑γ t rw [unitInterval.symm_symm] ** Qed
Path.refl_symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a : X ⊢ symm (refl a) = refl a ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a : X x✝ : ↑I ⊢ ↑(symm (refl a)) x✝ = ↑(refl a) x✝ rfl ** Qed
Path.symm_range X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ : Path x y a b : X γ : Path a b ⊢ range ↑(symm γ) = range ↑γ ext x case h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 γ✝ : Path x✝ y a b : X γ : Path a b x : X ⊢ x ∈ range ↑(symm γ) ↔ x ∈ range ↑γ simp only [mem_range, Path.symm, FunLike.coe, unitInterval.symm, SetCoe.exists, comp_apply, Subtype.coe_mk] case h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 γ✝ : Path x✝ y a b : X γ : Path a b x : X ⊢ (∃ x_1 h, ContinuousMap.toFun γ.toContinuousMap { val := 1 - x_1, property := (_ : 1 - ↑{ val := x_1, property := (_ : x_1 ∈ I) } ∈ I) } = x) ↔ ∃ x_1 h, ContinuousMap.toFun γ.toContinuousMap { val := x_1, property := h } = x constructor <;> rintro ⟨y, hy, hxy⟩ <;> refine' ⟨1 - y, mem_iff_one_sub_mem.mp hy, _⟩ <;> convert hxy case h.e'_2.h.e'_6.h.e'_3 X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 γ✝ : Path x✝ y✝ a b : X γ : Path a b x : X y : ℝ hy : y ∈ I hxy : ContinuousMap.toFun γ.toContinuousMap { val := y, property := hy } = x ⊢ 1 - (1 - y) = y simp ** Qed
Path.extend_zero X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y ⊢ extend γ 0 = x simp ** Qed
Path.extend_one X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y ⊢ extend γ 1 = y simp ** Qed
Path.trans_apply ** X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I ⊢ (if ↑t ≤ 1 / 2 then extend γ (2 * ↑t) else extend γ' (2 * ↑t - 1)) = if h : ↑t ≤ 1 / 2 then ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } else ↑γ' { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) } split_ifs <;> rw [extend_extends] Qed
Path.trans_symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z ⊢ symm (trans γ γ') = trans (symm γ') (symm γ) ext t case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I ⊢ ↑(symm (trans γ γ')) t = ↑(trans (symm γ') (symm γ)) t simp only [trans_apply, ← one_div, symm_apply, not_le, Function.comp_apply] ** case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I ⊢ (if h : ↑(σ t) ≤ 1 / 2 then ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } else ↑γ' { val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) }) = if h : ↑t ≤ 1 / 2 then ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) else ↑γ (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) split_ifs with h h₁ h₂ <;> rw [coe_symm_eq] at h case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ↑t ≤ 1 / 2 ⊢ ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) have ht : (t : ℝ) = 1 / 2 := by linarith case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ↑t ≤ 1 / 2 ht : ↑t = 1 / 2 ⊢ ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) norm_num [ht] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ↑t ≤ 1 / 2 ⊢ ↑t = 1 / 2 linarith case neg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ¬↑t ≤ 1 / 2 ⊢ ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑γ (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) refine' congr_arg _ (Subtype.ext _) case neg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ¬↑t ≤ 1 / 2 ⊢ ↑{ val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑(σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) norm_num [sub_sub_eq_add_sub, mul_sub] case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑γ' { val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) } = ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) refine' congr_arg _ (Subtype.ext _) case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑{ val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) } = ↑(σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) norm_num [mul_sub, h] case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ 2 - 2 * ↑t - 1 = 1 - 2 * ↑t ring case neg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ ↑γ' { val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) } = ↑γ (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) exfalso case neg.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ False linarith Qed
Path.refl_trans_refl X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a : X ⊢ trans (refl a) (refl a) = refl a ext case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a : X x✝ : ↑I ⊢ ↑(trans (refl a) (refl a)) x✝ = ↑(refl a) x✝ simp only [Path.trans, ite_self, one_div, Path.refl_extend] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a : X x✝ : ↑I ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => a) ∘ Subtype.val), source' := (_ : ContinuousMap.toFun (ContinuousMap.mk ((fun t => a) ∘ Subtype.val)) 0 = a), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk ((fun t => a) ∘ Subtype.val)) 1 = a) } x✝ = ↑(refl a) x✝ rfl ** Qed
Path.trans_range X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑(trans γ₁ γ₂) = range ↑γ₁ ∪ range ↑γ₂ rw [Path.trans] ** X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } = range ↑γ₁ ∪ range ↑γ₂ apply eq_of_subset_of_subset case a X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } ⊆ range ↑γ₁ ∪ range ↑γ₂ rintro x ⟨⟨t, ht0, ht1⟩, hxt⟩ case a.intro.mk.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂ by_cases h : t ≤ 1 / 2 case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂ left case pos.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ x ∈ range ↑γ₁ ** use ⟨2 * t, ⟨by linarith, by linarith⟩⟩ ** case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ ↑γ₁ { val := 2 * t, property := (_ : 0 ≤ 2 * t ∧ 2 * t ≤ 1) } = x rw [← γ₁.extend_extends] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ extend γ₁ (2 * t) = x rwa [coe_mk_mk, Function.comp_apply, if_pos h] at hxt X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ 0 ≤ 2 * t linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ 2 * t ≤ 1 linarith case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂ right case neg.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ x ∈ range ↑γ₂ ** use ⟨2 * t - 1, ⟨by linarith, by linarith⟩⟩ ** case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ ↑γ₂ { val := 2 * t - 1, property := (_ : 0 ≤ 2 * t - 1 ∧ 2 * t - 1 ≤ 1) } = x rw [← γ₂.extend_extends] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ extend γ₂ (2 * t - 1) = x rwa [coe_mk_mk, Function.comp_apply, if_neg h] at hxt X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ 0 ≤ 2 * t - 1 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ 2 * t - 1 ≤ 1 linarith case a X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑γ₁ ∪ range ↑γ₂ ⊆ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } rintro x (⟨⟨t, ht0, ht1⟩, hxt⟩ \| ⟨⟨t, ht0, ht1⟩, hxt⟩) case a.inl.intro.mk.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } use ⟨t / 2, ⟨by linarith, by linarith⟩⟩ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t / 2, property := (_ : 0 ≤ t / 2 ∧ t / 2 ≤ 1) } = x have : t / 2 ≤ 1 / 2 := (div_le_div_right (zero_lt_two : (0 : ℝ) < 2)).mpr ht1 case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x this : t / 2 ≤ 1 / 2 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t / 2, property := (_ : 0 ≤ t / 2 ∧ t / 2 ≤ 1) } = x rw [coe_mk_mk, Function.comp_apply, if_pos this, Subtype.coe_mk] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x this : t / 2 ≤ 1 / 2 ⊢ extend γ₁ (2 * (t / 2)) = x ring_nf case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x this : t / 2 ≤ 1 / 2 ⊢ extend γ₁ t = x rwa [γ₁.extend_extends] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ 0 ≤ t / 2 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ t / 2 ≤ 1 linarith case a.inr.intro.mk.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } by_cases h : t = 0 case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } use ⟨1 / 2, ⟨by linarith, by linarith⟩⟩ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) } = x rw [coe_mk_mk, Function.comp_apply, if_pos le_rfl, Subtype.coe_mk, mul_one_div_cancel (two_ne_zero' ℝ)] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ extend γ₁ 1 = x rw [γ₁.extend_one] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ b = x rwa [← γ₂.extend_extends, h, γ₂.extend_zero] at hxt X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ 0 ≤ 1 / 2 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ 1 / 2 ≤ 1 linarith case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } use ⟨(t + 1) / 2, ⟨by linarith, by linarith⟩⟩ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x replace h : t ≠ 0 := h case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x have ht0 := lt_of_le_of_ne ht0 h.symm case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x have : ¬(t + 1) / 2 ≤ 1 / 2 := by rw [not_le] linarith case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t this : ¬(t + 1) / 2 ≤ 1 / 2 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x rw [coe_mk_mk, Function.comp_apply, Subtype.coe_mk, if_neg this] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t this : ¬(t + 1) / 2 ≤ 1 / 2 ⊢ extend γ₂ (2 * ((t + 1) / 2) - 1) = x ring_nf case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t this : ¬(t + 1) / 2 ≤ 1 / 2 ⊢ extend γ₂ t = x rwa [γ₂.extend_extends] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ 0 ≤ (t + 1) / 2 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ (t + 1) / 2 ≤ 1 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t ⊢ ¬(t + 1) / 2 ≤ 1 / 2 rw [not_le] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t ⊢ 1 / 2 < (t + 1) / 2 linarith Qed
Path.map_coe X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f ⊢ ↑(map γ h) = f ∘ ↑γ ext t case h X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f t : ↑I ⊢ ↑(map γ h) t = (f ∘ ↑γ) t rfl ** Qed
Path.map_trans X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f ⊢ map (trans γ γ') h = trans (map γ h) (map γ' h) ext t case a.h X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f t : ↑I ⊢ ↑(map (trans γ γ') h) t = ↑(trans (map γ h) (map γ' h)) t rw [trans_apply, map_coe, Function.comp_apply, trans_apply] ** case a.h X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f t : ↑I ⊢ f (if h : ↑t ≤ 1 / 2 then ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } else ↑γ' { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) = if h_1 : ↑t ≤ 1 / 2 then ↑(map γ h) { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } else ↑(map γ' h) { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) } split_ifs <;> rfl Qed
Path.map_id X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y ⊢ map γ (_ : Continuous id) = γ ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y x✝ : ↑I ⊢ ↑(map γ (_ : Continuous id)) x✝ = ↑γ x✝ rfl ** Qed
Path.map_map X : Type u_1 Y✝ : Type u_2 inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝¹ : TopologicalSpace Y Z : Type u_5 inst✝ : TopologicalSpace Z f : X → Y hf : Continuous f g : Y → Z hg : Continuous g ⊢ map (map γ hf) hg = map γ (_ : Continuous (g ∘ f)) ext case a.h X : Type u_1 Y✝ : Type u_2 inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝¹ : TopologicalSpace Y Z : Type u_5 inst✝ : TopologicalSpace Z f : X → Y hf : Continuous f g : Y → Z hg : Continuous g x✝ : ↑I ⊢ ↑(map (map γ hf) hg) x✝ = ↑(map γ (_ : Continuous (g ∘ f))) x✝ rfl ** Qed
Path.continuous_uncurry_extend_of_continuous_family X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ✝ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b : ι → X γ : (t : ι) → Path (a t) (b t) h : Continuous ↿γ ⊢ Continuous ↿fun t => extend (γ t) refine' h.comp (continuous_id.prod_map continuous_projIcc) X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ✝ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b : ι → X γ : (t : ι) → Path (a t) (b t) h : Continuous ↿γ ⊢ 0 ≤ 1 exact zero_le_one ** Qed

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formal stringlengths 41 427k	informal stringclasses 1 value
continuousAt_sign_of_ne_zero α : Type u_1 inst✝³ : Zero α inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α h : a ≠ 0 ⊢ ContinuousAt (↑SignType.sign) a rcases h.lt_or_lt with (h_neg \| h_pos) case inl α : Type u_1 inst✝³ : Zero α inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α h : a ≠ 0 h_neg : a < 0 ⊢ ContinuousAt (↑SignType.sign) a exact continuousAt_sign_of_neg h_neg case inr α : Type u_1 inst✝³ : Zero α inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α h : a ≠ 0 h_pos : 0 < a ⊢ ContinuousAt (↑SignType.sign) a exact continuousAt_sign_of_pos h_pos ** Qed
dense_irrational ⊢ Dense {x \| Irrational x} refine' Real.isTopologicalBasis_Ioo_rat.dense_iff.2 _ ⊢ ∀ (o : Set ℝ), o ∈ ⋃ a, ⋃ b, ⋃ (_ : a < b), {Ioo ↑a ↑b} → Set.Nonempty o → Set.Nonempty (o ∩ {x \| Irrational x}) simp only [gt_iff_lt, Rat.cast_lt, not_lt, ge_iff_le, Rat.cast_le, mem_iUnion, mem_singleton_iff, exists_prop, forall_exists_index, and_imp] ⊢ ∀ (o : Set ℝ) (x x_1 : ℚ), x < x_1 → o = Ioo ↑x ↑x_1 → Set.Nonempty o → Set.Nonempty (o ∩ {x \| Irrational x}) rintro _ a b hlt rfl _ a b : ℚ hlt : a < b a✝ : Set.Nonempty (Ioo ↑a ↑b) ⊢ Set.Nonempty (Ioo ↑a ↑b ∩ {x \| Irrational x}) rw [inter_comm] a b : ℚ hlt : a < b a✝ : Set.Nonempty (Ioo ↑a ↑b) ⊢ Set.Nonempty ({x \| Irrational x} ∩ Ioo ↑a ↑b) exact exists_irrational_btwn (Rat.cast_lt.2 hlt) ** Qed
Irrational.eventually_forall_le_dist_cast_div x : ℝ hx : Irrational x n : ℕ ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) := ((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.closedEmbedding_coe_real.isClosedMap).closed_range ** x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) ** have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by rintro ⟨m, rfl⟩ simp at hx ** x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with ⟨ε, ε0, hε⟩ case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ ⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n) refine' (ge_mem_nhds ε0).mono fun δ hδ m => not_lt.1 fun hlt => _ case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ δ : ℝ hδ : δ ≤ ε m : ℤ hlt : dist x (↑m / ↑n) < δ ⊢ False rw [dist_comm] at hlt case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ δ : ℝ hδ : δ ≤ ε m : ℤ hlt : dist (↑m / ↑n) x < δ ⊢ False refine' hε (ball_subset_ball hδ hlt) ⟨m, _⟩ case intro.intro x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) B : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m ε : ℝ ε0 : ε > 0 hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ δ : ℝ hδ : δ ≤ ε m : ℤ hlt : dist (↑m / ↑n) x < δ ⊢ (fun m => (↑n)⁻¹ * ↑m) m = ↑m / ↑n simp [div_eq_inv_mul] x : ℝ hx : Irrational x n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) ⊢ ¬x ∈ range fun m => (↑n)⁻¹ * ↑m rintro ⟨m, rfl⟩ case intro n : ℕ A : IsClosed (range fun m => (↑n)⁻¹ * ↑m) m : ℤ hx : Irrational ((fun m => (↑n)⁻¹ * ↑m) m) ⊢ False simp at hx Qed
Irrational.eventually_forall_le_dist_cast_rat_of_den_le x : ℝ hx : Irrational x n : ℕ ε : ℝ H : ∀ (k : ℕ), k ≤ n → ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k) r : ℚ hr : r.den ≤ n ⊢ ε ≤ dist x ↑r simpa only [Rat.cast_def] using H r.den hr r.num ** Qed
ContinuousOpenMap.cancel_left F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β →CO γ f₁ f₂ : α →CO β hg : Injective ↑g h : comp g f₁ = comp g f₂ a : α ⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a) rw [← comp_apply, h, comp_apply] ** Qed
OnePoint.compl_image_coe X : Type u_1 s : Set X ⊢ (some '' s)ᶜ = some '' sᶜ ∪ {∞} rw [coe_injective.compl_image_eq, compl_range_coe] ** Qed
OnePoint.ne_infty_iff_exists X : Type u_1 x : OnePoint X ⊢ x ≠ ∞ ↔ ∃ y, ↑y = x induction x using OnePoint.rec <;> simp ** Qed
OnePoint.not_mem_range_coe_iff X : Type u_1 x : OnePoint X ⊢ ¬x ∈ range some ↔ x = ∞ rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff] ** Qed
OnePoint.coe_preimage_infty X : Type u_1 ⊢ some ⁻¹' {∞} = ∅ ext case h X : Type u_1 x✝ : X ⊢ x✝ ∈ some ⁻¹' {∞} ↔ x✝ ∈ ∅ simp ** Qed
OnePoint.isOpen_iff_of_mem' X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s ⊢ IsOpen s ↔ IsCompact (some ⁻¹' s)ᶜ ∧ IsOpen (some ⁻¹' s) simp [isOpen_def, h] ** Qed
OnePoint.isOpen_iff_of_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s ⊢ IsOpen s ↔ IsClosed (some ⁻¹' s)ᶜ ∧ IsCompact (some ⁻¹' s)ᶜ simp only [isOpen_iff_of_mem' h, isClosed_compl_iff, and_comm] ** Qed
OnePoint.isOpen_iff_of_not_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ¬∞ ∈ s ⊢ IsOpen s ↔ IsOpen (some ⁻¹' s) simp [isOpen_def, h] ** Qed
OnePoint.isClosed_iff_of_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s ⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) have : ∞ ∉ sᶜ := fun H => H h X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ∞ ∈ s this : ¬∞ ∈ sᶜ ⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) rw [← isOpen_compl_iff, isOpen_iff_of_not_mem this, ← isOpen_compl_iff, preimage_compl] ** Qed
OnePoint.isClosed_iff_of_not_mem X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X h : ¬∞ ∈ s ⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) ∧ IsCompact (some ⁻¹' s) rw [← isOpen_compl_iff, isOpen_iff_of_mem (mem_compl h), ← preimage_compl, compl_compl] ** Qed
OnePoint.isOpen_image_coe X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ IsOpen (some '' s) ↔ IsOpen s rw [isOpen_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective] ** Qed
OnePoint.isOpen_compl_image_coe X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ IsOpen (some '' s)ᶜ ↔ IsClosed s ∧ IsCompact s rw [isOpen_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective] X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ ∞ ∈ (some '' s)ᶜ exact infty_not_mem_image_coe ** Qed
OnePoint.isClosed_image_coe X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X ⊢ IsClosed (some '' s) ↔ IsClosed s ∧ IsCompact s rw [← isOpen_compl_iff, isOpen_compl_image_coe] ** Qed
OnePoint.isClosed_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ IsClosed {∞} rw [← compl_range_coe, isClosed_compl_iff] X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ IsOpen (range some) exact isOpen_range_coe ** Qed
OnePoint.nhdsWithin_compl_infty_eq X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ 𝓝[{∞}ᶜ] ∞ = map some (coclosedCompact X) refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _ case refine'_1 X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ ∀ (i : Set (OnePoint X)), ∞ ∈ i ∧ IsOpen i → ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ i ∩ {∞}ᶜ rintro s ⟨hs, hso⟩ case refine'_1.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t : Set X s : Set (OnePoint X) hs : ∞ ∈ s hso : IsOpen s ⊢ ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ s ∩ {∞}ᶜ refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩ case refine'_1.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t : Set X s : Set (OnePoint X) hs : ∞ ∈ s hso : IsOpen s ⊢ some '' (some ⁻¹' s)ᶜᶜ ⊆ s ∩ {∞}ᶜ simp [Subset.rfl] case refine'_2 X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ ∀ (i' : Set X), IsClosed i' ∧ IsCompact i' → ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' i'ᶜ rintro s ⟨h₁, h₂⟩ case refine'_2.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X h₁ : IsClosed s h₂ : IsCompact s ⊢ ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' sᶜ refine' ⟨_, ⟨mem_compl infty_not_mem_image_coe, isOpen_compl_image_coe.2 ⟨h₁, h₂⟩⟩, _⟩ case refine'_2.intro X : Type u_1 inst✝ : TopologicalSpace X s✝ : Set (OnePoint X) t s : Set X h₁ : IsClosed s h₂ : IsCompact s ⊢ (some '' s)ᶜ ∩ {∞}ᶜ ⊆ some '' sᶜ simp [compl_image_coe, ← diff_eq, subset_preimage_image] ** Qed
OnePoint.nhds_infty_eq X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ 𝓝 ∞ = map some (coclosedCompact X) ⊔ pure ∞ rw [← nhdsWithin_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure] ** Qed
OnePoint.hasBasis_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ HasBasis (𝓝 ∞) (fun s => IsClosed s ∧ IsCompact s) fun s => some '' sᶜ ∪ {∞} rw [nhds_infty_eq] X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ HasBasis (map some (coclosedCompact X) ⊔ pure ∞) (fun s => IsClosed s ∧ IsCompact s) fun s => some '' sᶜ ∪ {∞} exact (hasBasis_coclosedCompact.map _).sup_pure _ ** Qed
OnePoint.comap_coe_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X ⊢ comap some (𝓝 ∞) = coclosedCompact X simp [nhds_infty_eq, comap_sup, comap_map coe_injective] ** Qed
OnePoint.le_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X f : Filter (OnePoint X) ⊢ f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed s → IsCompact s → some '' sᶜ ∪ {∞} ∈ f simp only [hasBasis_nhds_infty.ge_iff, and_imp] ** Qed
OnePoint.ultrafilter_le_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X f : Ultrafilter (OnePoint X) ⊢ ↑f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed s → IsCompact s → ¬some '' s ∈ f simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] ** Qed
OnePoint.tendsto_nhds_infty' X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X α : Type u_2 f : OnePoint X → α l : Filter α ⊢ Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l simp [nhds_infty_eq, and_comm] ** Qed
OnePoint.tendsto_nhds_infty X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X α : Type u_2 f : OnePoint X → α l : Filter α ⊢ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l ↔ ∀ (s : Set α), s ∈ l → f ∞ ∈ s ∧ ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s simp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and, and_assoc, exists_prop] ** Qed
OnePoint.continuousAt_infty X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X Y : Type u_2 inst✝ : TopologicalSpace Y f : OnePoint X → Y ⊢ Tendsto (f ∘ some) (coclosedCompact X) (𝓝 (f ∞)) ↔ ∀ (s : Set Y), s ∈ 𝓝 (f ∞) → ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s simp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc] ** Qed
OnePoint.continuousAt_coe X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X Y : Type u_2 inst✝ : TopologicalSpace Y f : OnePoint X → Y x : X ⊢ ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt] X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X Y : Type u_2 inst✝ : TopologicalSpace Y f : OnePoint X → Y x : X ⊢ Tendsto (f ∘ some) (𝓝 x) (𝓝 (f ↑x)) ↔ Tendsto (f ∘ some) (𝓝 x) (𝓝 ((f ∘ some) x)) rfl ** Qed
OnePoint.denseRange_coe X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝ : NoncompactSpace X ⊢ DenseRange some rw [DenseRange, ← compl_infty] X : Type u_1 inst✝¹ : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝ : NoncompactSpace X ⊢ Dense {∞}ᶜ exact dense_compl_singleton _ ** Qed
OnePoint.inseparable_iff X : Type u_1 inst✝ : TopologicalSpace X s : Set (OnePoint X) t : Set X x y : OnePoint X ⊢ Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x', x = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y' induction x using OnePoint.rec <;> induction y using OnePoint.rec <;> simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl] ** Qed
OnePoint.not_continuous_cofiniteTopology_of_symm X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X ⊢ ¬Continuous ↑CofiniteTopology.of.symm inhabit X X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X inhabited_h : Inhabited X ⊢ ¬Continuous ↑CofiniteTopology.of.symm simp only [continuous_iff_continuousAt, ContinuousAt, not_forall] X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X inhabited_h : Inhabited X ⊢ ∃ x, ¬Tendsto (↑CofiniteTopology.of.symm) (𝓝 x) (𝓝 (↑CofiniteTopology.of.symm x)) use CofiniteTopology.of ↑(default : X) case h X : Type u_1 inst✝² : TopologicalSpace X s : Set (OnePoint X) t : Set X inst✝¹ : Infinite X inst✝ : DiscreteTopology X inhabited_h : Inhabited X ⊢ ¬Tendsto (↑CofiniteTopology.of.symm) (𝓝 (↑CofiniteTopology.of ↑default)) (𝓝 (↑CofiniteTopology.of.symm (↑CofiniteTopology.of ↑default))) simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using (finite_singleton ((default : X) : OnePoint X)).infinite_compl ** Qed
Complex.subfield_eq_of_closed K : Subfield ℂ hc : IsClosed ↑K ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ suffices range (ofReal' : ℝ → ℂ) ⊆ K by rw [range_subset_iff, ← coe_algebraMap] at this have := (Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top (Subfield.toIntermediateField K this).toSubalgebra simp_rw [← SetLike.coe_set_eq] at this ⊢ convert this using 2 simp only [RingHom.coe_fieldRange, Algebra.coe_bot, coe_algebraMap] rfl K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ ↑K suffices range (ofReal' : ℝ → ℂ) ⊆ closure (Set.range ((ofReal' : ℝ → ℂ) ∘ ((↑) : ℚ → ℝ))) by refine' subset_trans this _ rw [← IsClosed.closure_eq hc] apply closure_mono rintro _ ⟨_, rfl⟩ simp only [Function.comp_apply, ofReal_rat_cast, SetLike.mem_coe, SubfieldClass.coe_rat_mem] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) nth_rw 1 [range_comp] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ closure (ofReal' '' range Rat.cast) refine' subset_trans _ (image_closure_subset_closure_image continuous_ofReal) K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ ofReal' '' closure (range Rat.cast) rw [DenseRange.closure_range Rat.denseEmbedding_coe_real.dense] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ ofReal' '' univ simp only [image_univ] K : Subfield ℂ hc : IsClosed ↑K ⊢ range ofReal' ⊆ range ofReal' rfl K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ ↑K ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ rw [range_subset_iff, ← coe_algebraMap] at this K : Subfield ℂ hc : IsClosed ↑K this : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ have := (Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top (Subfield.toIntermediateField K this).toSubalgebra K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : (Subfield.toIntermediateField K this✝).toSubalgebra = ⊥ ∨ (Subfield.toIntermediateField K this✝).toSubalgebra = ⊤ ⊢ K = RingHom.fieldRange ofReal ∨ K = ⊤ simp_rw [← SetLike.coe_set_eq] at this ⊢ K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊥ ∨ ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊤ ⊢ ↑K = ↑(RingHom.fieldRange ofReal) ∨ ↑K = ↑⊤ convert this using 2 case h.e'_1.h.e'_3 K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊥ ∨ ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊤ ⊢ ↑(RingHom.fieldRange ofReal) = ↑⊥ simp only [RingHom.coe_fieldRange, Algebra.coe_bot, coe_algebraMap] case h.e'_1.h.e'_3 K : Subfield ℂ hc : IsClosed ↑K this✝ : ∀ (y : ℝ), ↑(algebraMap ℝ ℂ) y ∈ ↑K this : ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊥ ∨ ↑(Subfield.toIntermediateField K this✝).toSubalgebra = ↑⊤ ⊢ range ↑ofReal = range ofReal' rfl K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ range ofReal' ⊆ ↑K refine' subset_trans this _ K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ closure (range (ofReal' ∘ Rat.cast)) ⊆ ↑K rw [← IsClosed.closure_eq hc] K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ closure (range (ofReal' ∘ Rat.cast)) ⊆ closure ↑K apply closure_mono case h K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) ⊢ range (ofReal' ∘ Rat.cast) ⊆ ↑K rintro _ ⟨_, rfl⟩ case h.intro K : Subfield ℂ hc : IsClosed ↑K this : range ofReal' ⊆ closure (range (ofReal' ∘ Rat.cast)) w✝ : ℚ ⊢ (ofReal' ∘ Rat.cast) w✝ ∈ ↑K simp only [Function.comp_apply, ofReal_rat_cast, SetLike.mem_coe, SubfieldClass.coe_rat_mem] ** Qed
Complex.uniformContinuous_ringHom_eq_id_or_conj ** K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) letI : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) letI : TopologicalRing K.topologicalClosure := Subring.instTopologicalRing K.topologicalClosure.toSubring K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) set ι : K → K.topologicalClosure := ⇑(Subfield.inclusion K.le_topologicalClosure) K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) have ui : UniformInducing ι := ⟨by erw [uniformity_subtype, uniformity_subtype, Filter.comap_comap] congr ⟩ K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) let di := ui.denseInducing (?_ : DenseRange ι) K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ⊢ Filter.comap (fun x => (ι x.1, ι x.2)) (uniformity { x // x ∈ Subfield.topologicalClosure K }) = uniformity { x // x ∈ K } erw [uniformity_subtype, uniformity_subtype, Filter.comap_comap] K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ⊢ Filter.comap ((fun q => (↑q.1, ↑q.2)) ∘ fun x => (ι x.1, ι x.2)) (uniformity ℂ) = Filter.comap (fun q => (↑q.1, ↑q.2)) (uniformity ℂ) congr case refine_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) let extψ := DenseInducing.extendRingHom ui di.dense hc case refine_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) haveI hψ := (uniformContinuous_uniformly_extend ui di.dense hc).continuous case refine_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) cases' Complex.subfield_eq_of_closed (Subfield.isClosed_topologicalClosure K) with h h case refine_2.inl K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) left case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal ⊢ ψ.toFun = ↑(Subfield.subtype K) let j := RingEquiv.subfieldCongr h case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ⊢ ψ.toFun = ↑(Subfield.subtype K) let ψ₁ := RingHom.comp extψ (RingHom.comp j.symm.toRingHom ofReal.rangeRestrict) case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) ⊢ ψ.toFun = ↑(Subfield.subtype K) have hψ₁ : Continuous ψ₁ := by simpa only [RingHom.coe_comp] using hψ.comp ((continuous_algebraMap ℝ ℂ).subtype_mk _) case refine_2.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ ⊢ ψ.toFun = ↑(Subfield.subtype K) ext1 x case refine_2.inl.h.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ofReal.rangeRestrict r = j (ι x) K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } ⊢ ∃ r, ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) obtain ⟨r, hr⟩ := SetLike.coe_mem (j (ι x)) case intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑ofReal r = ↑(↑j (ι x)) ⊢ ∃ r, ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) exact ⟨r, Subtype.ext hr⟩ K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) ⊢ Continuous ↑ψ₁ simpa only [RingHom.coe_comp] using hψ.comp ((continuous_algebraMap ℝ ℂ).subtype_mk _) case refine_2.inl.h.h.intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x have := RingHom.congr_fun (ringHom_eq_ofReal_of_continuous hψ₁) r case refine_2.inl.h.h.intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑ψ₁ r = ↑ofReal r ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x erw [RingHom.comp_apply, RingHom.comp_apply, hr, RingEquiv.toRingHom_eq_coe] at this case refine_2.inl.h.h.intro K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ OneHom.toFun (↑↑ψ) x = ↑(Subfield.subtype K) x convert this using 1 case h.e'_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ OneHom.toFun (↑↑ψ) x = ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) exact (DenseInducing.extend_eq di hc.continuous _).symm case h.e'_3 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ ↑(Subfield.subtype K) x = ↑ofReal r rw [← ofReal.coe_rangeRestrict, hr] case h.e'_3 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this✝ : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = RingHom.fieldRange ofReal j : { x // x ∈ Subfield.topologicalClosure K } ≃+* { x // x ∈ RingHom.fieldRange ofReal } := RingEquiv.subfieldCongr h ψ₁ : ℝ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm j)) (RingHom.rangeRestrict ofReal)) hψ₁ : Continuous ↑ψ₁ x : { x // x ∈ K } r : ℝ hr : ↑(RingHom.rangeRestrict ofReal) r = ↑j (ι x) this : ↑extψ (↑↑(RingEquiv.symm j) (↑j (ι x))) = ↑ofReal r ⊢ ↑(Subfield.subtype K) x = ↑(↑j (ι x)) rfl case refine_2.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) let ψ₁ := RingHom.comp extψ (RingHom.comp (RingEquiv.subfieldCongr h).symm.toRingHom (@Subfield.topEquiv ℂ _).symm.toRingHom) case refine_2.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) have hψ₁ : Continuous ψ₁ := by simpa only [RingHom.coe_comp] using hψ.comp (continuous_id.subtype_mk _) case refine_2.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) cases' ringHom_eq_id_or_conj_of_continuous hψ₁ with h h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) ⊢ Continuous ↑ψ₁ simpa only [RingHom.coe_comp] using hψ.comp (continuous_id.subtype_mk _) case refine_2.inr.inl K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) left case refine_2.inr.inl.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ ⊢ ψ.toFun = ↑(Subfield.subtype K) ext1 z case refine_2.inr.inl.h.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = ↑(Subfield.subtype K) z convert RingHom.congr_fun h z using 1 case h.e'_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = RingHom.id ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = ↑ψ₁ ↑z exact (DenseInducing.extend_eq di hc.continuous z).symm case refine_2.inr.inr K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ ⊢ ψ.toFun = ↑(Subfield.subtype K) ∨ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) right case refine_2.inr.inr.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ ⊢ ψ.toFun = ↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K) ext1 z case refine_2.inr.inr.h.h K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = (↑(starRingEnd ℂ) ∘ ↑(Subfield.subtype K)) z convert RingHom.congr_fun h z using 1 case h.e'_2 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι di : DenseInducing ι := UniformInducing.denseInducing ui ?refine_1 extψ : { x // x ∈ Subfield.topologicalClosure K } →+* ℂ := DenseInducing.extendRingHom ui (_ : DenseRange ι) hc hψ : Continuous (DenseInducing.extend (_ : DenseInducing ι) ↑ψ) h✝ : Subfield.topologicalClosure K = ⊤ ψ₁ : ℂ →+* ℂ := RingHom.comp extψ (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.subfieldCongr h✝))) (RingEquiv.toRingHom (RingEquiv.symm Subfield.topEquiv))) hψ₁ : Continuous ↑ψ₁ h : ψ₁ = starRingEnd ℂ z : { x // x ∈ K } ⊢ OneHom.toFun (↑↑ψ) z = ↑ψ₁ ↑z exact (DenseInducing.extend_eq di hc.continuous z).symm case refine_1 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι ⊢ DenseRange ι let j : { x // x ∈ closure (id '' { x \| (K : Set ℂ) x }) } → (K.topologicalClosure : Set ℂ) := fun x => ⟨x, by convert x.prop simp only [id.def, Set.image_id'] rfl ⟩ case refine_1 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } ⊢ DenseRange ι convert DenseRange.comp (Function.Surjective.denseRange _) (DenseEmbedding.subtype denseEmbedding_id (K : Set ℂ)).dense (by continuity : Continuous j) case refine_1 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } ⊢ Function.Surjective j rintro ⟨y, hy⟩ case refine_1.mk K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ ∃ a, j a = { val := y, property := hy } use ⟨y, by convert hy simp only [id.def, Set.image_id'] rfl ⟩ K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι x : { x // x ∈ closure (id '' {x \| ↑K x}) } ⊢ ↑x ∈ ↑(Subfield.topologicalClosure K) convert x.prop case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι x : { x // x ∈ closure (id '' {x \| ↑K x}) } ⊢ ↑(Subfield.topologicalClosure K) = closure (id '' {x \| ↑K x}) simp only [id.def, Set.image_id'] case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι x : { x // x ∈ closure (id '' {x \| ↑K x}) } ⊢ ↑(Subfield.topologicalClosure K) = closure {x \| ↑K x} rfl K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } ⊢ Continuous j continuity K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ y ∈ closure (id '' {x \| ↑K x}) convert hy case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ closure (id '' {x \| ↑K x}) = ↑(Subfield.topologicalClosure K) simp only [id.def, Set.image_id'] case h.e'_5 K : Subfield ℂ ψ : { x // x ∈ K } →+* ℂ hc : UniformContinuous ↑ψ this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk this : TopologicalRing { x // x ∈ Subfield.topologicalClosure K } := Subring.instTopologicalRing (Subfield.topologicalClosure K).toSubring ι : { x // x ∈ K } → { x // x ∈ Subfield.topologicalClosure K } := ↑(Subfield.inclusion (_ : K ≤ Subfield.topologicalClosure K)) ui : UniformInducing ι j : { x // x ∈ closure (id '' {x \| ↑K x}) } → ↑↑(Subfield.topologicalClosure K) := fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(Subfield.topologicalClosure K)) } y : ℂ hy : y ∈ ↑(Subfield.topologicalClosure K) ⊢ closure {x \| ↑K x} = ↑(Subfield.topologicalClosure K) rfl Qed
continuous_right_toIcoMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 ⊢ ContinuousWithinAt (toIcoMod hp a) (Ici x) x intro s h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 h : s ∈ 𝓝 (toIcoMod hp a x) ⊢ s ∈ Filter.map (toIcoMod hp a) (𝓝[Ici x] x) rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 h : s ∈ 𝓝 (toIcoMod hp a x) ⊢ ∃ u, u ∈ 𝓝 x ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s haveI : Nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩ 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 h : s ∈ 𝓝 (toIcoMod hp a x) this : Nontrivial 𝕜 ⊢ ∃ u, u ∈ 𝓝 x ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s simp_rw [mem_nhds_iff_exists_Ioo_subset] at h ⊢ 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 h : ∃ l u, toIcoMod hp a x ∈ Ioo l u ∧ Ioo l u ⊆ s ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s obtain ⟨l, u, hxI, hIs⟩ := h case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s let d := toIcoDiv hp a x • p case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s have hd := toIcoMod_mem_Ico hp a x case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s simp_rw [subset_def, mem_inter_iff] case intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) ⊢ ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ ∀ (x : 𝕜), x ∈ Ioo l u_1 → x ∈ u) ∧ ∀ (x_1 : 𝕜), x_1 ∈ u ∧ x_1 ∈ Ici x → x_1 ∈ toIcoMod hp a ⁻¹' s refine' ⟨_, ⟨l + d, min (a + p) u + d, _, fun x => id⟩, fun y => _⟩ <;> simp_rw [← sub_mem_Ioo_iff_left, mem_Ioo, lt_min_iff] case intro.intro.intro.refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) ⊢ l < x - toIcoDiv hp a x • p ∧ x - toIcoDiv hp a x • p < a + p ∧ x - toIcoDiv hp a x • p < u exact ⟨hxI.1, hd.2, hxI.2⟩ case intro.intro.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 ⊢ (l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u) ∧ y ∈ Ici x → y ∈ toIcoMod hp a ⁻¹' s rintro ⟨h, h'⟩ case intro.intro.intro.refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ y ∈ toIcoMod hp a ⁻¹' s apply hIs case intro.intro.intro.refine'_2.intro.a 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ toIcoMod hp a y ∈ Ioo l u rw [← toIcoMod_sub_zsmul, (toIcoMod_eq_self _).2] case intro.intro.intro.refine'_2.intro.a 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ y - ?intro.intro.intro.refine'_2.intro.a.m • p ∈ Ioo l u case intro.intro.intro.refine'_2.intro.a 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ y - ?intro.intro.intro.refine'_2.intro.a.m • p ∈ Ico a (a + p) case intro.intro.intro.refine'_2.intro.a.m 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ ℤ case intro.intro.intro.refine'_2.intro.a.m 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 s : Set 𝕜 this : Nontrivial 𝕜 l u : 𝕜 hxI : toIcoMod hp a x ∈ Ioo l u hIs : Ioo l u ⊆ s d : 𝕜 := toIcoDiv hp a x • p hd : toIcoMod hp a x ∈ Ico a (a + p) y : 𝕜 h : l < y - toIcoDiv hp a x • p ∧ y - toIcoDiv hp a x • p < a + p ∧ y - toIcoDiv hp a x • p < u h' : y ∈ Ici x ⊢ ℤ exacts [⟨h.1, h.2.2⟩, ⟨hd.1.trans (sub_le_sub_right h' _), h.2.1⟩] ** Qed
continuous_left_toIocMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 ⊢ ContinuousWithinAt (toIocMod hp a) (Iic x) x rw [(funext fun y => Eq.trans (by rw [neg_neg]) <\| toIocMod_neg _ _ _ : toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 ⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 this : ContinuousNeg 𝕜 ⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x exact (continuous_sub_left _).continuousAt.comp_continuousWithinAt <\| (continuous_right_toIcoMod _ _ _).comp continuous_neg.continuousWithinAt fun y => neg_le_neg 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x y : 𝕜 ⊢ toIocMod hp a y = toIocMod hp a (- -y) rw [neg_neg] ** Qed
toIcoMod_eventuallyEq_toIocMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 hx : ↑x ≠ ↑a ⊢ IsOpen {x \| (fun x => toIcoMod hp a x = toIocMod hp a x) x} rw [Ico_eq_locus_Ioc_eq_iUnion_Ioo] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : Archimedean 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p a x : 𝕜 hx : ↑x ≠ ↑a ⊢ IsOpen (⋃ z, Ioo (a + z • p) (a + p + z • p)) exact isOpen_iUnion fun i => isOpen_Ioo ** Qed
AddCircle.coe_eq_zero_iff 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p x : 𝕜 ⊢ ↑x = 0 ↔ ∃ n, n • p = x simp [AddSubgroup.mem_zmultiples_iff] ** Qed
AddCircle.coe_eq_zero_of_pos_iff 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p x : 𝕜 hx : 0 < x ⊢ ↑x = 0 ↔ ∃ n, n • p = x rw [coe_eq_zero_iff] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p x : 𝕜 hx : 0 < x ⊢ (∃ n, n • p = x) ↔ ∃ n, n • p = x constructor <;> rintro ⟨n, rfl⟩ case mp.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n • p ⊢ ∃ n_1, n_1 • p = n • p replace hx : 0 < n case mp.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n ⊢ ∃ n_1, n_1 • p = n • p exact ⟨n.toNat, by rw [← coe_nat_zsmul, Int.toNat_of_nonneg hx.le]⟩ case hx 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n • p ⊢ 0 < n contrapose! hx case hx 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : n ≤ 0 ⊢ n • p ≤ 0 simpa only [← neg_nonneg, ← zsmul_neg, zsmul_neg'] using zsmul_nonneg hp.le (neg_nonneg.2 hx) 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℤ hx : 0 < n ⊢ Int.toNat n • p = n • p rw [← coe_nat_zsmul, Int.toNat_of_nonneg hx.le] case mpr.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℕ hx : 0 < n • p ⊢ ∃ n_1, n_1 • p = n • p exact ⟨(n : ℤ), by simp⟩ 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p : 𝕜 hp : 0 < p n : ℕ hx : 0 < n • p ⊢ ↑n • p = n • p simp ** Qed
AddCircle.coe_add_period 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedAddCommGroup 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p x : 𝕜 ⊢ ↑(x + p) = ↑x rw [coe_add, ← eq_sub_iff_add_eq', sub_self, coe_period] ** Qed
AddCircle.coe_eq_coe_iff_of_mem_Ico 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) ⊢ ↑x = ↑y ↔ x = y refine' ⟨fun h => _, by tauto⟩ 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑x = ↑y ⊢ x = y suffices (⟨x, hx⟩ : Ico a (a + p)) = ⟨y, hy⟩ by exact Subtype.mk.inj this 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑x = ↑y ⊢ { val := x, property := hx } = { val := y, property := hy } apply_fun equivIco p a at h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y ⊢ { val := x, property := hx } = { val := y, property := hy } rw [← (equivIco p a).right_inv ⟨x, hx⟩, ← (equivIco p a).right_inv ⟨y, hy⟩] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑(equivIco p a) ↑x = ↑(equivIco p a) ↑y ⊢ Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := x, property := hx }) = Equiv.toFun (equivIco p a) (Equiv.invFun (equivIco p a) { val := y, property := hy }) exact h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) ⊢ x = y → ↑x = ↑y tauto 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x y : 𝕜 hx : x ∈ Ico a (a + p) hy : y ∈ Ico a (a + p) h : ↑x = ↑y this : { val := x, property := hx } = { val := y, property := hy } ⊢ x = y exact Subtype.mk.inj this ** Qed
AddCircle.liftIco_coe_apply 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) ⊢ liftIco p a f ↑x = f x have : (equivIco p a) x = ⟨x, hx⟩ := by rw [Equiv.apply_eq_iff_eq_symm_apply] rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) this : ↑(equivIco p a) ↑x = { val := x, property := hx } ⊢ liftIco p a f ↑x = f x rw [liftIco, comp_apply, this] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) this : ↑(equivIco p a) ↑x = { val := x, property := hx } ⊢ restrict (Ico a (a + p)) f { val := x, property := hx } = f x rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) ⊢ ↑(equivIco p a) ↑x = { val := x, property := hx } rw [Equiv.apply_eq_iff_eq_symm_apply] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico a (a + p) ⊢ ↑x = ↑(equivIco p a).symm { val := x, property := hx } rfl ** Qed
AddCircle.liftIoc_coe_apply 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) ⊢ liftIoc p a f ↑x = f x have : (equivIoc p a) x = ⟨x, hx⟩ := by rw [Equiv.apply_eq_iff_eq_symm_apply] rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) this : ↑(equivIoc p a) ↑x = { val := x, property := hx } ⊢ liftIoc p a f ↑x = f x rw [liftIoc, comp_apply, this] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) this : ↑(equivIoc p a) ↑x = { val := x, property := hx } ⊢ restrict (Ioc a (a + p)) f { val := x, property := hx } = f x rfl 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) ⊢ ↑(equivIoc p a) ↑x = { val := x, property := hx } rw [Equiv.apply_eq_iff_eq_symm_apply] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ioc a (a + p) ⊢ ↑x = ↑(equivIoc p a).symm { val := x, property := hx } rfl ** Qed
AddCircle.continuousAt_equivIco 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p hx : x ≠ ↑a ⊢ ContinuousAt (↑(equivIco p a)) x induction x using QuotientAddGroup.induction_on' case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ ContinuousAt ↑(equivIco p a) ↑z✝ rw [ContinuousAt, Filter.Tendsto, QuotientAddGroup.nhds_eq, Filter.map_map] case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ Filter.map (↑(equivIco p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 (↑(equivIco p a) ↑z✝) exact (continuousAt_toIcoMod hp.out a hx).codRestrict _ ** Qed
AddCircle.continuousAt_equivIoc 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p hx : x ≠ ↑a ⊢ ContinuousAt (↑(equivIoc p a)) x induction x using QuotientAddGroup.induction_on' case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ ContinuousAt ↑(equivIoc p a) ↑z✝ rw [ContinuousAt, Filter.Tendsto, QuotientAddGroup.nhds_eq, Filter.map_map] case H 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 x : AddCircle p z✝ : 𝕜 hx : ↑z✝ ≠ ↑a ⊢ Filter.map (↑(equivIoc p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 (↑(equivIoc p a) ↑z✝) exact (continuousAt_toIocMod hp.out a hx).codRestrict _ ** Qed
AddCircle.coe_image_Ico_eq 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ QuotientAddGroup.mk '' Ico a (a + p) = univ rw [image_eq_range] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ (range fun x => ↑↑x) = univ exact (equivIco p a).symm.range_eq_univ ** Qed
AddCircle.coe_image_Ioc_eq 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ QuotientAddGroup.mk '' Ioc a (a + p) = univ rw [image_eq_range] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p : 𝕜 hp : Fact (0 < p) a : 𝕜 inst✝ : Archimedean 𝕜 ⊢ (range fun x => ↑↑x) = univ exact (equivIoc p a).symm.range_eq_univ ** Qed
AddCircle.addOrderOf_period_div 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < n ⊢ addOrderOf ↑(p / ↑n) = n rw [addOrderOf_eq_iff h] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < n ⊢ n • ↑(p / ↑n) = 0 ∧ ∀ (m : ℕ), m < n → 0 < m → m • ↑(p / ↑n) ≠ 0 replace h : 0 < (n : 𝕜) := Nat.cast_pos.2 h 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n ⊢ n • ↑(p / ↑n) = 0 ∧ ∀ (m : ℕ), m < n → 0 < m → m • ↑(p / ↑n) ≠ 0 refine' ⟨_, fun m hn h0 => _⟩ <;> simp only [Ne, ← coe_nsmul, nsmul_eq_mul] ** case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m ⊢ ¬↑(↑m * (p / ↑n)) = 0 rw [coe_eq_zero_of_pos_iff p hp.out (mul_pos (Nat.cast_pos.2 h0) <\| div_pos hp.out h)] case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m ⊢ ¬∃ n_1, n_1 • p = ↑m * (p / ↑n) rintro ⟨k, hk⟩ case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m k : ℕ hk : k • p = ↑m * (p / ↑n) ⊢ False rw [mul_div, eq_div_iff h.ne', nsmul_eq_mul, mul_right_comm, ← Nat.cast_mul, (mul_left_injective₀ hp.out.ne').eq_iff, Nat.cast_inj, mul_comm] at hk case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n m : ℕ hn : m < n h0 : 0 < m k : ℕ hk : n * k = m ⊢ False exact (Nat.le_of_dvd h0 ⟨_, hk.symm⟩).not_lt hn case refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < ↑n ⊢ ↑(↑n * (p / ↑n)) = 0 rw [mul_div_cancel' _ h.ne', coe_period] Qed
AddCircle.gcd_mul_addOrderOf_div_eq ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ Nat.gcd m n * addOrderOf ↑(↑m / ↑n * p) = n rw [mul_comm_div, ← nsmul_eq_mul, coe_nsmul, addOrderOf_nsmul''] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ Nat.gcd m n * (addOrderOf ↑(p / ↑n) / Nat.gcd (addOrderOf ↑(p / ↑n)) m) = n rw [addOrderOf_period_div hn, Nat.gcd_comm, Nat.mul_div_cancel'] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ Nat.gcd n m ∣ n exact n.gcd_dvd_left m case h 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ IsOfFinAddOrder ↑(p / ↑n) rw [← addOrderOf_pos_iff, addOrderOf_period_div hn] case h 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n m : ℕ hn : 0 < n ⊢ 0 < n exact hn Qed
AddCircle.addOrderOf_div_of_gcd_eq_one ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) m n : ℕ hn : 0 < n h : Nat.gcd m n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = n convert gcd_mul_addOrderOf_div_eq p m hn case h.e'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) m n : ℕ hn : 0 < n h : Nat.gcd m n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = Nat.gcd m n * addOrderOf ↑(↑m / ↑n * p) rw [h, one_mul] Qed
AddCircle.addOrderOf_div_of_gcd_eq_one' ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) m : ℤ n : ℕ hn : 0 < n h : Nat.gcd (Int.natAbs m) n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = n induction m case ofNat 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd (Int.natAbs (Int.ofNat a✝)) n = 1 ⊢ addOrderOf ↑(↑(Int.ofNat a✝) / ↑n * p) = n simp only [Int.ofNat_eq_coe, Int.cast_ofNat, Int.natAbs_ofNat] at h ⊢ case ofNat 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd a✝ n = 1 ⊢ addOrderOf ↑(↑a✝ / ↑n * p) = n exact addOrderOf_div_of_gcd_eq_one hn h case negSucc 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd (Int.natAbs (Int.negSucc a✝)) n = 1 ⊢ addOrderOf ↑(↑(Int.negSucc a✝) / ↑n * p) = n simp only [Int.cast_negSucc, neg_div, neg_mul, coe_neg, addOrderOf_neg] case negSucc 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n a✝ : ℕ h : Nat.gcd (Int.natAbs (Int.negSucc a✝)) n = 1 ⊢ addOrderOf ↑(↑(a✝ + 1) / ↑n * p) = n exact addOrderOf_div_of_gcd_eq_one hn h Qed
AddCircle.addOrderOf_coe_rat ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ ⊢ addOrderOf ↑(↑q * p) = q.den have : (↑(q.den : ℤ) : 𝕜) ≠ 0 := by norm_cast exact q.pos.ne.symm 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ this : ↑↑q.den ≠ 0 ⊢ addOrderOf ↑(↑q * p) = q.den rw [← @Rat.num_den q, Rat.cast_mk_of_ne_zero _ _ this, Int.cast_ofNat, Rat.num_den, addOrderOf_div_of_gcd_eq_one' q.pos q.reduced] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ ⊢ ↑↑q.den ≠ 0 norm_cast 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q✝ : 𝕜 hp : Fact (0 < p) q : ℚ ⊢ ¬q.den = 0 exact q.pos.ne.symm Qed
AddCircle.addOrderOf_eq_pos_iff ** 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n ⊢ addOrderOf u = n ↔ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = u refine' ⟨QuotientAddGroup.induction_on' u fun k hk => _, _⟩ case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k have h0 := addOrderOf_nsmul_eq_zero (k : AddCircle p) case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n h0 : addOrderOf ↑k • ↑k = 0 ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k rw [hk, ← coe_nsmul, coe_eq_zero_iff] at h0 case refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n h0 : ∃ n_1, n_1 • p = n • k ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k obtain ⟨a, ha⟩ := h0 case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : a • p = n • k ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k have h0 : (_ : 𝕜) ≠ 0 := Nat.cast_ne_zero.2 h.ne' case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : a • p = n • k h0 : ↑n ≠ 0 ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k rw [nsmul_eq_mul, mul_comm, ← div_eq_iff h0, ← a.ediv_add_emod' n, add_smul, add_div, zsmul_eq_mul, Int.cast_mul, Int.cast_ofNat, mul_assoc, ← mul_div, mul_comm _ p, mul_div_cancel p h0] at ha case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k have han : _ = a % n := Int.toNat_of_nonneg (Int.emod_nonneg _ <\| by exact_mod_cast h.ne') case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k ** have he : (↑(↑((a % n).toNat) / ↑n * p) : AddCircle p) = k ** case refine'_2.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ ∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = ↑k refine' ⟨(a % n).toNat, _, _, he⟩ case refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n ⊢ (∃ m, m < n ∧ Nat.gcd m n = 1 ∧ ↑(↑m / ↑n * p) = u) → addOrderOf u = n rintro ⟨m, _, h₁, rfl⟩ case refine'_1.intro.intro.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ h : 0 < n m : ℕ left✝ : m < n h₁ : Nat.gcd m n = 1 ⊢ addOrderOf ↑(↑m / ↑n * p) = n exact addOrderOf_div_of_gcd_eq_one h h₁ 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 ⊢ ↑n ≠ 0 exact_mod_cast h.ne' case he 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n ⊢ ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k convert congr_arg (QuotientAddGroup.mk : 𝕜 → (AddCircle p)) ha using 1 case h.e'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n ⊢ ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑(↑(a / ↑n) * p + (a % ↑n) • p / ↑n) rw [coe_add, ← Int.cast_ofNat, han, zsmul_eq_mul, mul_div_right_comm, eq_comm, add_left_eq_self, ←zsmul_eq_mul, coe_zsmul, coe_period, smul_zero] case refine'_2.intro.refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ Int.toNat (a % ↑n) < n rw [← Int.ofNat_lt, han] case refine'_2.intro.refine'_1 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ a % ↑n < ↑n exact Int.emod_lt_of_pos _ (Int.ofNat_lt.2 h) case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k ⊢ Nat.gcd (Int.toNat (a % ↑n)) n = 1 have := (gcd_mul_addOrderOf_div_eq p (Int.toNat (a % ↑n)) h).trans ((congr_arg addOrderOf he).trans hk).symm case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k this : Nat.gcd (Int.toNat (a % ↑n)) n * addOrderOf ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = addOrderOf ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) ⊢ Nat.gcd (Int.toNat (a % ↑n)) n = 1 rw [he, Nat.mul_left_eq_self_iff] at this case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k this : Nat.gcd (Int.toNat (a % ↑n)) n = 1 ⊢ Nat.gcd (Int.toNat (a % ↑n)) n = 1 exact this case refine'_2.intro.refine'_2 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p n : ℕ h : 0 < n k : 𝕜 hk : addOrderOf ↑k = n a : ℤ ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k h0 : ↑n ≠ 0 han : ↑(Int.toNat (a % ↑n)) = a % ↑n he : ↑(↑(Int.toNat (a % ↑n)) / ↑n * p) = ↑k this : Nat.gcd (Int.toNat (a % ↑n)) n * addOrderOf ↑k = addOrderOf ↑k ⊢ 0 < addOrderOf ↑k rwa [hk] Qed
AddCircle.card_addOrderOf_eq_totient 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ ⊢ Nat.card { u // addOrderOf u = n } = Nat.totient n rcases n.eq_zero_or_pos with (rfl \| hn) case inl 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) ⊢ Nat.card { u // addOrderOf u = 0 } = Nat.totient 0 simp only [Nat.totient_zero, addOrderOf_eq_zero_iff] case inl 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 rcases em (∃ u : AddCircle p, ¬IsOfFinAddOrder u) with (⟨u, hu⟩ \| h) case inl.inl.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 have : Infinite { u : AddCircle p // ¬IsOfFinAddOrder u } := by erw [infinite_coe_iff] exact infinite_not_isOfFinAddOrder hu case inl.inl.intro 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u this : Infinite { u // ¬IsOfFinAddOrder u } ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 exact Nat.card_eq_zero_of_infinite 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u ⊢ Infinite { u // ¬IsOfFinAddOrder u } erw [infinite_coe_iff] 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) u : AddCircle p hu : ¬IsOfFinAddOrder u ⊢ Set.Infinite fun u => IsOfFinAddOrder u → False exact infinite_not_isOfFinAddOrder hu case inl.inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) h : ¬∃ u, ¬IsOfFinAddOrder u ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 have : IsEmpty { u : AddCircle p // ¬IsOfFinAddOrder u } := by simpa using h case inl.inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) h : ¬∃ u, ¬IsOfFinAddOrder u this : IsEmpty { u // ¬IsOfFinAddOrder u } ⊢ Nat.card { u // ¬IsOfFinAddOrder u } = 0 exact Nat.card_of_isEmpty 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) h : ¬∃ u, ¬IsOfFinAddOrder u ⊢ IsEmpty { u // ¬IsOfFinAddOrder u } simpa using h case inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : n > 0 ⊢ Nat.card { u // addOrderOf u = n } = Nat.totient n rw [← coe_setOf, Nat.card_congr (setAddOrderOfEquiv p hn), n.totient_eq_card_lt_and_coprime] case inr 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : n > 0 ⊢ Nat.card ↑{m \| m < n ∧ Nat.gcd m n = 1} = Nat.card ↑{m \| m < n ∧ Nat.Coprime n m} simp only [Nat.gcd_comm] ** Qed
AddCircle.finite_setOf_add_order_eq 𝕜 : Type u_1 B : Type u_2 inst✝² : LinearOrderedField 𝕜 inst✝¹ : TopologicalSpace 𝕜 inst✝ : OrderTopology 𝕜 p q : 𝕜 hp : Fact (0 < p) n : ℕ hn : 0 < n ⊢ Nat.card ↑{u \| addOrderOf u = n} ≠ 0 simpa only [coe_setOf, card_addOrderOf_eq_totient p] using (Nat.totient_pos hn).ne' ** Qed
AddCircle.equivIccQuot_comp_mk_eq_toIocMod 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 ⊢ ↑(equivIccQuot p a) ∘ Quotient.mk'' = fun x => Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } rw [equivIccQuot_comp_mk_eq_toIcoMod] 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 ⊢ (fun x => Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) }) = fun x => Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } funext x case h 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 ⊢ Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) } = Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } by_cases a ≡ x [PMOD p] case pos 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 h : a ≡ x [PMOD p] ⊢ Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) } = Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } simp_rw [(modEq_iff_toIcoMod_eq_left hp.out).1 h, (modEq_iff_toIocMod_eq_right hp.out).1 h] case pos 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 h : a ≡ x [PMOD p] ⊢ Quot.mk (EndpointIdent p a) { val := a, property := (_ : (fun x => x ∈ Icc a (a + p)) a) } = Quot.mk (EndpointIdent p a) { val := a + p, property := (_ : (fun x => x ∈ Icc a (a + p)) (a + p)) } exact Quot.sound EndpointIdent.mk case neg 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 x : 𝕜 h : ¬a ≡ x [PMOD p] ⊢ Quot.mk (EndpointIdent p a) { val := toIcoMod (_ : 0 < p) a x, property := (_ : toIcoMod (_ : 0 < p) a x ∈ Icc a (a + p)) } = Quot.mk (EndpointIdent p a) { val := toIocMod (_ : 0 < p) a x, property := (_ : toIocMod (_ : 0 < p) a x ∈ Icc a (a + p)) } simp_rw [(not_modEq_iff_toIcoMod_eq_toIocMod hp.out).1 h] ** Qed
AddCircle.liftIco_eq_lift_Icc 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 f : 𝕜 → B h : f a = f (a + p) ⊢ ∀ (a_1 b : ↑(Icc a (a + p))), EndpointIdent p a a_1 b → restrict (Icc a (a + p)) f a_1 = restrict (Icc a (a + p)) f b rintro _ _ ⟨_⟩ case mk 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 f : 𝕜 → B h : f a = f (a + p) ⊢ restrict (Icc a (a + p)) f { val := a, property := (_ : a ∈ Icc a (a + p)) } = restrict (Icc a (a + p)) f { val := a + p, property := (_ : a + p ∈ Icc a (a + p)) } exact h ** Qed
AddCircle.liftIco_continuous 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f a = f (a + p) hc : ContinuousOn f (Icc a (a + p)) ⊢ Continuous (liftIco p a f) rw [liftIco_eq_lift_Icc hf] 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f a = f (a + p) hc : ContinuousOn f (Icc a (a + p)) ⊢ Continuous (Quot.lift (restrict (Icc a (a + p)) f) (_ : ∀ (a_1 b : ↑(Icc a (a + p))), EndpointIdent p a a_1 b → restrict (Icc a (a + p)) f a_1 = restrict (Icc a (a + p)) f b) ∘ ↑(equivIccQuot p a)) refine' Continuous.comp _ (homeoIccQuot p a).continuous_toFun 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f a = f (a + p) hc : ContinuousOn f (Icc a (a + p)) ⊢ Continuous (Quot.lift (restrict (Icc a (a + p)) f) (_ : ∀ (a_1 b : ↑(Icc a (a + p))), EndpointIdent p a a_1 b → restrict (Icc a (a + p)) f a_1 = restrict (Icc a (a + p)) f b)) exact continuous_coinduced_dom.mpr (continuousOn_iff_continuous_restrict.mp hc) ** Qed
AddCircle.liftIco_zero_coe_apply 𝕜 : Type u_1 B : Type u_2 inst✝³ : LinearOrderedAddCommGroup 𝕜 inst✝² : TopologicalSpace 𝕜 inst✝¹ : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝ : Archimedean 𝕜 f : 𝕜 → B x : 𝕜 hx : x ∈ Ico 0 p ⊢ x ∈ Ico 0 (0 + p) rwa [zero_add] ** Qed
AddCircle.liftIco_zero_continuous 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f 0 = f p hc : ContinuousOn f (Icc 0 p) ⊢ f 0 = f (0 + p) rwa [zero_add] 𝕜 : Type u_1 B : Type u_2 inst✝⁴ : LinearOrderedAddCommGroup 𝕜 inst✝³ : TopologicalSpace 𝕜 inst✝² : OrderTopology 𝕜 p a : 𝕜 hp : Fact (0 < p) inst✝¹ : Archimedean 𝕜 inst✝ : TopologicalSpace B f : 𝕜 → B hf : f 0 = f p hc : ContinuousOn f (Icc 0 p) ⊢ ContinuousOn f (Icc 0 (0 + p)) rwa [zero_add] ** Qed
ContinuousMap.continuous_isUnit_unit X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) ⊢ Continuous fun x => IsUnit.unit (_ : IsUnit (↑f x)) refine' continuous_induced_rng.2 (Continuous.prod_mk f.continuous (MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => _))) X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X ⊢ ContinuousAt (fun x => ↑((fun x => IsUnit.unit (_ : IsUnit (↑f x))) x)⁻¹) x have := NormedRing.inverse_continuousAt (h x).unit X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X this : ContinuousAt Ring.inverse ↑(IsUnit.unit (_ : IsUnit (↑f x))) ⊢ ContinuousAt (fun x => ↑((fun x => IsUnit.unit (_ : IsUnit (↑f x))) x)⁻¹) x simp only X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X this : ContinuousAt Ring.inverse ↑(IsUnit.unit (_ : IsUnit (↑f x))) ⊢ ContinuousAt (fun x => ↑(IsUnit.unit (_ : IsUnit (↑f x)))⁻¹) x simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢ X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) x : X this : ContinuousAt Ring.inverse (↑f x) ⊢ ContinuousAt (fun x => Ring.inverse (↑f x)) x exact this.comp (f.continuousAt x) ** Qed
ContinuousMap.isUnit_iff_forall_isUnit X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) ⊢ ↑(↑unitsLift (unitsOfForallIsUnit h)) = f ext case h X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedRing R inst✝ : CompleteSpace R f : C(X, R) h : ∀ (x : X), IsUnit (↑f x) a✝ : X ⊢ ↑↑(↑unitsLift (unitsOfForallIsUnit h)) a✝ = ↑f a✝ rfl ** Qed
ContinuousMap.isUnit_iff_forall_ne_zero X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedField 𝕜 inst✝ : CompleteSpace 𝕜 f : C(X, 𝕜) ⊢ IsUnit f ↔ ∀ (x : X), ↑f x ≠ 0 simp_rw [f.isUnit_iff_forall_isUnit, isUnit_iff_ne_zero] ** Qed
ContinuousMap.spectrum_eq_range X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedField 𝕜 inst✝ : CompleteSpace 𝕜 f : C(X, 𝕜) ⊢ spectrum 𝕜 f = Set.range ↑f ext x case h X : Type u_1 M : Type u_2 R : Type u_3 𝕜 : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : NormedField 𝕜 inst✝ : CompleteSpace 𝕜 f : C(X, 𝕜) x : 𝕜 ⊢ x ∈ spectrum 𝕜 f ↔ x ∈ Set.range ↑f simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, coe_sub, Pi.sub_apply, algebraMap_apply, Algebra.id.smul_eq_mul, mul_one, Classical.not_not, Set.mem_range, sub_eq_zero, @eq_comm _ x _] ** Qed
edist_div_right M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : DivInvMonoid M inst✝¹ : PseudoEMetricSpace M inst✝ : IsometricSMul Mᵐᵒᵖ M a b c : M ⊢ edist (a / c) (b / c) = edist a b simp only [div_eq_mul_inv, edist_mul_right] ** Qed
edist_inv_inv M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : PseudoEMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G ⊢ edist a⁻¹ b⁻¹ = edist a b rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_right_inv, one_mul, inv_mul_cancel_right, edist_comm] ** Qed
edist_inv M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : PseudoEMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G x y : G ⊢ edist x⁻¹ y = edist x y⁻¹ rw [← edist_inv_inv, inv_inv] ** Qed
edist_div_left M : Type u G : Type v X : Type w inst✝⁶ : PseudoEMetricSpace X inst✝⁵ : Group G inst✝⁴ : MulAction G X inst✝³ : IsometricSMul G X inst✝² : PseudoEMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b c : G ⊢ edist (a / b) (a / c) = edist b c rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv] ** Qed
EMetric.preimage_smul_ball M : Type u G : Type v X : Type w inst✝³ : PseudoEMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ≥0∞ ⊢ (fun x x_1 => x • x_1) c ⁻¹' ball x r = ball (c⁻¹ • x) r rw [preimage_smul, smul_ball] ** Qed
EMetric.preimage_smul_closedBall M : Type u G : Type v X : Type w inst✝³ : PseudoEMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ≥0∞ ⊢ (fun x x_1 => x • x_1) c ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r rw [preimage_smul, smul_closedBall] ** Qed
EMetric.preimage_mul_right_ball ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b * a⁻¹) r exact preimage_smul_ball (MulOpposite.op a) b r Qed
EMetric.preimage_mul_right_closedBall ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoEMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoEMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ≥0∞ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b * a⁻¹) r exact preimage_smul_closedBall (MulOpposite.op a) b r Qed
dist_div_right M : Type u G : Type v X : Type w inst✝² : DivInvMonoid M inst✝¹ : PseudoMetricSpace M inst✝ : IsometricSMul Mᵐᵒᵖ M a b c : M ⊢ dist (a / c) (b / c) = dist a b simp only [div_eq_mul_inv, dist_mul_right] ** Qed
nndist_div_right M : Type u G : Type v X : Type w inst✝² : DivInvMonoid M inst✝¹ : PseudoMetricSpace M inst✝ : IsometricSMul Mᵐᵒᵖ M a b c : M ⊢ nndist (a / c) (b / c) = nndist a b simp only [div_eq_mul_inv, nndist_mul_right] ** Qed
dist_div_left M : Type u G : Type v X : Type w inst✝³ : Group G inst✝² : PseudoMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b c : G ⊢ dist (a / b) (a / c) = dist b c simp [div_eq_mul_inv] ** Qed
nndist_div_left M : Type u G : Type v X : Type w inst✝³ : Group G inst✝² : PseudoMetricSpace G inst✝¹ : IsometricSMul G G inst✝ : IsometricSMul Gᵐᵒᵖ G a b c : G ⊢ nndist (a / b) (a / c) = nndist b c simp [div_eq_mul_inv] ** Qed
Metric.preimage_smul_ball M : Type u G : Type v X : Type w inst✝³ : PseudoMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ ⊢ (fun x x_1 => x • x_1) c ⁻¹' ball x r = ball (c⁻¹ • x) r rw [preimage_smul, smul_ball] ** Qed
Metric.preimage_smul_closedBall M : Type u G : Type v X : Type w inst✝³ : PseudoMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ ⊢ (fun x x_1 => x • x_1) c ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r rw [preimage_smul, smul_closedBall] ** Qed
Metric.preimage_smul_sphere M : Type u G : Type v X : Type w inst✝³ : PseudoMetricSpace X inst✝² : Group G inst✝¹ : MulAction G X inst✝ : IsometricSMul G X c : G x : X r : ℝ ⊢ (fun x x_1 => x • x_1) c ⁻¹' sphere x r = sphere (c⁻¹ • x) r rw [preimage_smul, smul_sphere] ** Qed
Metric.preimage_mul_right_ball ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' ball b r = ball (b * a⁻¹) r exact preimage_smul_ball (MulOpposite.op a) b r Qed
Metric.preimage_mul_right_closedBall ** M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r rw [div_eq_mul_inv] M : Type u G : Type v X : Type w inst✝⁵ : PseudoMetricSpace X inst✝⁴ : Group G inst✝³ : MulAction G X inst✝² : IsometricSMul G X inst✝¹ : PseudoMetricSpace G inst✝ : IsometricSMul Gᵐᵒᵖ G a b : G r : ℝ ⊢ (fun x => x * a) ⁻¹' closedBall b r = closedBall (b * a⁻¹) r exact preimage_smul_closedBall (MulOpposite.op a) b r Qed
ContinuousMap.sup'_coe α : Type u_1 β : Type u_2 γ : Type u_3 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : LinearOrder γ inst✝ : OrderClosedTopology γ ι : Type u_4 s : Finset ι H : Finset.Nonempty s f : ι → C(β, γ) ⊢ ↑(Finset.sup' s H f) = Finset.sup' s H fun a => ↑(f a) ext case h α : Type u_1 β : Type u_2 γ : Type u_3 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : LinearOrder γ inst✝ : OrderClosedTopology γ ι : Type u_4 s : Finset ι H : Finset.Nonempty s f : ι → C(β, γ) x✝ : β ⊢ ↑(Finset.sup' s H f) x✝ = Finset.sup' s H (fun a => ↑(f a)) x✝ simp [sup'_apply] ** Qed
exists_subset_iUnion_ball_radius_lt α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with ⟨v, hsv, hvc, hcv⟩ case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i have := fun i => exists_lt_subset_ball (hvc i) (hcv i) case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) this : ∀ (i : ι), ∃ r', r' < r i ∧ v i ⊆ ball (c i) r' ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i choose r' hlt hsub using this case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) r' : ι → ℝ hlt : ∀ (i : ι), r' i < r i hsub : ∀ (i : ι), v i ⊆ ball (c i) (r' i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i exact ⟨r', hsv.trans <\| iUnion_mono <\| hsub, hlt⟩ ** Qed
exists_subset_iUnion_ball_radius_pos_lt α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with ⟨v, hsv, hvc, hcv⟩ case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) have := fun i => exists_pos_lt_subset_ball (hr i) (hvc i) (hcv i) case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) this : ∀ (i : ι), ∃ r', r' ∈ Ioo 0 (r i) ∧ v i ⊆ ball (c i) r' ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) choose r' hlt hsub using this case intro.intro.intro α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r✝ : ℝ s : Set α r : ι → ℝ hr : ∀ (i : ι), 0 < r i hs : IsClosed s uf : ∀ (x : α), x ∈ s → Set.Finite {i \| x ∈ ball (c i) (r i)} us : s ⊆ ⋃ i, ball (c i) (r i) v : ι → Set α hsv : s ⊆ iUnion v hvc : ∀ (i : ι), IsClosed (v i) hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i) r' : ι → ℝ hlt : ∀ (i : ι), r' i ∈ Ioo 0 (r i) hsub : ∀ (i : ι), v i ⊆ ball (c i) (r' i) ⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Ioo 0 (r i) exact ⟨r', hsv.trans <\| iUnion_mono hsub, hlt⟩ ** Qed
exists_locallyFinite_subset_iUnion_ball_radius_lt α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) have : ∀ x ∈ s, (𝓝 x).HasBasis (fun r : ℝ => 0 < r ∧ r < R x) fun r => ball x r := fun x hx => nhds_basis_uniformity (uniformity_basis_dist_lt (hR x hx)) α : Type u ι : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c : ι → α x : α r : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x this : ∀ (x : α), x ∈ s → Filter.HasBasis (𝓝 x) (fun r => 0 < r ∧ r < R x) fun r => ball x r ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs this with ⟨ι, c, r', hr', hsub', hfin⟩ case intro.intro.intro.intro.intro α : Type u ι✝ : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c✝ : ι✝ → α x : α r : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x this : ∀ (x : α), x ∈ s → Filter.HasBasis (𝓝 x) (fun r => 0 < r ∧ r < R x) fun r => ball x r ι : Type u c : ι → α r' : ι → ℝ hr' : ∀ (a : ι), c a ∈ s ∧ 0 < r' a ∧ r' a < R (c a) hsub' : s ⊆ ⋃ a, ball (c a) (r' a) hfin : LocallyFinite fun a => ball (c a) (r' a) ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) rcases exists_subset_iUnion_ball_radius_pos_lt (fun i => (hr' i).2.1) hs (fun x _ => hfin.point_finite x) hsub' with ⟨r, hsub, hlt⟩ case intro.intro.intro.intro.intro.intro.intro α : Type u ι✝ : Type v inst✝¹ : MetricSpace α inst✝ : ProperSpace α c✝ : ι✝ → α x : α r✝ : ℝ s : Set α hs : IsClosed s R : α → ℝ hR : ∀ (x : α), x ∈ s → 0 < R x this : ∀ (x : α), x ∈ s → Filter.HasBasis (𝓝 x) (fun r => 0 < r ∧ r < R x) fun r => ball x r ι : Type u c : ι → α r' : ι → ℝ hr' : ∀ (a : ι), c a ∈ s ∧ 0 < r' a ∧ r' a < R (c a) hsub' : s ⊆ ⋃ a, ball (c a) (r' a) hfin : LocallyFinite fun a => ball (c a) (r' a) r : ι → ℝ hsub : s ⊆ ⋃ i, ball (c i) (r i) hlt : ∀ (i : ι), r i ∈ Ioo 0 (r' i) ⊢ ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) exact ⟨ι, c, r, r', fun i => ⟨(hr' i).1, (hlt i).1, (hlt i).2, (hr' i).2.2⟩, hfin, hsub⟩ ** Qed
Path.ext X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 ⊢ ∀ {γ₁ γ₂ : Path x y}, ↑γ₁ = ↑γ₂ → γ₁ = γ₂ rintro ⟨⟨x, h11⟩, h12, h13⟩ ⟨⟨x, h21⟩, h22, h23⟩ rfl case mk.mk.mk.mk X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 x : ↑I → X h11 : Continuous x h12 : ContinuousMap.toFun (ContinuousMap.mk x) 0 = x✝ h13 : ContinuousMap.toFun (ContinuousMap.mk x) 1 = y h21 : Continuous ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 } h22 : ContinuousMap.toFun (ContinuousMap.mk ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 }) 0 = x✝ h23 : ContinuousMap.toFun (ContinuousMap.mk ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 }) 1 = y ⊢ { toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 } = { toContinuousMap := ContinuousMap.mk ↑{ toContinuousMap := ContinuousMap.mk x, source' := h12, target' := h13 }, source' := h22, target' := h23 } rfl ** Qed
Path.refl_range X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a : X ⊢ range ↑(refl a) = {a} simp [Path.refl, CoeFun.coe] ** Qed
Path.symm_symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y ⊢ symm (symm γ) = γ ext t case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y t : ↑I ⊢ ↑(symm (symm γ)) t = ↑γ t show γ (σ (σ t)) = γ t case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y t : ↑I ⊢ ↑γ (σ (σ t)) = ↑γ t rw [unitInterval.symm_symm] ** Qed
Path.refl_symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a : X ⊢ symm (refl a) = refl a ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y a : X x✝ : ↑I ⊢ ↑(symm (refl a)) x✝ = ↑(refl a) x✝ rfl ** Qed
Path.symm_range X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ : Path x y a b : X γ : Path a b ⊢ range ↑(symm γ) = range ↑γ ext x case h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 γ✝ : Path x✝ y a b : X γ : Path a b x : X ⊢ x ∈ range ↑(symm γ) ↔ x ∈ range ↑γ simp only [mem_range, Path.symm, FunLike.coe, unitInterval.symm, SetCoe.exists, comp_apply, Subtype.coe_mk] case h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y z : X ι : Type u_3 γ✝ : Path x✝ y a b : X γ : Path a b x : X ⊢ (∃ x_1 h, ContinuousMap.toFun γ.toContinuousMap { val := 1 - x_1, property := (_ : 1 - ↑{ val := x_1, property := (_ : x_1 ∈ I) } ∈ I) } = x) ↔ ∃ x_1 h, ContinuousMap.toFun γ.toContinuousMap { val := x_1, property := h } = x constructor <;> rintro ⟨y, hy, hxy⟩ <;> refine' ⟨1 - y, mem_iff_one_sub_mem.mp hy, _⟩ <;> convert hxy case h.e'_2.h.e'_6.h.e'_3 X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x✝ y✝ z : X ι : Type u_3 γ✝ : Path x✝ y✝ a b : X γ : Path a b x : X y : ℝ hy : y ∈ I hxy : ContinuousMap.toFun γ.toContinuousMap { val := y, property := hy } = x ⊢ 1 - (1 - y) = y simp ** Qed
Path.extend_zero X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y ⊢ extend γ 0 = x simp ** Qed
Path.extend_one X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ : Path x y ⊢ extend γ 1 = y simp ** Qed
Path.trans_apply ** X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I ⊢ (if ↑t ≤ 1 / 2 then extend γ (2 * ↑t) else extend γ' (2 * ↑t - 1)) = if h : ↑t ≤ 1 / 2 then ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } else ↑γ' { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) } split_ifs <;> rw [extend_extends] Qed
Path.trans_symm X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z ⊢ symm (trans γ γ') = trans (symm γ') (symm γ) ext t case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I ⊢ ↑(symm (trans γ γ')) t = ↑(trans (symm γ') (symm γ)) t simp only [trans_apply, ← one_div, symm_apply, not_le, Function.comp_apply] ** case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I ⊢ (if h : ↑(σ t) ≤ 1 / 2 then ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } else ↑γ' { val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) }) = if h : ↑t ≤ 1 / 2 then ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) else ↑γ (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) split_ifs with h h₁ h₂ <;> rw [coe_symm_eq] at h case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ↑t ≤ 1 / 2 ⊢ ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) have ht : (t : ℝ) = 1 / 2 := by linarith case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ↑t ≤ 1 / 2 ht : ↑t = 1 / 2 ⊢ ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) norm_num [ht] X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ↑t ≤ 1 / 2 ⊢ ↑t = 1 / 2 linarith case neg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ¬↑t ≤ 1 / 2 ⊢ ↑γ { val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑γ (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) refine' congr_arg _ (Subtype.ext _) case neg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ↑(σ t) ≤ 1 / 2 h : 1 - ↑t ≤ 1 / 2 h₁ : ¬↑t ≤ 1 / 2 ⊢ ↑{ val := 2 * ↑(σ t), property := (_ : 2 * ↑(σ t) ∈ I) } = ↑(σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) norm_num [sub_sub_eq_add_sub, mul_sub] case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑γ' { val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) } = ↑γ' (σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) refine' congr_arg _ (Subtype.ext _) case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑{ val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) } = ↑(σ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) }) norm_num [mul_sub, h] case pos X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ 2 - 2 * ↑t - 1 = 1 - 2 * ↑t ring case neg X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ ↑γ' { val := 2 * ↑(σ t) - 1, property := (_ : 2 * ↑(σ t) - 1 ∈ I) } = ↑γ (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) exfalso case neg.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z t : ↑I h✝ : ¬↑(σ t) ≤ 1 / 2 h : ¬1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ False linarith Qed
Path.refl_trans_refl X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a : X ⊢ trans (refl a) (refl a) = refl a ext case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a : X x✝ : ↑I ⊢ ↑(trans (refl a) (refl a)) x✝ = ↑(refl a) x✝ simp only [Path.trans, ite_self, one_div, Path.refl_extend] case a.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a : X x✝ : ↑I ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => a) ∘ Subtype.val), source' := (_ : ContinuousMap.toFun (ContinuousMap.mk ((fun t => a) ∘ Subtype.val)) 0 = a), target' := (_ : ContinuousMap.toFun (ContinuousMap.mk ((fun t => a) ∘ Subtype.val)) 1 = a) } x✝ = ↑(refl a) x✝ rfl ** Qed
Path.trans_range X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑(trans γ₁ γ₂) = range ↑γ₁ ∪ range ↑γ₂ rw [Path.trans] ** X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } = range ↑γ₁ ∪ range ↑γ₂ apply eq_of_subset_of_subset case a X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } ⊆ range ↑γ₁ ∪ range ↑γ₂ rintro x ⟨⟨t, ht0, ht1⟩, hxt⟩ case a.intro.mk.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂ by_cases h : t ≤ 1 / 2 case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂ left case pos.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ x ∈ range ↑γ₁ ** use ⟨2 * t, ⟨by linarith, by linarith⟩⟩ ** case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ ↑γ₁ { val := 2 * t, property := (_ : 0 ≤ 2 * t ∧ 2 * t ≤ 1) } = x rw [← γ₁.extend_extends] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ extend γ₁ (2 * t) = x rwa [coe_mk_mk, Function.comp_apply, if_pos h] at hxt X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ 0 ≤ 2 * t linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≤ 1 / 2 ⊢ 2 * t ≤ 1 linarith case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂ right case neg.h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ x ∈ range ↑γ₂ ** use ⟨2 * t - 1, ⟨by linarith, by linarith⟩⟩ ** case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ ↑γ₂ { val := 2 * t - 1, property := (_ : 0 ≤ 2 * t - 1 ∧ 2 * t - 1 ≤ 1) } = x rw [← γ₂.extend_extends] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ extend γ₂ (2 * t - 1) = x rwa [coe_mk_mk, Function.comp_apply, if_neg h] at hxt X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ 0 ≤ 2 * t - 1 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t ≤ 1 / 2 ⊢ 2 * t - 1 ≤ 1 linarith case a X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x y z : X✝ ι : Type u_3 γ : Path x y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c ⊢ range ↑γ₁ ∪ range ↑γ₂ ⊆ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } rintro x (⟨⟨t, ht0, ht1⟩, hxt⟩ \| ⟨⟨t, ht0, ht1⟩, hxt⟩) case a.inl.intro.mk.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } use ⟨t / 2, ⟨by linarith, by linarith⟩⟩ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t / 2, property := (_ : 0 ≤ t / 2 ∧ t / 2 ≤ 1) } = x have : t / 2 ≤ 1 / 2 := (div_le_div_right (zero_lt_two : (0 : ℝ) < 2)).mpr ht1 case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x this : t / 2 ≤ 1 / 2 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := t / 2, property := (_ : 0 ≤ t / 2 ∧ t / 2 ≤ 1) } = x rw [coe_mk_mk, Function.comp_apply, if_pos this, Subtype.coe_mk] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x this : t / 2 ≤ 1 / 2 ⊢ extend γ₁ (2 * (t / 2)) = x ring_nf case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x this : t / 2 ≤ 1 / 2 ⊢ extend γ₁ t = x rwa [γ₁.extend_extends] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ 0 ≤ t / 2 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₁ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ t / 2 ≤ 1 linarith case a.inr.intro.mk.intro X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } by_cases h : t = 0 case pos X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } use ⟨1 / 2, ⟨by linarith, by linarith⟩⟩ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) } = x rw [coe_mk_mk, Function.comp_apply, if_pos le_rfl, Subtype.coe_mk, mul_one_div_cancel (two_ne_zero' ℝ)] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ extend γ₁ 1 = x rw [γ₁.extend_one] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ b = x rwa [← γ₂.extend_extends, h, γ₂.extend_zero] at hxt X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ 0 ≤ 1 / 2 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t = 0 ⊢ 1 / 2 ≤ 1 linarith case neg X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ x ∈ range ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } use ⟨(t + 1) / 2, ⟨by linarith, by linarith⟩⟩ case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x replace h : t ≠ 0 := h case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x have ht0 := lt_of_le_of_ne ht0 h.symm case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x have : ¬(t + 1) / 2 ≤ 1 / 2 := by rw [not_le] linarith case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t this : ¬(t + 1) / 2 ≤ 1 / 2 ⊢ ↑{ toContinuousMap := ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val), source' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a), target' := (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } = x rw [coe_mk_mk, Function.comp_apply, Subtype.coe_mk, if_neg this] case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t this : ¬(t + 1) / 2 ≤ 1 / 2 ⊢ extend γ₂ (2 * ((t + 1) / 2) - 1) = x ring_nf case h X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t this : ¬(t + 1) / 2 ≤ 1 / 2 ⊢ extend γ₂ t = x rwa [γ₂.extend_extends] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ 0 ≤ (t + 1) / 2 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0 : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : ¬t = 0 ⊢ (t + 1) / 2 ≤ 1 linarith X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t ⊢ ¬(t + 1) / 2 ≤ 1 / 2 rw [not_le] X✝ : Type u_1 Y : Type u_2 inst✝² : TopologicalSpace X✝ inst✝¹ : TopologicalSpace Y x✝ y z : X✝ ι : Type u_3 γ : Path x✝ y X : Type u_4 inst✝ : TopologicalSpace X a b c : X γ₁ : Path a b γ₂ : Path b c x : X t : ℝ ht0✝ : 0 ≤ t ht1 : t ≤ 1 hxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x h : t ≠ 0 ht0 : 0 < t ⊢ 1 / 2 < (t + 1) / 2 linarith Qed
Path.map_coe X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f ⊢ ↑(map γ h) = f ∘ ↑γ ext t case h X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f t : ↑I ⊢ ↑(map γ h) t = (f ∘ ↑γ) t rfl ** Qed
Path.map_trans X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f ⊢ map (trans γ γ') h = trans (map γ h) (map γ' h) ext t case a.h X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f t : ↑I ⊢ ↑(map (trans γ γ') h) t = ↑(trans (map γ h) (map γ' h)) t rw [trans_apply, map_coe, Function.comp_apply, trans_apply] ** case a.h X : Type u_1 Y✝ : Type u_2 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y γ' : Path y z Y : Type u_4 inst✝ : TopologicalSpace Y f : X → Y h : Continuous f t : ↑I ⊢ f (if h : ↑t ≤ 1 / 2 then ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } else ↑γ' { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) }) = if h_1 : ↑t ≤ 1 / 2 then ↑(map γ h) { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } else ↑(map γ' h) { val := 2 * ↑t - 1, property := (_ : 2 * ↑t - 1 ∈ I) } split_ifs <;> rfl Qed
Path.map_id X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y ⊢ map γ (_ : Continuous id) = γ ext case a.h X : Type u_1 Y : Type u_2 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y x y z : X ι : Type u_3 γ✝ γ : Path x y x✝ : ↑I ⊢ ↑(map γ (_ : Continuous id)) x✝ = ↑γ x✝ rfl ** Qed
Path.map_map X : Type u_1 Y✝ : Type u_2 inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝¹ : TopologicalSpace Y Z : Type u_5 inst✝ : TopologicalSpace Z f : X → Y hf : Continuous f g : Y → Z hg : Continuous g ⊢ map (map γ hf) hg = map γ (_ : Continuous (g ∘ f)) ext case a.h X : Type u_1 Y✝ : Type u_2 inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y✝ x y z : X ι : Type u_3 γ✝ γ : Path x y Y : Type u_4 inst✝¹ : TopologicalSpace Y Z : Type u_5 inst✝ : TopologicalSpace Z f : X → Y hf : Continuous f g : Y → Z hg : Continuous g x✝ : ↑I ⊢ ↑(map (map γ hf) hg) x✝ = ↑(map γ (_ : Continuous (g ∘ f))) x✝ rfl ** Qed
Path.continuous_uncurry_extend_of_continuous_family X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ✝ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b : ι → X γ : (t : ι) → Path (a t) (b t) h : Continuous ↿γ ⊢ Continuous ↿fun t => extend (γ t) refine' h.comp (continuous_id.prod_map continuous_projIcc) X✝ : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X✝ inst✝² : TopologicalSpace Y x y z : X✝ ι✝ : Type u_3 γ✝ : Path x y X : Type u_4 ι : Type u_5 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace ι a b : ι → X γ : (t : ι) → Path (a t) (b t) h : Continuous ↿γ ⊢ 0 ≤ 1 exact zero_le_one ** Qed