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leandojo-lean4-formal-informal-strings-split

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formal stringlengths 41 427k	informal stringclasses 1 value
TopologicalSpace.Opens.map_eq X Y Z : TopCat f g : X ⟶ Y h : f = g ⊢ map f = map g subst h X Y Z : TopCat f : X ⟶ Y ⊢ map f = map f rfl ** Qed
TopologicalSpace.Opens.mapIso_hom_app X Y Z : TopCat f g : X ⟶ Y h : f = g U : Opens ↑Y ⊢ (map f).obj U = (map g).obj U rw [h] ** Qed
TopologicalSpace.Opens.mapIso_inv_app X Y Z : TopCat f g : X ⟶ Y h : f = g U : Opens ↑Y ⊢ (map g).obj U = (map f).obj U rw [h] ** Qed
TopologicalSpace.Opens.openEmbedding_obj_top X : TopCat U : Opens ↑X ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ⊤ = U ext1 case h X : TopCat U : Opens ↑X ⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ⊤) = ↑U exact Set.image_univ.trans Subtype.range_coe ** Qed
TopologicalSpace.Opens.inclusion_map_eq_top X : TopCat U : Opens ↑X ⊢ (map (inclusion U)).obj U = ⊤ ext1 case h X : TopCat U : Opens ↑X ⊢ ↑((map (inclusion U)).obj U) = ↑⊤ exact Subtype.coe_preimage_self _ ** Qed
TopologicalSpace.Opens.adjunction_counit_app_self X : TopCat U : Opens ↑X ⊢ (map (inclusion U) ⋙ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj U = (𝟭 (Opens ↑X)).obj U simp ** Qed
TopologicalSpace.Opens.inclusion_top_functor X : TopCat ⊢ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤)) = map (inclusionTopIso X).inv refine' CategoryTheory.Functor.ext _ _ case refine'_1 X : TopCat ⊢ ∀ (X_1 : Opens ↑((toTopCat X).obj ⊤)), (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj X_1 = (map (inclusionTopIso X).inv).obj X_1 intro U case refine'_1 X : TopCat U : Opens ↑((toTopCat X).obj ⊤) ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj U = (map (inclusionTopIso X).inv).obj U ext x case refine'_1.h.h X : TopCat U : Opens ↑((toTopCat X).obj ⊤) x : ↑X ⊢ x ∈ ↑((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj U) ↔ x ∈ ↑((map (inclusionTopIso X).inv).obj U) exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩ case refine'_2 X : TopCat ⊢ ∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y), (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).map f = eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj X_1 = (map (inclusionTopIso X).inv).obj X_1) ≫ (map (inclusionTopIso X).inv).map f ≫ eqToHom (_ : (map (inclusionTopIso X).inv).obj Y = (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj Y) intros U V f case refine'_2 X : TopCat U V : Opens ↑((toTopCat X).obj ⊤) f : U ⟶ V ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).map f = eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj U = (map (inclusionTopIso X).inv).obj U) ≫ (map (inclusionTopIso X).inv).map f ≫ eqToHom (_ : (map (inclusionTopIso X).inv).obj V = (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj V) apply Subsingleton.elim ** Qed
TopologicalSpace.Opens.functor_obj_map_obj X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y ⊢ (IsOpenMap.functor hf).obj ((map f).obj U) = (IsOpenMap.functor hf).obj ⊤ ⊓ U ext case h.h X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x✝ : ↑Y ⊢ x✝ ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) ↔ x✝ ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) constructor case h.h.mp X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x✝ : ↑Y ⊢ x✝ ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) → x✝ ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) rintro ⟨x, hx, rfl⟩ case h.h.mp.intro.intro X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x : (forget TopCat).obj X hx : x ∈ ↑((map f).obj U) ⊢ ↑f x ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) exact ⟨⟨x, trivial, rfl⟩, hx⟩ case h.h.mpr X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x✝ : ↑Y ⊢ x✝ ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) → x✝ ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) rintro ⟨⟨x, -, rfl⟩, hx⟩ case h.h.mpr.intro.intro.intro X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x : (forget TopCat).obj X hx : ↑f x ∈ ↑U ⊢ ↑f x ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) exact ⟨x, hx, rfl⟩ ** Qed
TopologicalSpace.Opens.functor_map_eq_inf X : TopCat U V : Opens ↑X ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((map (inclusion U)).obj V) = V ⊓ U ext1 case h X : TopCat U V : Opens ↑X ⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((map (inclusion U)).obj V)) = ↑(V ⊓ U) refine' Set.image_preimage_eq_inter_range.trans _ case h X : TopCat U V : Opens ↑X ⊢ (V.1 ∩ Set.range fun x => ↑(inclusion U) x) = ↑(V ⊓ U) erw [set_range_forget_map_inclusion U] case h X : TopCat U V : Opens ↑X ⊢ V.1 ∩ ↑U = ↑(V ⊓ U) rfl ** Qed
TopologicalSpace.Opens.adjunction_counit_map_functor X : TopCat U : Opens ↑X V : Opens { x // x ∈ U } ⊢ (map (inclusion U) ⋙ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) = (𝟭 (Opens ↑X)).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) dsimp X : TopCat U : Opens ↑X V : Opens { x // x ∈ U } ⊢ (IsOpenMap.functor (_ : IsOpenMap Subtype.val)).obj ((map (inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap Subtype.val)).obj V)) = (IsOpenMap.functor (_ : IsOpenMap Subtype.val)).obj V rw [map_functor_eq V] X : TopCat U : Opens ↑X V : Opens { x // x ∈ U } ⊢ (IsOpenMap.adjunction (_ : IsOpenMap ↑(inclusion U))).counit.app ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) = eqToHom (_ : (map (inclusion U) ⋙ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) = (𝟭 (Opens ↑X)).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V)) apply Subsingleton.elim ** Qed
Pretrivialization.continuousOn_continuousLinearMapCoordChange ** 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₁ := (compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)).continuous 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₂ := (ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)).continuous 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₃ := continuousOn_coordChange 𝕜₁ e₁' e₁ 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₄ := continuousOn_coordChange 𝕜₂ e₂ e₂' 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) refine' ((h₁.comp_continuousOn (h₄.mono _)).clm_comp (h₂.comp_continuousOn (h₃.mono _))).congr _ case refine'_1 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₂.baseSet ∩ e₂'.baseSet mfld_set_tac case refine'_2 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₁'.baseSet ∩ e₁.baseSet mfld_set_tac case refine'_3 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ EqOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (fun x => comp ((↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) x) ((↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) x)) (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) intro b _ case refine'_3 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) b : B a✝ : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊢ continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b = (fun x => comp ((↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) x) ((↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) x)) b ext L v case refine'_3.h.h 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) b : B a✝ : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(↑(continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L) v = ↑(↑((fun x => comp ((↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) x) ((↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) x)) b) L) v dsimp [continuousLinearMapCoordChange] case refine'_3.h.h 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) b : B a✝ : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (↑L (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (Trivialization.coordChangeL 𝕜₁ e₁' e₁ b))) v)) = ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (↑L (↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b) v)) rw [ContinuousLinearEquiv.symm_symm] Qed
Pretrivialization.continuousLinearMap_symm_apply' ** 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L = comp (Trivialization.symmL 𝕜₂ e₂ b) (comp L (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b)) rw [symm_apply] 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ cast (_ : Bundle.ContinuousLinearMap σ E₁ E₂ (↑(LocalEquiv.symm (continuousLinearMap σ e₁ e₂).toLocalEquiv) (b, L)).proj = Bundle.ContinuousLinearMap σ E₁ E₂ b) (↑(LocalEquiv.symm (continuousLinearMap σ e₁ e₂).toLocalEquiv) (b, L)).snd = comp (Trivialization.symmL 𝕜₂ e₂ b) (comp L (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b)) case hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ b ∈ (continuousLinearMap σ e₁ e₂).baseSet rfl case hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ b ∈ (continuousLinearMap σ e₁ e₂).baseSet exact hb Qed
Pretrivialization.continuousLinearMapCoordChange_apply ** 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ ⊢ ↑(continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L = (↑(continuousLinearMap σ e₁' e₂') { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }).2 ext v case h 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(↑(continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L) v = ↑(↑(continuousLinearMap σ e₁' e₂') { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }).2 v simp_rw [continuousLinearMapCoordChange, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.arrowCongrSL_apply, continuousLinearMap_apply, continuousLinearMap_symm_apply' σ e₁ e₂ hb.1, comp_apply, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.symm_symm, Trivialization.continuousLinearMapAt_apply, Trivialization.symmL_apply] case h 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (↑L (↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b) v)) = ↑(Trivialization.linearMapAt 𝕜₂ e₂' { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }.proj) (Trivialization.symm e₂ b (↑L (↑(Trivialization.linearMapAt 𝕜₁ e₁ b) (Trivialization.symm e₁' b v)))) rw [e₂.coordChangeL_apply e₂', e₁'.coordChangeL_apply e₁, e₁.coe_linearMapAt_of_mem hb.1.1, e₂'.coe_linearMapAt_of_mem hb.2.2] case h.hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ b ∈ e₁'.baseSet ∩ e₁.baseSet case h.hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ b ∈ e₂.baseSet ∩ e₂'.baseSet exacts [⟨hb.2.1, hb.1.1⟩, ⟨hb.1.2, hb.2.2⟩] Qed
FiberBundle.totalSpaceMk_closedEmbedding ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : TopologicalSpace B inst✝⁴ : TopologicalSpace F E : B → Type u_5 inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (b : B) → TopologicalSpace (E b) inst✝¹ : FiberBundle F E inst✝ : T1Space B x : B ⊢ IsClosed (range (TotalSpace.mk x)) rw [TotalSpace.range_mk] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : TopologicalSpace B inst✝⁴ : TopologicalSpace F E : B → Type u_5 inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (b : B) → TopologicalSpace (E b) inst✝¹ : FiberBundle F E inst✝ : T1Space B x : B ⊢ IsClosed (TotalSpace.proj ⁻¹' {x}) exact isClosed_singleton.preimage <\| continuous_proj F E ** Qed
FiberBundle.exists_trivialization_Icc_subset ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ⊢ ∃ e, Icc a b ⊆ e.baseSet obtain ⟨ea, hea⟩ : ∃ ea : Trivialization F (π F E), a ∈ ea.baseSet := ⟨trivializationAt F E a, mem_baseSet_trivializationAt F E a⟩ case intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet cases' lt_or_le b a with hab hab case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b ⊢ ∃ e, Icc a b ⊆ e.baseSet set s : Set B := { x ∈ Icc a b \| ∃ e : Trivialization F (π F E), Icc a x ⊆ e.baseSet } case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ⊢ ∃ e, Icc a b ⊆ e.baseSet have ha : a ∈ s := ⟨left_mem_Icc.2 hab, ea, by simp [hea]⟩ case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s ⊢ ∃ e, Icc a b ⊆ e.baseSet have sne : s.Nonempty := ⟨a, ha⟩ case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s ⊢ ∃ e, Icc a b ⊆ e.baseSet have hsb : b ∈ upperBounds s := fun x hx => hx.1.2 case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s ⊢ ∃ e, Icc a b ⊆ e.baseSet have sbd : BddAbove s := ⟨b, hsb⟩ case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s ⊢ ∃ e, Icc a b ⊆ e.baseSet set c := sSup s case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s ⊢ ∃ e, Icc a b ⊆ e.baseSet have hsc : IsLUB s c := isLUB_csSup sne sbd case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c ⊢ ∃ e, Icc a b ⊆ e.baseSet have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩ case intro.inr.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet cases' hc.2.eq_or_lt with heq hlt case intro.inr.intro.intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b ⊢ ∃ e, Icc a b ⊆ e.baseSet suffices : ∃ d ∈ Ioc c b, ∃ e : Trivialization F (π F E), Icc a d ⊆ e.baseSet case this ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet obtain ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, Ico c d ⊆ ec.baseSet := (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1 (mem_nhdsWithin_of_mem_nhds <\| IsOpen.mem_nhds ec.open_baseSet (hec ⟨hc.1, le_rfl⟩)) case this.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet have had : Ico a d ⊆ ec.baseSet := Ico_subset_Icc_union_Ico.trans (union_subset hec hd) case this.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet by_cases he : Disjoint (Iio d) (Ioi c) case intro.inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : b < a ⊢ ∃ e, Icc a b ⊆ e.baseSet exact ⟨ea, by simp []⟩ * ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : b < a ⊢ Icc a b ⊆ ea.baseSet ** simp [] * ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ⊢ Icc a a ⊆ ea.baseSet simp [hea] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ⊢ c ∈ s cases' hc.1.eq_or_lt with heq hlt case inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b hlt : a < c ⊢ c ∈ s refine ⟨hc, ?_⟩ case inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b hlt : a < c ⊢ ∃ e, Icc a c ⊆ e.baseSet obtain ⟨ec, hc⟩ : ∃ ec : Trivialization F (π F E), c ∈ ec.baseSet := ⟨trivializationAt F E c, mem_baseSet_trivializationAt F E c⟩ case inr.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet ⊢ ∃ e, Icc a c ⊆ e.baseSet obtain ⟨c', hc', hc'e⟩ : ∃ c' ∈ Ico a c, Ioc c' c ⊆ ec.baseSet := (mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset hlt).1 (mem_nhdsWithin_of_mem_nhds <\| IsOpen.mem_nhds ec.open_baseSet hc) case inr.intro.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet c' : B hc' : c' ∈ Ico a c hc'e : Ioc c' c ⊆ ec.baseSet ⊢ ∃ e, Icc a c ⊆ e.baseSet obtain ⟨d, ⟨hdab, ead, had⟩, hd⟩ : ∃ d ∈ s, d ∈ Ioc c' c := hsc.exists_between hc'.2 case inr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet c' : B hc' : c' ∈ Ico a c hc'e : Ioc c' c ⊆ ec.baseSet d : B hd : d ∈ Ioc c' c hdab : d ∈ Icc a b ead : Trivialization F TotalSpace.proj had : Icc a d ⊆ ead.baseSet ⊢ ∃ e, Icc a c ⊆ e.baseSet refine' ⟨ead.piecewiseLe ec d (had ⟨hdab.1, le_rfl⟩) (hc'e hd), subset_ite.2 _⟩ case inr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet c' : B hc' : c' ∈ Ico a c hc'e : Ioc c' c ⊆ ec.baseSet d : B hd : d ∈ Ioc c' c hdab : d ∈ Icc a b ead : Trivialization F TotalSpace.proj had : Icc a d ⊆ ead.baseSet ⊢ Icc a c ∩ Iic d ⊆ ead.baseSet ∧ Icc a c \ Iic d ⊆ (Trivialization.transFiberHomeomorph ec (Trivialization.coordChangeHomeomorph ec ead (_ : d ∈ ec.baseSet) (_ : d ∈ ead.baseSet))).baseSet refine' ⟨fun x hx => had ⟨hx.1.1, hx.2⟩, fun x hx => hc'e ⟨hd.1.trans (not_le.1 hx.2), hx.1.2⟩⟩ case inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b heq : a = c ⊢ c ∈ s rwa [← heq] case intro.inr.intro.intro.inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet heq : c = b ⊢ ∃ e, Icc a b ⊆ e.baseSet exact ⟨ec, heq ▸ hec⟩ case intro.inr.intro.intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b this : ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet rcases this with ⟨d, hdcb, hd⟩ case intro.inr.intro.intro.inr.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : ∃ e, Icc a d ⊆ e.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet exact ((hsc.1 ⟨⟨hc.1.trans hdcb.1.le, hdcb.2⟩, hd⟩).not_lt hdcb.1).elim case pos ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet obtain ⟨ed, hed⟩ : ∃ ed : Trivialization F (π F E), d ∈ ed.baseSet := ⟨trivializationAt F E d, mem_baseSet_trivializationAt F E d⟩ case pos.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ed : Trivialization F TotalSpace.proj hed : d ∈ ed.baseSet ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet refine' ⟨d, hdcb, (ec.restrOpen (Iio d) isOpen_Iio).disjointUnion (ed.restrOpen (Ioi c) isOpen_Ioi) (he.mono (inter_subset_right _ _) (inter_subset_right _ _)), fun x hx => _⟩ case pos.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ed : Trivialization F TotalSpace.proj hed : d ∈ ed.baseSet x : B hx : x ∈ Icc a d ⊢ x ∈ (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))) (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))) (_ : Disjoint (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))).baseSet (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet rcases hx.2.eq_or_lt with (rfl \| hxd) case pos.intro.inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b ed : Trivialization F TotalSpace.proj x : B hdcb : x ∈ Ioc c b hd : Ico c x ⊆ ec.baseSet had : Ico a x ⊆ ec.baseSet he : Disjoint (Iio x) (Ioi c) hed : x ∈ ed.baseSet hx : x ∈ Icc a x ⊢ x ∈ (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x))) (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))) (_ : Disjoint (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x))).baseSet (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet case pos.intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ed : Trivialization F TotalSpace.proj hed : d ∈ ed.baseSet x : B hx : x ∈ Icc a d hxd : x < d ⊢ x ∈ (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))) (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))) (_ : Disjoint (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))).baseSet (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet exacts [Or.inr ⟨hed, hdcb.1⟩, Or.inl ⟨had ⟨hx.1, hxd⟩, hxd⟩] case neg ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : ¬Disjoint (Iio d) (Ioi c) ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet rw [disjoint_left] at he case neg ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : ¬∀ ⦃a : B⦄, a ∈ Iio d → ¬a ∈ Ioi c ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet push_neg at he case neg ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Exists fun ⦃a⦄ => a ∈ Iio d ∧ a ∈ Ioi c ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet rcases he with ⟨d', hdd' : d' < d, hd'c⟩ case neg.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet d' : B hdd' : d' < d hd'c : d' ∈ Ioi c ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet exact ⟨d', ⟨hd'c, hdd'.le.trans hdcb.2⟩, ec, (Icc_subset_Ico_right hdd').trans had⟩ Qed
FiberBundleCore.mem_trivChange_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i j : ι p : B × F ⊢ p ∈ (trivChange Z i j).toLocalEquiv.source ↔ p.1 ∈ baseSet Z i ∩ baseSet Z j erw [mem_prod] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i j : ι p : B × F ⊢ p.1 ∈ baseSet Z i ∩ baseSet Z j ∧ p.2 ∈ univ ↔ p.1 ∈ baseSet Z i ∩ baseSet Z j simp ** Qed
FiberBundleCore.mem_localTrivAsLocalEquiv_target ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι p : B × F ⊢ p ∈ (localTrivAsLocalEquiv Z i).target ↔ p.1 ∈ baseSet Z i erw [mem_prod] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι p : B × F ⊢ p.1 ∈ baseSet Z i ∧ p.2 ∈ univ ↔ p.1 ∈ baseSet Z i simp only [and_true_iff, mem_univ] ** Qed
FiberBundleCore.localTrivAsLocalEquiv_trans ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι ⊢ LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j) ≈ (trivChange Z i j).toLocalEquiv constructor case left ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι ⊢ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source = (trivChange Z i j).toLocalEquiv.source ext x case left.h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι x : B × F ⊢ x ∈ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source ↔ x ∈ (trivChange Z i j).toLocalEquiv.source simp only [mem_localTrivAsLocalEquiv_target, mfld_simps] case left.h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι x : B × F ⊢ x.1 ∈ baseSet Z i ∧ ↑(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) x ∈ (localTrivAsLocalEquiv Z j).source ↔ x.1 ∈ baseSet Z i ∧ x.1 ∈ baseSet Z j rfl case right ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι ⊢ EqOn (↑(LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j))) (↑(trivChange Z i j).toLocalEquiv) (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source rintro ⟨x, v⟩ hx case right.mk ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι x : B v : F hx : x ∈ baseSet Z i ∧ x ∈ baseSet Z j ⊢ coordChange Z (indexAt Z x) j x (coordChange Z i (indexAt Z x) x v) = coordChange Z i j x v simp only [Z.coordChange_comp, hx, mem_inter_iff, and_self_iff, mem_baseSet_at] ** Qed
FiberBundleCore.open_source' ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ IsOpen (localTrivAsLocalEquiv Z i).source apply TopologicalSpace.GenerateOpen.basic case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ (localTrivAsLocalEquiv Z i).source ∈ ⋃ i, ⋃ s, ⋃ (_ : IsOpen s), {(localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' s} simp only [exists_prop, mem_iUnion, mem_singleton_iff] case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ ∃ i_1 i_2, IsOpen i_2 ∧ (localTrivAsLocalEquiv Z i).source = (localTrivAsLocalEquiv Z i_1).source ∩ ↑(localTrivAsLocalEquiv Z i_1) ⁻¹' i_2 refine ⟨i, Z.baseSet i ×ˢ univ, (Z.isOpen_baseSet i).prod isOpen_univ, ?_⟩ case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ (localTrivAsLocalEquiv Z i).source = (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ ext p case a.h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι p : TotalSpace Z ⊢ p ∈ (localTrivAsLocalEquiv Z i).source ↔ p ∈ (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ simp only [localTrivAsLocalEquiv_apply, prod_mk_mem_set_prod_eq, mem_inter_iff, and_self_iff, mem_localTrivAsLocalEquiv_source, and_true, mem_univ, mem_preimage] ** Qed
FiberBundleCore.continuous_const_section ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v ⊢ Continuous (let_fun this := fun x => { proj := x, snd := v }; this) refine continuous_iff_continuousAt.2 fun x => ?_ ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B ⊢ ContinuousAt (let_fun this := fun x => { proj := x, snd := v }; this) x have A : Z.baseSet (Z.indexAt x) ∈ 𝓝 x := IsOpen.mem_nhds (Z.isOpen_baseSet (Z.indexAt x)) (Z.mem_baseSet_at x) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ ContinuousAt (let_fun this := fun x => { proj := x, snd := v }; this) x refine ((Z.localTrivAt x).toLocalHomeomorph.continuousAt_iff_continuousAt_comp_left ?_).2 ?_ case refine_1 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ (let_fun this := fun x => { proj := x, snd := v }; this) ⁻¹' (localTrivAt Z x).toLocalHomeomorph.toLocalEquiv.source ∈ 𝓝 x exact A case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ ContinuousAt (↑(localTrivAt Z x).toLocalHomeomorph ∘ let_fun this := fun x => { proj := x, snd := v }; this) x apply continuousAt_id.prod case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x have : ContinuousOn (fun _ : B => v) (Z.baseSet (Z.indexAt x)) := continuousOn_const case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x this : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x)) ⊢ ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x refine (this.congr fun y hy ↦ ?_).continuousAt A case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x this : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x)) y : B hy : y ∈ baseSet Z (indexAt Z x) ⊢ coordChange Z (indexAt Z y) (indexAt Z x) y v = v exact h _ _ _ ⟨mem_baseSet_at _ _, hy⟩ ** Qed
FiberBundleCore.localTrivAt_apply ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a : F p : TotalSpace Z ⊢ ↑(localTrivAt Z p.proj) p = (p.proj, p.snd) rw [localTrivAt, localTriv_apply, coordChange_self] case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a : F p : TotalSpace Z ⊢ p.proj ∈ baseSet Z (indexAt Z p.proj) exact Z.mem_baseSet_at p.1 ** Qed
FiberBundleCore.mem_localTrivAt_baseSet ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b✝ : B a : F b : B ⊢ b ∈ (localTrivAt Z b).baseSet rw [localTrivAt, ← baseSet_at] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b✝ : B a : F b : B ⊢ b ∈ baseSet Z (indexAt Z b) exact Z.mem_baseSet_at b ** Qed
FiberBundleCore.mk_mem_localTrivAt_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a : F ⊢ { proj := b, snd := a } ∈ (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source simp only [mfld_simps] ** Qed
FiberPrebundle.continuous_symm_of_mem_pretrivializationAtlas ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas ⊢ ContinuousOn (↑(LocalEquiv.symm e.toLocalEquiv)) e.target refine' fun z H U h => preimage_nhdsWithin_coinduced' H (le_def.1 (nhds_mono _) U h) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas z : B × F H : z ∈ e.target U : Set (TotalSpace F E) h : U ∈ 𝓝 (↑(LocalEquiv.symm e.toLocalEquiv) z) ⊢ coinduced (fun x => ↑(LocalEquiv.symm e.toLocalEquiv) ↑x) inferInstance ≤ totalSpaceTopology a exact le_iSup₂ (α := TopologicalSpace (TotalSpace F E)) e he ** Qed
FiberPrebundle.isOpen_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e : Pretrivialization F TotalSpace.proj ⊢ IsOpen e.source refine isOpen_iSup_iff.mpr fun e' => isOpen_iSup_iff.mpr fun _ => ?_ ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj x✝ : e' ∈ a.pretrivializationAtlas ⊢ IsOpen e.source refine' isOpen_coinduced.mpr (isOpen_induced_iff.mpr ⟨e.target, e.open_target, _⟩) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj x✝ : e' ∈ a.pretrivializationAtlas ⊢ Subtype.val ⁻¹' e.target = Pretrivialization.setSymm e' ⁻¹' e.source ext ⟨x, hx⟩ case h.mk ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj x✝ : e' ∈ a.pretrivializationAtlas x : B × F hx : x ∈ e'.target ⊢ { val := x, property := hx } ∈ Subtype.val ⁻¹' e.target ↔ { val := x, property := hx } ∈ Pretrivialization.setSymm e' ⁻¹' e.source simp only [mem_preimage, Pretrivialization.setSymm, restrict, e.mem_target, e.mem_source, e'.proj_symm_apply hx] ** Qed
FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas ⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source) obtain ⟨u, hu1, hu2⟩ := continuousOn_iff'.mp (a.continuous_symm_of_mem_pretrivializationAtlas he') e.source (a.isOpen_source e) case intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a u : Set (B × F) hu1 : IsOpen u hu2 : ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source ∩ e'.target = u ∩ e'.target ⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source) rw [inter_comm, hu2] case intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a u : Set (B × F) hu1 : IsOpen u hu2 : ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source ∩ e'.target = u ∩ e'.target ⊢ IsOpen (u ∩ e'.target) exact hu1.inter e'.open_target ** Qed
FiberPrebundle.mem_pretrivializationAt_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B x : E b ⊢ { proj := b, snd := x } ∈ (pretrivializationAt a b).source simp only [(a.pretrivializationAt b).source_eq, mem_preimage, TotalSpace.proj] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B x : E b ⊢ b ∈ (pretrivializationAt a b).baseSet exact a.mem_base_pretrivializationAt b ** Qed
FiberPrebundle.continuous_totalSpaceMk ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B ⊢ Continuous (TotalSpace.mk b) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Continuous (TotalSpace.mk b) let e := a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas b) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj b : B this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a b ∈ a.pretrivializationAtlas) ⊢ Continuous (TotalSpace.mk b) rw [e.toLocalHomeomorph.continuous_iff_continuous_comp_left (a.totalSpaceMk_preimage_source b)] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj b : B this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a b ∈ a.pretrivializationAtlas) ⊢ Continuous (↑e.toLocalHomeomorph ∘ TotalSpace.mk b) exact continuous_iff_le_induced.mpr (le_antisymm_iff.mp (a.totalSpaceMk_inducing b).induced).1 ** Qed
FiberPrebundle.inducing_totalSpaceMk_of_inducing_comp ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (↑(pretrivializationAt a b) ∘ TotalSpace.mk b) ⊢ Inducing (TotalSpace.mk b) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (↑(pretrivializationAt a b) ∘ TotalSpace.mk b) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Inducing (TotalSpace.mk b) rw [← restrict_comp_codRestrict (a.mem_pretrivializationAt_source b)] at h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (restrict (pretrivializationAt a b).source ↑(pretrivializationAt a b) ∘ codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Inducing (TotalSpace.mk b) apply Inducing.of_codRestrict (a.mem_pretrivializationAt_source b) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (restrict (pretrivializationAt a b).source ↑(pretrivializationAt a b) ∘ codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Inducing (codRestrict (fun a => { proj := b, snd := a }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) refine inducing_of_inducing_compose ?_ (continuousOn_iff_continuous_restrict.mp (a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas b)).continuous_toFun) h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (restrict (pretrivializationAt a b).source ↑(pretrivializationAt a b) ∘ codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Continuous (codRestrict (fun a => { proj := b, snd := a }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) exact (a.continuous_totalSpaceMk b).codRestrict (a.mem_pretrivializationAt_source b) ** Qed
FiberPrebundle.continuous_proj ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj ⊢ Continuous TotalSpace.proj letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Continuous TotalSpace.proj letI := a.toFiberBundle ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a ⊢ Continuous TotalSpace.proj exact FiberBundle.continuous_proj F E ** Qed
FiberPrebundle.continuousOn_of_comp_right ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) ⊢ ContinuousOn f (TotalSpace.proj ⁻¹' s) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ ContinuousOn f (TotalSpace.proj ⁻¹' s) intro z hz ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s ⊢ ContinuousWithinAt f (TotalSpace.proj ⁻¹' s) z let e : Trivialization F (π F E) := a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas z.proj) ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ ContinuousWithinAt f (TotalSpace.proj ⁻¹' s) z refine' (e.continuousAt_of_comp_right _ ((hf z.proj hz).continuousAt (IsOpen.mem_nhds _ _))).continuousWithinAt case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ ↑e z ∈ (s ∩ (pretrivializationAt a z.proj).baseSet) ×ˢ univ refine' ⟨_, mem_univ _⟩ case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ (↑e z).1 ∈ s ∩ (pretrivializationAt a z.proj).baseSet rw [e.coe_fst] case refine'_1 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z.proj ∈ e.baseSet exact a.mem_base_pretrivializationAt z.proj case refine'_2 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ IsOpen ((s ∩ (pretrivializationAt a z.proj).baseSet) ×ˢ univ) exact (hs.inter (a.pretrivializationAt z.proj).open_baseSet).prod isOpen_univ case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z.proj ∈ s ∩ (pretrivializationAt a z.proj).baseSet exact ⟨hz, a.mem_base_pretrivializationAt z.proj⟩ case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z ∈ e.source rw [e.mem_source] case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z.proj ∈ e.baseSet exact a.mem_base_pretrivializationAt z.proj ** Qed
TopCat.Presheaf.toTypes_isSheaf X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf ⊢ ∃! s, IsGluing (presheafToTypes X T) U sf s choose index index_spec using fun x : ↑(iSup U) => Opens.mem_iSup.mp x.2 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) ⊢ ∃! s, IsGluing (presheafToTypes X T) U sf s let s : ∀ x : ↑(iSup U), T x := fun x => sf (index x) ⟨x.1, index_spec x⟩ X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } ⊢ ∃! s, IsGluing (presheafToTypes X T) U sf s refine' ⟨s, _, _⟩ case refine'_1 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } ⊢ (fun s => IsGluing (presheafToTypes X T) U sf s) s intro i case refine'_1 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } i : ι ⊢ ↑((presheafToTypes X T).map (leSupr U i).op) s = sf i funext x case refine'_1.h X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } i : ι x : { x // x ∈ (Opposite.op (U i)).unop } ⊢ ↑((presheafToTypes X T).map (leSupr U i).op) s x = sf i x exact congr_fun (hsf (index ⟨x, _⟩) i) ⟨x, ⟨index_spec ⟨x.1, _⟩, x.2⟩⟩ case refine'_2 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } ⊢ ∀ (y : (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (iSup U)))), (fun s => IsGluing (presheafToTypes X T) U sf s) y → y = s intro t ht case refine'_2 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } t : (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (iSup U))) ht : IsGluing (presheafToTypes X T) U sf t ⊢ t = s funext x case refine'_2.h X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } t : (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (iSup U))) ht : IsGluing (presheafToTypes X T) U sf t x : { x // x ∈ (Opposite.op (iSup U)).unop } ⊢ t x = s x exact congr_fun (ht (index x)) ⟨x.1, index_spec x⟩ ** Qed
TopCat.PrelocalPredicate.sheafifyOf X : TopCat T✝ T : ↑X → Type v P : PrelocalPredicate T U : Opens ↑X f : (x : { x // x ∈ U }) → T ↑x h : pred P f x : { x // x ∈ U } ⊢ pred P fun x => f ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U) }) x) convert h ** Qed
TopCat.subpresheafToTypes.isSheaf X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s let sf' : ∀ i : ι, (presheafToTypes X T).obj (op (U i)) := fun i => (sf i).val X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s have sf'_comp : (presheafToTypes X T).IsCompatible U sf' := fun i j => congr_arg Subtype.val (sf_comp i j) X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s obtain ⟨gl, gl_spec, gl_uniq⟩ := (sheafToTypes X T).existsUnique_gluing U sf' sf'_comp case intro.intro X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s refine' ⟨⟨gl, _⟩, _, _⟩ case intro.intro.refine'_1 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ PrelocalPredicate.pred P.toPrelocalPredicate gl apply P.locality case intro.intro.refine'_1.x X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ ∀ (x : { x // x ∈ (op (iSup U)).unop }), ∃ V x i, PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) rintro ⟨x, mem⟩ case intro.intro.refine'_1.x.mk X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop ⊢ ∃ V x i, PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) choose i hi using Opens.mem_iSup.mp mem case intro.intro.refine'_1.x.mk X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop i : ι hi : x ∈ U i ⊢ ∃ V x i, PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) use U i, hi, Opens.leSupr U i case h X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop i : ι hi : x ∈ U i ⊢ PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) convert (sf i).property using 1 case h.e'_5.h X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop i : ι hi : x ∈ U i e_4✝ : U i = (op (U i)).unop ⊢ (fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x)) = ↑(sf i) exact gl_spec i case intro.intro.refine'_2 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ (fun s => IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s) { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } exact fun i => Subtype.ext (gl_spec i) case intro.intro.refine'_3 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ ∀ (y : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U)))), (fun s => IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s) y → y = { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } intro gl' hgl' case intro.intro.refine'_3 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl gl' : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U))) hgl' : IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf gl' ⊢ gl' = { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } refine Subtype.ext ?_ case intro.intro.refine'_3 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl gl' : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U))) hgl' : IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf gl' ⊢ ↑gl' = ↑{ val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } exact gl_uniq gl'.1 fun i => congr_arg Subtype.val (hgl' i) ** Qed
TopCat.stalkToFiber_germ X : TopCat T : ↑X → Type v P : LocalPredicate T U : Opens ↑X x : { x // x ∈ U } f : (Sheaf.presheaf (subsheafToTypes P)).obj (op U) ⊢ stalkToFiber P (↑x) (Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) x f) = ↑f x dsimp [Presheaf.germ, stalkToFiber] X : TopCat T : ↑X → Type v P : LocalPredicate T U : Opens ↑X x : { x // x ∈ U } f : (Sheaf.presheaf (subsheafToTypes P)).obj (op U) ⊢ colimit.desc ((OpenNhds.inclusion ↑x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) { pt := T ↑x, ι := NatTrans.mk fun U_1 f => ↑f { val := ↑x, property := (_ : ↑x ∈ U_1.unop.obj) } } (colimit.ι ((OpenNhds.inclusion ↑x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) (op { obj := U, property := (_ : ↑x ∈ U) }) f) = ↑f x cases x case mk X : TopCat T : ↑X → Type v P : LocalPredicate T U : Opens ↑X f : (Sheaf.presheaf (subsheafToTypes P)).obj (op U) val✝ : ↑X property✝ : val✝ ∈ U ⊢ colimit.desc ((OpenNhds.inclusion ↑{ val := val✝, property := property✝ }).op ⋙ subpresheafToTypes P.toPrelocalPredicate) { pt := T ↑{ val := val✝, property := property✝ }, ι := NatTrans.mk fun U_1 f => ↑f { val := ↑{ val := val✝, property := property✝ }, property := (_ : ↑{ val := val✝, property := property✝ } ∈ U_1.unop.obj) } } (colimit.ι ((OpenNhds.inclusion ↑{ val := val✝, property := property✝ }).op ⋙ subpresheafToTypes P.toPrelocalPredicate) (op { obj := U, property := (_ : ↑{ val := val✝, property := property✝ } ∈ U) }) f) = ↑f { val := val✝, property := property✝ } simp ** Qed
TopCat.stalkToFiber_surjective X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t t : T x ⊢ ∃ a, stalkToFiber P x a = t rcases w t with ⟨U, f, h, rfl⟩ case intro.intro.intro X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t U : OpenNhds x f : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y h : PrelocalPredicate.pred P.toPrelocalPredicate f ⊢ ∃ a, stalkToFiber P x a = f { val := x, property := (_ : x ∈ U.obj) } fconstructor case intro.intro.intro.w X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t U : OpenNhds x f : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y h : PrelocalPredicate.pred P.toPrelocalPredicate f ⊢ Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x exact (subsheafToTypes P).presheaf.germ ⟨x, U.2⟩ ⟨f, h⟩ case intro.intro.intro.h X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t U : OpenNhds x f : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y h : PrelocalPredicate.pred P.toPrelocalPredicate f ⊢ stalkToFiber P x (Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ U.obj) } { val := f, property := h }) = f { val := x, property := (_ : x ∈ U.obj) } exact stalkToFiber_germ _ U.1 ⟨x, U.2⟩ ⟨f, h⟩ ** Qed
TopCat.stalkToFiber_injective X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV ⊢ tU = tV let Q : ∃ (W : (OpenNhds x)ᵒᵖ) (s : ∀ w : (unop W).1, T w) (hW : P.pred s), tU = (subsheafToTypes P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ ∧ tV = (subsheafToTypes P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ := ?_ case refine_1 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV ⊢ ∃ W s hW, tU = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } obtain ⟨U, ⟨fU, hU⟩, rfl⟩ := jointly_surjective'.{v, v} tU case refine_1.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU }) = stalkToFiber P x tV ⊢ ∃ W s hW, colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU } = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } obtain ⟨V, ⟨fV, hV⟩, rfl⟩ := jointly_surjective'.{v, v} tV case refine_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV Q : ∃ W s hW, tU = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } := ?refine_1 ⊢ tU = tV choose W s hW e using Q case refine_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV W : (OpenNhds x)ᵒᵖ s : (w : { x_1 // x_1 ∈ W.unop.obj }) → T ↑w hW : PrelocalPredicate.pred P.toPrelocalPredicate s e : tU = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ⊢ tU = tV exact e.1.trans e.2.symm case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU }) = stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) V { val := fV, property := hV }) ⊢ ∃ W s hW, colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU } = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) V { val := fV, property := hV } = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } dsimp case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU }) = stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) V { val := fV, property := hV }) ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } simp only [stalkToFiber, Types.Colimit.ι_desc_apply'] at h case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } specialize w (unop U) (unop V) fU hU fV hV h case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } w : ∃ W iU iV, ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } rcases w with ⟨W, iU, iV, w⟩ case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } refine' ⟨op W, fun w => fU (iU w : (unop U).1), P.res _ _ hU, _⟩ case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_1 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ (op W).unop.obj ⟶ U.unop.obj case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } rcases W with ⟨W, m⟩ case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_1.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : Opens ↑X m : x ∈ W iU : { obj := W, property := m } ⟶ U.unop iV : { obj := W, property := m } ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ { obj := W, property := m }.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ (op { obj := W, property := m }).unop.obj ⟶ U.unop.obj exact iU case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } exact ⟨colimit_sound iU.op (Subtype.eq rfl), colimit_sound iV.op (Subtype.eq (funext w).symm)⟩ ** Qed
TopCat.epi_iff_surjective X Y : TopCat f : X ⟶ Y ⊢ Epi f ↔ Function.Surjective ↑f suffices Epi f ↔ Epi ((forget TopCat).map f) by rw [this, CategoryTheory.epi_iff_surjective] rfl X Y : TopCat f : X ⟶ Y ⊢ Epi f ↔ Epi ((forget TopCat).map f) constructor X Y : TopCat f : X ⟶ Y this : Epi f ↔ Epi ((forget TopCat).map f) ⊢ Epi f ↔ Function.Surjective ↑f rw [this, CategoryTheory.epi_iff_surjective] X Y : TopCat f : X ⟶ Y this : Epi f ↔ Epi ((forget TopCat).map f) ⊢ Function.Surjective ((forget TopCat).map f) ↔ Function.Surjective ↑f rfl case mp X Y : TopCat f : X ⟶ Y ⊢ Epi f → Epi ((forget TopCat).map f) intro case mp X Y : TopCat f : X ⟶ Y a✝ : Epi f ⊢ Epi ((forget TopCat).map f) infer_instance case mpr X Y : TopCat f : X ⟶ Y ⊢ Epi ((forget TopCat).map f) → Epi f apply Functor.epi_of_epi_map ** Qed
TopCat.mono_iff_injective X Y : TopCat f : X ⟶ Y ⊢ Mono f ↔ Function.Injective ↑f suffices Mono f ↔ Mono ((forget TopCat).map f) by rw [this, CategoryTheory.mono_iff_injective] rfl X Y : TopCat f : X ⟶ Y ⊢ Mono f ↔ Mono ((forget TopCat).map f) constructor X Y : TopCat f : X ⟶ Y this : Mono f ↔ Mono ((forget TopCat).map f) ⊢ Mono f ↔ Function.Injective ↑f rw [this, CategoryTheory.mono_iff_injective] X Y : TopCat f : X ⟶ Y this : Mono f ↔ Mono ((forget TopCat).map f) ⊢ Function.Injective ((forget TopCat).map f) ↔ Function.Injective ↑f rfl case mp X Y : TopCat f : X ⟶ Y ⊢ Mono f → Mono ((forget TopCat).map f) intro case mp X Y : TopCat f : X ⟶ Y a✝ : Mono f ⊢ Mono ((forget TopCat).map f) infer_instance case mpr X Y : TopCat f : X ⟶ Y ⊢ Mono ((forget TopCat).map f) → Mono f apply Functor.mono_of_mono_map ** Qed
TopCat.isSheaf_of_isLimit C : Type u inst✝² : Category.{v, u} C J : Type v inst✝¹ : SmallCategory J inst✝ : HasLimits C X : TopCat F : J ⥤ Presheaf C X H : ∀ (j : J), Presheaf.IsSheaf (F.obj j) c : Cone F hc : IsLimit c ⊢ Presheaf.IsSheaf c.pt let F' : J ⥤ Sheaf C X := { obj := fun j => ⟨F.obj j, H j⟩ map := fun f => ⟨F.map f⟩ } C : Type u inst✝² : Category.{v, u} C J : Type v inst✝¹ : SmallCategory J inst✝ : HasLimits C X : TopCat F : J ⥤ Presheaf C X H : ∀ (j : J), Presheaf.IsSheaf (F.obj j) c : Cone F hc : IsLimit c F' : J ⥤ Sheaf C X := CategoryTheory.Functor.mk { obj := fun j => { val := F.obj j, cond := (_ : Presheaf.IsSheaf (F.obj j)) }, map := fun {X_1 Y} f => { val := F.map f } } ⊢ Presheaf.IsSheaf c.pt let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _ C : Type u inst✝² : Category.{v, u} C J : Type v inst✝¹ : SmallCategory J inst✝ : HasLimits C X : TopCat F : J ⥤ Presheaf C X H : ∀ (j : J), Presheaf.IsSheaf (F.obj j) c : Cone F hc : IsLimit c F' : J ⥤ Sheaf C X := CategoryTheory.Functor.mk { obj := fun j => { val := F.obj j, cond := (_ : Presheaf.IsSheaf (F.obj j)) }, map := fun {X_1 Y} f => { val := F.map f } } e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun x => Iso.refl ((F' ⋙ Sheaf.forget C X).obj x) ⊢ Presheaf.IsSheaf c.pt exact Presheaf.isSheaf_of_iso ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e) (limit F').2 ** Qed
BoundedContinuousFunction.dist_set_exists F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩ case intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C✝ C : ℝ hC : ∀ ⦃x : β⦄, x ∈ range ↑f ∪ range ↑g → ∀ ⦃y : β⦄, y ∈ range ↑f ∪ range ↑g → dist x y ≤ C ⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C refine' ⟨max 0 C, le_max_left _ _, fun x => (hC _ _).trans (le_max_right _ _)⟩ <;> [left; right] <;> apply mem_range_self ** Qed
BoundedContinuousFunction.dist_lt_of_nonempty_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C ⊢ dist f g < C have c : Continuous fun x => dist (f x) (g x) := by continuity F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C c : Continuous fun x => dist (↑f x) (↑g x) ⊢ dist f g < C obtain ⟨x, -, le⟩ := IsCompact.exists_isMaxOn isCompact_univ Set.univ_nonempty (Continuous.continuousOn c) case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x✝ : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C c : Continuous fun x => dist (↑f x) (↑g x) x : α le : IsMaxOn (fun x => dist (↑f x) (↑g x)) univ x ⊢ dist f g < C exact lt_of_le_of_lt (dist_le_iff_of_nonempty.mpr fun y => le trivial) (w x) F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C ⊢ Continuous fun x => dist (↑f x) (↑g x) continuity ** Qed
BoundedContinuousFunction.dist_lt_iff_of_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C ⊢ dist f g < C ↔ ∀ (x : α), dist (↑f x) (↑g x) < C fconstructor case mp F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C ⊢ dist f g < C → ∀ (x : α), dist (↑f x) (↑g x) < C intro w x case mp F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C w : dist f g < C x : α ⊢ dist (↑f x) (↑g x) < C exact lt_of_le_of_lt (dist_coe_le_dist x) w case mpr F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C ⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C by_cases h : Nonempty α case pos F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : Nonempty α ⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C exact dist_lt_of_nonempty_compact case neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C rintro - case neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ dist f g < C convert C0 case h.e'_3 F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ dist f g = 0 apply le_antisymm _ dist_nonneg' F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ dist f g ≤ 0 rw [dist_eq] F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} ≤ 0 exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempty.intro x))⟩ ** Qed
BoundedContinuousFunction.nndist_eq F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} = ↑(sInf {C \| ∀ (x : α), nndist (↑f x) (↑g x) ≤ C}) rw [val_eq_coe, coe_sInf, coe_image] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} = sInf {x \| ∃ h, { val := x, property := h } ∈ {C \| ∀ (x : α), nndist (↑f x) (↑g x) ≤ C}} simp_rw [mem_setOf_eq, ← NNReal.coe_le_coe, coe_mk, exists_prop, coe_nndist] ** Qed
BoundedContinuousFunction.dist_zero_of_empty F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : IsEmpty α ⊢ dist f g = 0 rw [(ext isEmptyElim : f = g), dist_self] ** Qed
BoundedContinuousFunction.dist_eq_iSup F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ dist f g = ⨆ x, dist (↑f x) (↑g x) cases isEmpty_or_nonempty α case inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ h✝ : Nonempty α ⊢ dist f g = ⨆ x, dist (↑f x) (↑g x) refine' (dist_le_iff_of_nonempty.mpr <\| le_ciSup _).antisymm (ciSup_le dist_coe_le_dist) case inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ h✝ : Nonempty α ⊢ BddAbove (range fun x => dist (↑f x) (↑g x)) exact dist_set_exists.imp fun C hC => forall_range_iff.2 hC.2 case inl F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ h✝ : IsEmpty α ⊢ dist f g = ⨆ x, dist (↑f x) (↑g x) rw [iSup_of_empty', Real.sSup_empty, dist_zero_of_empty] ** Qed
BoundedContinuousFunction.nndist_eq_iSup F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ ⨆ x, dist (↑f x) (↑g x) = ↑(⨆ x, nndist (↑f x) (↑g x)) simp_rw [val_eq_coe, coe_iSup, coe_nndist] ** Qed
BoundedContinuousFunction.inducing_coeFn F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ Inducing (↑UniformFun.ofFun ∘ FunLike.coe) rw [inducing_iff_nhds] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ ∀ (a : α →ᵇ β), 𝓝 a = comap (↑UniformFun.ofFun ∘ FunLike.coe) (𝓝 ((↑UniformFun.ofFun ∘ FunLike.coe) a)) refine' fun f => eq_of_forall_le_iff fun l => _ F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ f : α →ᵇ β l : Filter (α →ᵇ β) ⊢ l ≤ 𝓝 f ↔ l ≤ comap (↑UniformFun.ofFun ∘ FunLike.coe) (𝓝 ((↑UniformFun.ofFun ∘ FunLike.coe) f)) rw [← tendsto_iff_comap, ← tendsto_id', tendsto_iff_tendstoUniformly, UniformFun.tendsto_iff_tendstoUniformly] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ f : α →ᵇ β l : Filter (α →ᵇ β) ⊢ TendstoUniformly (fun i => ↑(id i)) (↑f) l ↔ TendstoUniformly (↑UniformFun.ofFun ∘ FunLike.coe) ((↑UniformFun.ofFun ∘ FunLike.coe) f) l rfl ** Qed
BoundedContinuousFunction.lipschitz_comp ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g✝ : α →ᵇ β x✝ : α C✝ : ℝ G : β → γ C : ℝ≥0 H : LipschitzWith C G f g : α →ᵇ β x : α ⊢ ↑C * dist (↑f x) (↑g x) ≤ ↑C * dist f g gcongr case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g✝ : α →ᵇ β x✝ : α C✝ : ℝ G : β → γ C : ℝ≥0 H : LipschitzWith C G f g : α →ᵇ β x : α ⊢ dist (↑f x) (↑g x) ≤ dist f g apply dist_coe_le_dist Qed
BoundedContinuousFunction.dist_extend_extend F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β ⊢ dist (extend f g₁ h₁) (extend f g₂ h₂) = max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) refine' le_antisymm ((dist_le <\| le_max_iff.2 <\| Or.inl dist_nonneg).2 fun x => _) (max_le _ _) case refine'_1 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : δ ⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) rcases _root_.em (∃ y, f y = x) with (⟨x, rfl⟩ \| hx) case refine'_1.inl.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) simp only [extend_apply] case refine'_1.inl.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑g₁ x) (↑g₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) exact (dist_coe_le_dist x).trans (le_max_left _ _) case refine'_1.inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : δ hx : ¬∃ y, ↑f y = x ⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) simp only [extend_apply' hx] case refine'_1.inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : δ hx : ¬∃ y, ↑f y = x ⊢ dist (↑h₁ x) (↑h₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) lift x to ((range f)ᶜ : Set δ) using hx case refine'_1.inr.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : { x // x ∈ (range ↑f)ᶜ } ⊢ dist (↑h₁ ↑x) (↑h₂ ↑x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) calc dist (h₁ x) (h₂ x) = dist (h₁.restrict (range f)ᶜ x) (h₂.restrict (range f)ᶜ x) := rfl _ ≤ dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ) := (dist_coe_le_dist x) _ ≤ _ := le_max_right _ _ case refine'_2 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β ⊢ dist g₁ g₂ ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) refine' (dist_le dist_nonneg).2 fun x => _ case refine'_2 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑g₁ x) (↑g₂ x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) rw [← extend_apply f g₁ h₁, ← extend_apply f g₂ h₂] case refine'_2 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) exact dist_coe_le_dist _ case refine'_3 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β ⊢ dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) refine' (dist_le dist_nonneg).2 fun x => _ case refine'_3 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : ↑(range ↑f)ᶜ ⊢ dist (↑(restrict h₁ (range ↑f)ᶜ) x) (↑(restrict h₂ (range ↑f)ᶜ) x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) calc dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop] _ ≤ _ := dist_coe_le_dist _ F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : ↑(range ↑f)ᶜ ⊢ dist (↑h₁ ↑x) (↑h₂ ↑x) = dist (↑(extend f g₁ h₁) ↑x) (↑(extend f g₂ h₂) ↑x) rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop] ** Qed
BoundedContinuousFunction.isometry_extend F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ h : δ →ᵇ β g₁ g₂ : α →ᵇ β ⊢ dist (extend f g₁ h) (extend f g₂ h) = dist g₁ g₂ simp [dist_nonneg] ** Qed
BoundedContinuousFunction.arzela_ascoli₁ F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : Equicontinuous fun x => ↑↑x ⊢ IsCompact A simp_rw [Equicontinuous, Metric.equicontinuousAt_iff_pair] at H F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ⊢ IsCompact A refine' isCompact_of_totallyBounded_isClosed _ closed F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ⊢ TotallyBounded A refine' totallyBounded_of_finite_discretization fun ε ε0 => _ F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε rcases exists_between ε0 with ⟨ε₁, ε₁0, εε₁⟩ case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε let ε₂ := ε₁ / 2 / 2 case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε have ε₂0 : ε₂ > 0 := half_pos (half_pos ε₁0) case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε have : ∀ x : α, ∃ U, x ∈ U ∧ IsOpen U ∧ ∀ (y) (_ : y ∈ U) (z) (_ : z ∈ U) {f : α →ᵇ β}, f ∈ A → dist (f y) (f z) < ε₂ := fun x => let ⟨U, nhdsU, hU⟩ := H x _ ε₂0 let ⟨V, VU, openV, xV⟩ := _root_.mem_nhds_iff.1 nhdsU ⟨V, xV, openV, fun y hy z hz f hf => hU y (VU hy) z (VU hz) ⟨f, hf⟩⟩ case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 this : ∀ (x : α), ∃ U, x ∈ U ∧ IsOpen U ∧ ∀ (y : α), y ∈ U → ∀ (z : α), z ∈ U → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε choose U hU using this case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε rcases isCompact_univ.elim_finite_subcover_image (fun x _ => (hU x).2.1) fun x _ => mem_biUnion (mem_univ _) (hU x).1 with ⟨tα, _, ⟨_⟩, htα⟩ case intro.intro.intro.intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝ : Fintype ↑tα ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε rcases @finite_cover_balls_of_compact β _ _ isCompact_univ _ ε₂0 with ⟨tβ, _, ⟨_⟩, htβ⟩ case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε choose F hF using fun y => show ∃ z ∈ tβ, dist y z < ε₂ by simpa using htβ (mem_univ y) case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε refine' ⟨tα → tβ, by infer_instance, fun f a => ⟨F (f.1 a), (hF (f.1 a)).1⟩, _⟩ case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ ⊢ ∀ (x y : ↑A), (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) x = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) y → dist ↑x ↑y < ε rintro ⟨f, hf⟩ ⟨g, hg⟩ f_eq_g case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.mk F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } ⊢ dist ↑{ val := f, property := hf } ↑{ val := g, property := hg } < ε refine' lt_of_le_of_lt ((dist_le <\| le_of_lt ε₁0).2 fun x => _) εε₁ case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.mk F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x : α ⊢ dist (↑↑{ val := f, property := hf } x) (↑↑{ val := g, property := hg } x) ≤ ε₁ obtain ⟨x', x'tα, hx'⟩ := mem_iUnion₂.1 (htα (mem_univ x)) F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ y : β ⊢ ∃ z, z ∈ tβ ∧ dist y z < ε₂ simpa using htβ (mem_univ y) F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ ⊢ Fintype (↑tα → ↑tβ) infer_instance F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑f x) (↑f x') + dist (↑g x) (↑g x') + dist (↑f x') (↑g x') ≤ ε₂ + ε₂ + ε₁ / 2 refine' le_of_lt (add_lt_add (add_lt_add _ _) _) case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑f x) (↑f x') < ε₂ exact (hU x').2.2 _ hx' _ (hU x').1 hf case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑g x) (↑g x') < ε₂ exact (hU x').2.2 _ hx' _ (hU x').1 hg case refine'_3 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑f x') (↑g x') < ε₁ / 2 have F_f_g : F (f x') = F (g x') := (congr_arg (fun f : tα → tβ => (f ⟨x', x'tα⟩ : β)) f_eq_g : _) case refine'_3 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' F_f_g : F (↑f x') = F (↑g x') ⊢ dist (↑f x') (↑g x') < ε₁ / 2 calc dist (f x') (g x') ≤ dist (f x') (F (f x')) + dist (g x') (F (f x')) := dist_triangle_right _ _ _ _ = dist (f x') (F (f x')) + dist (g x') (F (g x')) := by rw [F_f_g] _ < ε₂ + ε₂ := (add_lt_add (hF (f x')).2 (hF (g x')).2) _ = ε₁ / 2 := add_halves _ F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' F_f_g : F (↑f x') = F (↑g x') ⊢ dist (↑f x') (F (↑f x')) + dist (↑g x') (F (↑f x')) = dist (↑f x') (F (↑f x')) + dist (↑g x') (F (↑g x')) rw [F_f_g] F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ ε₂ + ε₂ + ε₁ / 2 = ε₁ rw [add_halves, add_halves] ** Qed
BoundedContinuousFunction.arzela_ascoli₂ F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x ⊢ IsCompact A have M : LipschitzWith 1 (↑) := LipschitzWith.subtype_val s F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val ⊢ IsCompact A let F : (α →ᵇ s) → α →ᵇ β := comp (↑) M F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M ⊢ IsCompact A refine' IsCompact.of_isClosed_subset ((_ : IsCompact (F ⁻¹' A)).image (continuous_comp M)) closed fun f hf => _ case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M ⊢ IsCompact (F ⁻¹' A) haveI : CompactSpace s := isCompact_iff_compactSpace.1 hs case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M this : CompactSpace ↑s ⊢ IsCompact (F ⁻¹' A) refine' arzela_ascoli₁ _ (continuous_iff_isClosed.1 (continuous_comp M) _ closed) _ case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M this : CompactSpace ↑s ⊢ Equicontinuous fun x => ↑↑x rw [uniformEmbedding_subtype_val.toUniformInducing.equicontinuous_iff] case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M this : CompactSpace ↑s ⊢ Equicontinuous ((fun x x_1 => x ∘ x_1) Subtype.val ∘ fun x => ↑↑x) exact H.comp (A.restrictPreimage F) case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf : f ∈ A ⊢ f ∈ comp Subtype.val M '' (F ⁻¹' A) let g := codRestrict s f fun x => in_s f x hf case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) ⊢ f ∈ comp Subtype.val M '' (F ⁻¹' A) rw [show f = F g by ext; rfl] at hf ⊢ case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf✝ : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) hf : F g ∈ A ⊢ F g ∈ comp Subtype.val M '' (F ⁻¹' A) exact ⟨g, hf, rfl⟩ F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf✝ : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) hf : F g ∈ A ⊢ f = F g ext case h F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf✝ : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) hf : F g ∈ A x✝ : α ⊢ ↑f x✝ = ↑(F g) x✝ rfl ** Qed
BoundedContinuousFunction.coe_nsmulRec F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : AddMonoid β inst✝ : LipschitzAdd β f g : α →ᵇ β x : α C : ℝ ⊢ ↑(nsmulRec 0 f) = 0 • ↑f rw [nsmulRec, zero_smul, coe_zero] F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : AddMonoid β inst✝ : LipschitzAdd β f g : α →ᵇ β x : α C : ℝ n : ℕ ⊢ ↑(nsmulRec (n + 1) f) = (n + 1) • ↑f rw [nsmulRec, succ_nsmul, coe_add, coe_nsmulRec n] ** Qed
BoundedContinuousFunction.sum_apply F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : AddCommMonoid β inst✝ : LipschitzAdd β ι : Type u_2 s : Finset ι f : ι → α →ᵇ β a : α ⊢ ↑(∑ i in s, f i) a = ∑ i in s, ↑(f i) a simp ** Qed
BoundedContinuousFunction.norm_eq F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ f : α →ᵇ β ⊢ ‖f‖ = sInf {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} simp [norm_def, BoundedContinuousFunction.dist_eq] ** Qed
BoundedContinuousFunction.norm_eq_of_nonempty F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ h : Nonempty α ⊢ ‖f‖ = sInf {C \| ∀ (x : α), ‖↑f x‖ ≤ C} obtain ⟨a⟩ := h case intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α ⊢ ‖f‖ = sInf {C \| ∀ (x : α), ‖↑f x‖ ≤ C} rw [norm_eq] case intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} = sInf {C \| ∀ (x : α), ‖↑f x‖ ≤ C} congr case intro.e_a F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α ⊢ {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} = {C \| ∀ (x : α), ‖↑f x‖ ≤ C} ext case intro.e_a.h F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α x✝ : ℝ ⊢ x✝ ∈ {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} ↔ x✝ ∈ {C \| ∀ (x : α), ‖↑f x‖ ≤ C} simp only [mem_setOf_eq, and_iff_right_iff_imp] case intro.e_a.h F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α x✝ : ℝ ⊢ (∀ (x : α), ‖↑f x‖ ≤ x✝) → 0 ≤ x✝ exact fun h' => le_trans (norm_nonneg (f a)) (h' a) ** Qed
BoundedContinuousFunction.norm_coe_le_norm F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x✝ : α C : ℝ x : α ⊢ ‖↑f x‖ = dist (↑f x) (↑0 x) simp [dist_zero_right] ** Qed
BoundedContinuousFunction.norm_le F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ C0 : 0 ≤ C ⊢ ‖f‖ ≤ C ↔ ∀ (x : α), ‖↑f x‖ ≤ C simpa using @dist_le _ _ _ _ f 0 _ C0 ** Qed
BoundedContinuousFunction.norm_le_of_nonempty F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : Nonempty α f : α →ᵇ β M : ℝ ⊢ ‖f‖ ≤ M ↔ ∀ (x : α), ‖↑f x‖ ≤ M simp_rw [norm_def, ← dist_zero_right] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : Nonempty α f : α →ᵇ β M : ℝ ⊢ dist f 0 ≤ M ↔ ∀ (x : α), dist (↑f x) 0 ≤ M exact dist_le_iff_of_nonempty ** Qed
BoundedContinuousFunction.norm_lt_iff_of_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α f : α →ᵇ β M : ℝ M0 : 0 < M ⊢ ‖f‖ < M ↔ ∀ (x : α), ‖↑f x‖ < M simp_rw [norm_def, ← dist_zero_right] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α f : α →ᵇ β M : ℝ M0 : 0 < M ⊢ dist f 0 < M ↔ ∀ (x : α), dist (↑f x) 0 < M exact dist_lt_iff_of_compact M0 ** Qed
BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α f : α →ᵇ β M : ℝ ⊢ ‖f‖ < M ↔ ∀ (x : α), ‖↑f x‖ < M simp_rw [norm_def, ← dist_zero_right] F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α f : α →ᵇ β M : ℝ ⊢ dist f 0 < M ↔ ∀ (x : α), dist (↑f x) 0 < M exact dist_lt_iff_of_nonempty_compact ** Qed
BoundedContinuousFunction.norm_normComp F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖normComp f‖ = ‖f‖ simp only [norm_eq, coe_normComp, norm_norm, Function.comp] ** Qed
BoundedContinuousFunction.norm_eq_iSup_norm F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖f‖ = ⨆ x, ‖↑f x‖ simp_rw [norm_def, dist_eq_iSup, coe_zero, Pi.zero_apply, dist_zero_right] ** Qed
BoundedContinuousFunction.coe_zsmulRec F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ n : ℕ ⊢ ↑(zsmulRec (Int.ofNat n) f) = Int.ofNat n • ↑f rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmulRec, coe_nat_zsmul] F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ n : ℕ ⊢ ↑(zsmulRec (Int.negSucc n) f) = Int.negSucc n • ↑f rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmulRec] ** Qed
BoundedContinuousFunction.nnnorm_eq_iSup_nnnorm F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ⨆ x, ‖↑f x‖ = ↑(⨆ x, ‖↑f x‖₊) simp_rw [val_eq_coe, NNReal.coe_iSup, coe_nnnorm] ** Qed
BoundedContinuousFunction.abs_diff_coe_le_dist F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖↑f x - ↑g x‖ ≤ dist f g rw [dist_eq_norm] F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖↑f x - ↑g x‖ ≤ ‖f - g‖ exact (f - g).norm_coe_le_norm x ** Qed
BoundedContinuousFunction.norm_compContinuous_le ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g✝ : α →ᵇ β x : α C : ℝ inst✝ : TopologicalSpace γ f : α →ᵇ β g : C(γ, α) ⊢ ↑1 * dist f 0 ≤ ‖f‖ rw [NNReal.coe_one, one_mul, dist_zero_right] Qed
BoundedContinuousFunction.coe_npowRec F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α R : Type u_2 inst✝ : SeminormedRing R f : α →ᵇ R ⊢ ↑(npowRec 0 f) = ↑f ^ 0 rw [npowRec, pow_zero, coe_one] F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α R : Type u_2 inst✝ : SeminormedRing R f : α →ᵇ R n : ℕ ⊢ ↑(npowRec (n + 1) f) = ↑f ^ (n + 1) rw [npowRec, pow_succ, coe_mul, coe_npowRec f n] ** Qed
BoundedContinuousFunction.algebraMap_apply F : Type u_1 α : Type u β : Type v γ : Type w 𝕜 : Type u_2 inst✝⁵ : NormedField 𝕜 inst✝⁴ : TopologicalSpace α inst✝³ : SeminormedAddCommGroup β inst✝² : NormedSpace 𝕜 β inst✝¹ : NormedRing γ inst✝ : NormedAlgebra 𝕜 γ f g : α →ᵇ γ x : α c k : 𝕜 a : α ⊢ ↑(↑(algebraMap 𝕜 (α →ᵇ γ)) k) a = k • 1 rw [Algebra.algebraMap_eq_smul_one] F : Type u_1 α : Type u β : Type v γ : Type w 𝕜 : Type u_2 inst✝⁵ : NormedField 𝕜 inst✝⁴ : TopologicalSpace α inst✝³ : SeminormedAddCommGroup β inst✝² : NormedSpace 𝕜 β inst✝¹ : NormedRing γ inst✝ : NormedAlgebra 𝕜 γ f g : α →ᵇ γ x : α c k : 𝕜 a : α ⊢ ↑(k • 1) a = k • 1 rfl ** Qed
BoundedContinuousFunction.NNReal.upper_bound F : Type u_1 α✝ : Type u β : Type v γ : Type w α : Type u_2 inst✝ : TopologicalSpace α f : α →ᵇ ℝ≥0 x : α ⊢ ↑f x ≤ nndist f 0 have key : nndist (f x) ((0 : α →ᵇ ℝ≥0) x) ≤ nndist f 0 := @dist_coe_le_dist α ℝ≥0 _ _ f 0 x F : Type u_1 α✝ : Type u β : Type v γ : Type w α : Type u_2 inst✝ : TopologicalSpace α f : α →ᵇ ℝ≥0 x : α key : nndist (↑f x) (↑0 x) ≤ nndist f 0 ⊢ ↑f x ≤ nndist f 0 simp only [coe_zero, Pi.zero_apply] at key F : Type u_1 α✝ : Type u β : Type v γ : Type w α : Type u_2 inst✝ : TopologicalSpace α f : α →ᵇ ℝ≥0 x : α key : nndist (↑f x) 0 ≤ nndist f 0 ⊢ ↑f x ≤ nndist f 0 rwa [NNReal.nndist_zero_eq_val' (f x)] at key ** Qed
BoundedContinuousFunction.self_eq_nnrealPart_sub_nnrealPart_neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ ⊢ ↑f = toReal ∘ ↑(nnrealPart f) - toReal ∘ ↑(nnrealPart (-f)) funext x case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ ↑f x = (toReal ∘ ↑(nnrealPart f) - toReal ∘ ↑(nnrealPart (-f))) x dsimp case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ ↑f x = max (↑f x) 0 - max (-↑f x) 0 simp only [max_zero_sub_max_neg_zero_eq_self] ** Qed
BoundedContinuousFunction.abs_self_eq_nnrealPart_add_nnrealPart_neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ ⊢ abs ∘ ↑f = toReal ∘ ↑(nnrealPart f) + toReal ∘ ↑(nnrealPart (-f)) funext x case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ (abs ∘ ↑f) x = (toReal ∘ ↑(nnrealPart f) + toReal ∘ ↑(nnrealPart (-f))) x dsimp case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ \|↑f x\| = max (↑f x) 0 + max (-↑f x) 0 simp only [max_zero_add_max_neg_zero_eq_abs_self] ** Qed
SetTheory.PGame.impartialAux_def G : PGame ⊢ ImpartialAux G ↔ G ≈ -G ∧ (∀ (i : LeftMoves G), ImpartialAux (moveLeft G i)) ∧ ∀ (j : RightMoves G), ImpartialAux (moveRight G j) rw [ImpartialAux] ** Qed
SetTheory.PGame.impartial_def G : PGame ⊢ Impartial G ↔ G ≈ -G ∧ (∀ (i : LeftMoves G), Impartial (moveLeft G i)) ∧ ∀ (j : RightMoves G), Impartial (moveRight G j) simpa only [impartial_iff_aux] using impartialAux_def ** Qed
SetTheory.PGame.Impartial.impartial_congr G H : PGame e : G ≡r H ⊢ ∀ [inst : Impartial G], Impartial H intro h G H : PGame e : G ≡r H h : Impartial G ⊢ Impartial H exact impartial_def.2 ⟨Equiv.trans e.symm.equiv (Equiv.trans (neg_equiv_self G) (neg_equiv_neg_iff.2 e.equiv)), fun i => impartial_congr (e.moveLeftSymm i), fun j => impartial_congr (e.moveRightSymm j)⟩ G H : PGame x✝ : ∀ (y : (_ : PGame) ×' PGame), (invImage (fun a => PSigma.casesOn a fun G snd => (G, snd)) Prod.instWellFoundedRelationProd).1 y { fst := G, snd := H } → y.1 ≡r y.2 → ∀ [inst : Impartial y.1], Impartial y.2 e : G ≡r H h : Impartial G j : RightMoves H ⊢ (invImage (fun a => PSigma.casesOn a fun G snd => (G, snd)) Prod.instWellFoundedRelationProd).1 { fst := moveRight G (↑(Relabelling.rightMovesEquiv e).symm j), snd := moveRight H j } { fst := G, snd := H } pgame_wf_tac ** Qed
SetTheory.PGame.Impartial.nonpos G : PGame inst✝ : Impartial G h : 0 < G ⊢ False have h' := neg_lt_neg_iff.2 h G : PGame inst✝ : Impartial G h : 0 < G h' : -G < -0 ⊢ False rw [neg_zero, lt_congr_left (Equiv.symm (neg_equiv_self G))] at h' G : PGame inst✝ : Impartial G h : 0 < G h' : G < 0 ⊢ False exact (h.trans h').false ** Qed
SetTheory.PGame.Impartial.nonneg G : PGame inst✝ : Impartial G h : G < 0 ⊢ False have h' := neg_lt_neg_iff.2 h G : PGame inst✝ : Impartial G h : G < 0 h' : -0 < -G ⊢ False rw [neg_zero, lt_congr_right (Equiv.symm (neg_equiv_self G))] at h' G : PGame inst✝ : Impartial G h : G < 0 h' : 0 < G ⊢ False exact (h.trans h').false ** Qed
SetTheory.PGame.Impartial.equiv_or_fuzzy_zero G : PGame inst✝ : Impartial G ⊢ G ≈ 0 ∨ G ‖ 0 rcases lt_or_equiv_or_gt_or_fuzzy G 0 with (h \| h \| h \| h) case inl G : PGame inst✝ : Impartial G h : G < 0 ⊢ G ≈ 0 ∨ G ‖ 0 exact ((nonneg G) h).elim case inr.inl G : PGame inst✝ : Impartial G h : G ≈ 0 ⊢ G ≈ 0 ∨ G ‖ 0 exact Or.inl h case inr.inr.inl G : PGame inst✝ : Impartial G h : 0 < G ⊢ G ≈ 0 ∨ G ‖ 0 exact ((nonpos G) h).elim case inr.inr.inr G : PGame inst✝ : Impartial G h : G ‖ 0 ⊢ G ≈ 0 ∨ G ‖ 0 exact Or.inr h ** Qed
SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero G : PGame inst✝ : Impartial G H : PGame ⊢ H ≈ G ↔ H + G ≈ 0 rw [Game.PGame.equiv_iff_game_eq, ← @add_right_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add, Game.PGame.equiv_iff_game_eq] G : PGame inst✝ : Impartial G H : PGame ⊢ Quotient.mk setoid (H + G) = 0 ↔ Quotient.mk setoid (H + G) = Quotient.mk setoid 0 rfl ** Qed
SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero' G : PGame inst✝ : Impartial G H : PGame ⊢ G ≈ H ↔ G + H ≈ 0 rw [Game.PGame.equiv_iff_game_eq, ← @add_left_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add, Game.PGame.equiv_iff_game_eq] G : PGame inst✝ : Impartial G H : PGame ⊢ 0 = Quotient.mk setoid (G + H) ↔ Quotient.mk setoid (G + H) = Quotient.mk setoid 0 exact ⟨Eq.symm, Eq.symm⟩ ** Qed
SetTheory.PGame.Impartial.le_zero_iff G✝ : PGame inst✝¹ : Impartial G✝ G : PGame inst✝ : Impartial G ⊢ G ≤ 0 ↔ 0 ≤ G rw [← zero_le_neg_iff, le_congr_right (neg_equiv_self G)] ** Qed
SetTheory.PGame.Impartial.lf_zero_iff G✝ : PGame inst✝¹ : Impartial G✝ G : PGame inst✝ : Impartial G ⊢ G ⧏ 0 ↔ 0 ⧏ G rw [← zero_lf_neg_iff, lf_congr_right (neg_equiv_self G)] ** Qed
SetTheory.PGame.Impartial.forall_leftMoves_fuzzy_iff_equiv_zero G : PGame inst✝ : Impartial G ⊢ (∀ (i : LeftMoves G), moveLeft G i ‖ 0) ↔ G ≈ 0 refine' ⟨fun hb => _, fun hp i => _⟩ case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (i : LeftMoves G), moveLeft G i ‖ 0 ⊢ G ≈ 0 rw [equiv_zero_iff_le G, le_zero_lf] case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (i : LeftMoves G), moveLeft G i ‖ 0 ⊢ ∀ (i : LeftMoves G), moveLeft G i ⧏ 0 exact fun i => (hb i).1 case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : LeftMoves G ⊢ moveLeft G i ‖ 0 rw [fuzzy_zero_iff_lf] case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : LeftMoves G ⊢ moveLeft G i ⧏ 0 exact hp.1.moveLeft_lf i ** Qed
SetTheory.PGame.Impartial.forall_rightMoves_fuzzy_iff_equiv_zero G : PGame inst✝ : Impartial G ⊢ (∀ (j : RightMoves G), moveRight G j ‖ 0) ↔ G ≈ 0 refine' ⟨fun hb => _, fun hp i => _⟩ case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (j : RightMoves G), moveRight G j ‖ 0 ⊢ G ≈ 0 rw [equiv_zero_iff_ge G, zero_le_lf] case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (j : RightMoves G), moveRight G j ‖ 0 ⊢ ∀ (j : RightMoves G), 0 ⧏ moveRight G j exact fun i => (hb i).2 case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : RightMoves G ⊢ moveRight G i ‖ 0 rw [fuzzy_zero_iff_gf] case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : RightMoves G ⊢ 0 ⧏ moveRight G i exact hp.2.lf_moveRight i ** Qed
SetTheory.PGame.Impartial.exists_left_move_equiv_iff_fuzzy_zero G : PGame inst✝ : Impartial G ⊢ (∃ i, moveLeft G i ≈ 0) ↔ G ‖ 0 refine' ⟨fun ⟨i, hi⟩ => (fuzzy_zero_iff_gf G).2 (lf_of_le_moveLeft hi.2), fun hn => _⟩ G : PGame inst✝ : Impartial G hn : G ‖ 0 ⊢ ∃ i, moveLeft G i ≈ 0 rw [fuzzy_zero_iff_gf G, zero_lf_le] at hn G : PGame inst✝ : Impartial G hn : ∃ i, 0 ≤ moveLeft G i ⊢ ∃ i, moveLeft G i ≈ 0 cases' hn with i hi case intro G : PGame inst✝ : Impartial G i : LeftMoves G hi : 0 ≤ moveLeft G i ⊢ ∃ i, moveLeft G i ≈ 0 exact ⟨i, (equiv_zero_iff_ge _).2 hi⟩ ** Qed
SetTheory.PGame.Impartial.exists_right_move_equiv_iff_fuzzy_zero G : PGame inst✝ : Impartial G ⊢ (∃ j, moveRight G j ≈ 0) ↔ G ‖ 0 refine' ⟨fun ⟨i, hi⟩ => (fuzzy_zero_iff_lf G).2 (lf_of_moveRight_le hi.1), fun hn => _⟩ G : PGame inst✝ : Impartial G hn : G ‖ 0 ⊢ ∃ j, moveRight G j ≈ 0 rw [fuzzy_zero_iff_lf G, lf_zero_le] at hn G : PGame inst✝ : Impartial G hn : ∃ j, moveRight G j ≤ 0 ⊢ ∃ j, moveRight G j ≈ 0 cases' hn with i hi case intro G : PGame inst✝ : Impartial G i : RightMoves G hi : moveRight G i ≤ 0 ⊢ ∃ j, moveRight G j ≈ 0 exact ⟨i, (equiv_zero_iff_le _).2 hi⟩ ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ IsSheaf F ↔ IsSheafOpensLeCover F refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _ C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ (∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) ↔ IsSheafOpensLeCover F constructor case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ (∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) → IsSheafOpensLeCover F intro h ι U case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι✝ : Type w U✝ : ι✝ → Opens ↑X Y : Opens ↑X hY : Y = iSup U✝ h : ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) ι : Type w U : ι → Opens ↑X ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone U)))) rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr] case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι✝ : Type w U✝ : ι✝ → Opens ↑X Y : Opens ↑X hY : Y = iSup U✝ h : ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) ι : Type w U : ι → Opens ↑X ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U (iSup U))).arrows)))) apply h case mp.a C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι✝ : Type w U✝ : ι✝ → Opens ↑X Y : Opens ↑X hY : Y = iSup U✝ h : ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) ι : Type w U : ι → Opens ↑X ⊢ Sieve.generate (presieveOfCoveringAux U (iSup U)) ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) (iSup U) apply presieveOfCovering.mem_grothendieckTopology case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ IsSheafOpensLeCover F → ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) intro h Y S case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y ⊢ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) rw [← Sieve.generate_sieve S] case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y ⊢ Sieve.generate S.arrows ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate S.arrows).arrows)))) intro hS case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y hS : Sieve.generate S.arrows ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate S.arrows).arrows)))) rw [← (isLimitOpensLeEquivGenerate₂ F S.1 hS).nonempty_congr] case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y hS : Sieve.generate S.arrows ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone (coveringOfPresieve Y S.arrows))))) apply h ** Qed
TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 ⊢ IsLocallySurjective T ↔ ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) constructor <;> intro hT case mp C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T ⊢ ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) intro x g case mp C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X g : (forget C).obj ((stalkFunctor C x).obj 𝒢) ⊢ ∃ a, ↑((stalkFunctor C x).map T) a = g obtain ⟨U, hxU, t, rfl⟩ := 𝒢.germ_exist x g case mp.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) ⊢ ∃ a, ↑((stalkFunctor C x).map T) a = ↑(germ 𝒢 { val := x, property := hxU }) t rcases hT U t x hxU with ⟨V, ι, ⟨s, h_eq⟩, hxV⟩ case mp.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) V : Opens ↑X ι : V ⟶ U hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) h_eq : ↑(T.app (op V)) s = ↑(𝒢.map ι.op) t ⊢ ∃ a, ↑((stalkFunctor C x).map T) a = ↑(germ 𝒢 { val := x, property := hxU }) t use ℱ.germ ⟨x, hxV⟩ s case h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) V : Opens ↑X ι : V ⟶ U hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) h_eq : ↑(T.app (op V)) s = ↑(𝒢.map ι.op) t ⊢ ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = ↑(germ 𝒢 { val := x, property := hxU }) t convert stalkFunctor_map_germ_apply V ⟨x, hxV⟩ T s using 1 case h.e'_3.h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) V : Opens ↑X ι : V ⟶ U hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) h_eq : ↑(T.app (op V)) s = ↑(𝒢.map ι.op) t e_1✝ : (forget C).obj ((stalkFunctor C x).obj 𝒢) = (forget C).obj (Limits.colim.obj ((OpenNhds.inclusion ↑{ val := x, property := hxV }).op ⋙ 𝒢)) ⊢ ↑(germ 𝒢 { val := x, property := hxU }) t = ↑(Limits.colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxV }).op ⋙ 𝒢) (op { obj := V, property := (_ : ↑{ val := x, property := hxV } ∈ V) })) (↑(T.app (op V)) s) simpa [h_eq] using (germ_res_apply 𝒢 ι ⟨x, hxV⟩ t).symm case mpr C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) ⊢ IsLocallySurjective T intro U t x hxU case mpr C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 set t_x := 𝒢.germ ⟨x, hxU⟩ t with ht_x case mpr C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 obtain ⟨s_x, hs_x : ((stalkFunctor C x).map T) s_x = t_x⟩ := hT x t_x case mpr.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t s_x : (forget C).obj ((stalkFunctor C x).obj ℱ) hs_x : ↑((stalkFunctor C x).map T) s_x = t_x ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 obtain ⟨V, hxV, s, rfl⟩ := ℱ.germ_exist x s_x case mpr.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 have key_W := 𝒢.germ_eq x hxV hxU (T.app _ s) t <\| by convert hs_x using 1 symm convert stalkFunctor_map_germ_apply _ _ _ s case mpr.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x key_W : ∃ W _m iU iV, ↑(𝒢.map iU.op) (↑(T.app (op V)) s) = ↑(𝒢.map iV.op) t ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 obtain ⟨W, hxW, hWV, hWU, h_eq⟩ := key_W case mpr.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x W : Opens ↑X hxW : x ∈ W hWV : W ⟶ V hWU : W ⟶ U h_eq : ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) = ↑(𝒢.map hWU.op) t ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 refine' ⟨W, hWU, ⟨ℱ.map hWV.op s, _⟩, hxW⟩ case mpr.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x W : Opens ↑X hxW : x ∈ W hWV : W ⟶ V hWU : W ⟶ U h_eq : ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) = ↑(𝒢.map hWU.op) t ⊢ ↑(T.app (op W)) (↑(ℱ.map hWV.op) s) = ↑(𝒢.map hWU.op) t convert h_eq using 1 case h.e'_2 C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x W : Opens ↑X hxW : x ∈ W hWV : W ⟶ V hWU : W ⟶ U h_eq : ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) = ↑(𝒢.map hWU.op) t ⊢ ↑(T.app (op W)) (↑(ℱ.map hWV.op) s) = ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) simp only [← comp_apply, T.naturality] C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x ⊢ ↑(germ 𝒢 { val := x, property := hxV }) (↑(T.app (op V)) s) = ↑(germ 𝒢 { val := x, property := hxU }) t convert hs_x using 1 case h.e'_2.h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x e_1✝ : (fun x_1 => (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxV })) (↑(T.app (op V)) s) = (fun x_1 => (forget C).obj ((stalkFunctor C x).obj 𝒢)) (↑(germ ℱ { val := x, property := hxV }) s) ⊢ ↑(germ 𝒢 { val := x, property := hxV }) (↑(T.app (op V)) s) = ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) symm case h.e'_2.h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x e_1✝ : (fun x_1 => (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxV })) (↑(T.app (op V)) s) = (fun x_1 => (forget C).obj ((stalkFunctor C x).obj 𝒢)) (↑(germ ℱ { val := x, property := hxV }) s) ⊢ ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = ↑(germ 𝒢 { val := x, property := hxV }) (↑(T.app (op V)) s) convert stalkFunctor_map_germ_apply _ _ _ s ** Qed
TopCat.Presheaf.isGluing_iff_pairwise C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) ⊢ IsGluing F U sf s ↔ ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i refine ⟨fun h ↦ ?_, fun h i ↦ h (op <\| Pairwise.single i)⟩ C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s ⊢ ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i rintro (i\|⟨i,j⟩) case mk.single C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s i : ι ⊢ (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app { unop := Pairwise.single i } s = objPairwiseOfFamily sf { unop := Pairwise.single i } exact h i case mk.pair C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s i j : ι ⊢ (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app { unop := Pairwise.pair i j } s = objPairwiseOfFamily sf { unop := Pairwise.pair i j } rw [← (F.mapCone (Pairwise.cocone U).op).w (op <\| Pairwise.Hom.left i j)] case mk.pair C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s i j : ι ⊢ ((F.mapCone (Cocone.op (Pairwise.cocone U))).π.app { unop := Pairwise.single i } ≫ ((Pairwise.diagram U).op ⋙ F).map (op (Pairwise.Hom.left i j))) s = objPairwiseOfFamily sf { unop := Pairwise.pair i j } exact congr_arg _ (h i) ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing_types C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X ⊢ IsSheaf F ↔ IsSheafUniqueGluing F simp_rw [isSheaf_iff_isSheafPairwiseIntersections, IsSheafPairwiseIntersections, Types.isLimit_iff, IsSheafUniqueGluing, isGluing_iff_pairwise] C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X ⊢ (∀ ⦃ι : Type x⦄ (U : ι → Opens ↑X) (s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j), s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) → ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i) ↔ ∀ ⦃ι : Type x⦄ (U : ι → Opens ↑X) (sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i)))), IsCompatible F U sf → ∃! s, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i refine forall₂_congr fun ι U ↦ ⟨fun h sf cpt ↦ ?_, fun h s hs ↦ ?_⟩ case refine_1 C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X h : ∀ (s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j), s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) → ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) cpt : IsCompatible F U sf ⊢ ∃! s, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i exact h _ cpt.sectionPairwise.prop case refine_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X h : ∀ (sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i)))), IsCompatible F U sf → ∃! s, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) ⊢ ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i specialize h (fun i ↦ s <\| op <\| Pairwise.single i) fun i j ↦ (hs <\| op <\| Pairwise.Hom.left i j).trans (hs <\| op <\| Pairwise.Hom.right i j).symm case refine_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i ⊢ ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i convert h case h.e'_2.h.h.h.h.e'_3.h.e C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i e_1✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt = (forget (Type u)).obj (F.obj (op (iSup U))) x✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt a✝ : (CategoryTheory.Pairwise ι)ᵒᵖ ⊢ s = objPairwiseOfFamily fun i => s (op (Pairwise.single i)) ext (i\|⟨i,j⟩) case h.e'_2.h.h.h.h.e'_3.h.e.h.mk.single C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i e_1✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt = (forget (Type u)).obj (F.obj (op (iSup U))) x✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt a✝ : (CategoryTheory.Pairwise ι)ᵒᵖ i : ι ⊢ s { unop := Pairwise.single i } = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) { unop := Pairwise.single i } rfl case h.e'_2.h.h.h.h.e'_3.h.e.h.mk.pair C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i e_1✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt = (forget (Type u)).obj (F.obj (op (iSup U))) x✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt a✝ : (CategoryTheory.Pairwise ι)ᵒᵖ i j : ι ⊢ s { unop := Pairwise.pair i j } = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) { unop := Pairwise.pair i j } exact (hs <\| op <\| Pairwise.Hom.left i j).symm ** Qed
TopCat.Sheaf.existsUnique_gluing' C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf ⊢ ∃! s, ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le) C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U ⊢ ∃! s, ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i obtain ⟨gl, gl_spec, gl_uniq⟩ := F.existsUnique_gluing U sf h case intro.intro C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ ∃! s, ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i refine' ⟨F.1.map (eqToHom V_eq_supr_U).op gl, _, _⟩ case intro.intro.refine'_1 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ (fun s => ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i) (↑(F.val.map (eqToHom V_eq_supr_U).op) gl) intro i case intro.intro.refine'_1 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map (iUV i).op) (↑(F.val.map (eqToHom V_eq_supr_U).op) gl) = sf i rw [← comp_apply, ← F.1.map_comp] case intro.intro.refine'_1 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map ((eqToHom V_eq_supr_U).op ≫ (iUV i).op)) gl = sf i exact gl_spec i case intro.intro.refine'_2 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op V))), (fun s => ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i) y → y = ↑(F.val.map (eqToHom V_eq_supr_U).op) gl intro gl' gl'_spec case intro.intro.refine'_2 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl gl' : (CategoryTheory.forget C).obj (F.val.obj (op V)) gl'_spec : ∀ (i : ι), ↑(F.val.map (iUV i).op) gl' = sf i ⊢ gl' = ↑(F.val.map (eqToHom V_eq_supr_U).op) gl convert congr_arg _ (gl_uniq (F.1.map (eqToHom V_eq_supr_U.symm).op gl') fun i => _) <;> rw [← comp_apply, ← F.1.map_comp] case h.e'_2 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl gl' : (CategoryTheory.forget C).obj (F.val.obj (op V)) gl'_spec : ∀ (i : ι), ↑(F.val.map (iUV i).op) gl' = sf i ⊢ gl' = ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (eqToHom V_eq_supr_U).op)) gl' rw [eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id, id_apply] case intro.intro.refine'_2.convert_3 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl gl' : (CategoryTheory.forget C).obj (F.val.obj (op V)) gl'_spec : ∀ (i : ι), ↑(F.val.map (iUV i).op) gl' = sf i i : ι ⊢ ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) gl' = sf i convert gl'_spec i ** Qed
TopCat.Sheaf.eq_of_locally_eq C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t ⊢ s = t let sf : ∀ i : ι, F.1.obj (op (U i)) := fun i => F.1.map (Opens.leSupr U i).op s C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s ⊢ s = t have sf_compatible : IsCompatible _ U sf := by intro i j simp_rw [← comp_apply, ← F.1.map_comp] rfl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf ⊢ s = t obtain ⟨gl, -, gl_uniq⟩ := F.existsUnique_gluing U sf sf_compatible case intro.intro C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ s = t trans gl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s ⊢ IsCompatible F.val U sf intro i j C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s i j : ι ⊢ ↑(F.val.map (infLELeft (U i) (U j)).op) (sf i) = ↑(F.val.map (infLERight (U i) (U j)).op) (sf j) simp_rw [← comp_apply, ← F.1.map_comp] C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s i j : ι ⊢ ↑(F.val.map ((leSupr U i).op ≫ (infLELeft (U i) (U j)).op)) s = ↑(F.val.map ((leSupr U j).op ≫ (infLERight (U i) (U j)).op)) s rfl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ s = gl apply gl_uniq case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ IsGluing F.val U sf s intro i case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map (leSupr U i).op) s = sf i rfl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ gl = t symm C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ t = gl apply gl_uniq case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ IsGluing F.val U sf t intro i case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map (leSupr U i).op) t = sf i rw [← h] ** Qed
TopCat.Sheaf.eq_of_locally_eq' C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t ⊢ s = t have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le) C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U ⊢ s = t suffices F.1.map (eqToHom V_eq_supr_U.symm).op s = F.1.map (eqToHom V_eq_supr_U.symm).op t by convert congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;> rw [← comp_apply, ← F.1.map_comp, eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id, id_apply] C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U ⊢ ↑(F.val.map (eqToHom (_ : iSup U = V)).op) s = ↑(F.val.map (eqToHom (_ : iSup U = V)).op) t apply eq_of_locally_eq case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U ⊢ ∀ (i : ι), ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) s) = ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) t) intro i case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U i : ι ⊢ ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) s) = ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) t) rw [← comp_apply, ← comp_apply, ← F.1.map_comp] case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U i : ι ⊢ ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) s = ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) t convert h i C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U this : ↑(F.val.map (eqToHom (_ : iSup U = V)).op) s = ↑(F.val.map (eqToHom (_ : iSup U = V)).op) t ⊢ s = t convert congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;> rw [← comp_apply, ← F.1.map_comp, eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id, id_apply] ** Qed
TopCat.Sheaf.eq_of_locally_eq₂ C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ s = t fapply F.eq_of_locally_eq' fun t : ULift Bool => if t.1 then U₁ else U₂ case iUV C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ (i : ULift.{v, 0} Bool) → (if i.down = true then U₁ else U₂) ⟶ V exact fun i => if h : i.1 then eqToHom (if_pos h) ≫ i₁ else eqToHom (if_neg h) ≫ i₂ case hcover C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ V ≤ ⨆ t, if t.down = true then U₁ else U₂ refine' le_trans hcover _ case hcover C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ U₁ ⊔ U₂ ≤ ⨆ t, if t.down = true then U₁ else U₂ rw [sup_le_iff] case hcover C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ (U₁ ≤ ⨆ t, if t.down = true then U₁ else U₂) ∧ U₂ ≤ ⨆ t, if t.down = true then U₁ else U₂ constructor case hcover.left C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ U₁ ≤ ⨆ t, if t.down = true then U₁ else U₂ convert le_iSup (fun t : ULift Bool => if t.1 then U₁ else U₂) (ULift.up true) case hcover.right C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ U₂ ≤ ⨆ t, if t.down = true then U₁ else U₂ convert le_iSup (fun t : ULift Bool => if t.1 then U₁ else U₂) (ULift.up false) case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ∀ (i : ULift.{v, 0} Bool), ↑(F.val.map (if h : i.down = true then eqToHom (_ : (if i.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if i.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : i.down = true then eqToHom (_ : (if i.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if i.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t rintro ⟨_ \| _⟩ case h.up.false C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t case h.up.true C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t any_goals exact h₁ case h.up.false C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t any_goals exact h₂ case h.up.true C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t exact h₁ case h.up.false C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t exact h₂ ** Qed
TopCat.Presheaf.SheafConditionEqualizerProducts.res_π C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X i : ι ⊢ res F U ≫ limit.π (Discrete.functor fun i => F.obj (op (U i))) { as := i } = F.map (leSupr U i).op rw [res, limit.lift_π, Fan.mk_π_app] ** Qed
TopCat.Presheaf.SheafConditionEqualizerProducts.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X ⊢ res F U ≫ leftRes F U = res F U ≫ rightRes F U dsimp [res, leftRes, rightRes] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X ⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.1 ≫ F.map (infLELeft (U p.1) (U p.2)).op) = (Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.2 ≫ F.map (infLERight (U p.1) (U p.2)).op refine limit.hom_ext (fun _ => ?_) C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.1 ≫ F.map (infLELeft (U p.1) (U p.2)).op) ≫ limit.π (Discrete.functor fun p => F.obj (op (U p.1 ⊓ U p.2))) x✝ = ((Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.2 ≫ F.map (infLERight (U p.1) (U p.2)).op) ≫ limit.π (Discrete.functor fun p => F.obj (op (U p.1 ⊓ U p.2))) x✝ simp only [limit.lift_π, limit.lift_π_assoc, Fan.mk_π_app, Category.assoc] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ F.map (leSupr U x✝.as.1).op ≫ F.map (infLELeft (U x✝.as.1) (U x✝.as.2)).op = F.map (leSupr U x✝.as.2).op ≫ F.map (infLERight (U x✝.as.1) (U x✝.as.2)).op rw [← F.map_comp] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ F.map ((leSupr U x✝.as.1).op ≫ (infLELeft (U x✝.as.1) (U x✝.as.2)).op) = F.map (leSupr U x✝.as.2).op ≫ F.map (infLERight (U x✝.as.1) (U x✝.as.2)).op rw [← F.map_comp] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ F.map ((leSupr U x✝.as.1).op ≫ (infLELeft (U x✝.as.1) (U x✝.as.2)).op) = F.map ((leSupr U x✝.as.2).op ≫ (infLERight (U x✝.as.1) (U x✝.as.2)).op) congr 1 ** Qed
SetTheory.PGame.Domineering.fst_pred_mem_erase_of_mem_right b : Board m : ℤ × ℤ h : m ∈ right b ⊢ (m.1 - 1, m.2) ∈ Finset.erase b m rw [mem_right] at h b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1 - 1, m.2) ∈ b ⊢ (m.1 - 1, m.2) ∈ Finset.erase b m apply Finset.mem_erase_of_ne_of_mem _ h.2 b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1 - 1, m.2) ∈ b ⊢ (m.1 - 1, m.2) ≠ m exact ne_of_apply_ne Prod.fst (pred_ne_self m.1) ** Qed
SetTheory.PGame.Domineering.snd_pred_mem_erase_of_mem_left b : Board m : ℤ × ℤ h : m ∈ left b ⊢ (m.1, m.2 - 1) ∈ Finset.erase b m rw [mem_left] at h b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1, m.2 - 1) ∈ b ⊢ (m.1, m.2 - 1) ∈ Finset.erase b m apply Finset.mem_erase_of_ne_of_mem _ h.2 b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1, m.2 - 1) ∈ b ⊢ (m.1, m.2 - 1) ≠ m exact ne_of_apply_ne Prod.snd (pred_ne_self m.2) ** Qed
SetTheory.PGame.Domineering.card_of_mem_left b : Board m : ℤ × ℤ h : m ∈ left b ⊢ 2 ≤ Finset.card b have w₁ : m ∈ b := (Finset.mem_inter.1 h).1 b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b ⊢ 2 ≤ Finset.card b have w₂ : (m.1, m.2 - 1) ∈ b.erase m := snd_pred_mem_erase_of_mem_left h b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b w₂ : (m.1, m.2 - 1) ∈ Finset.erase b m ⊢ 2 ≤ Finset.card b have i₁ := Finset.card_erase_lt_of_mem w₁ b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b w₂ : (m.1, m.2 - 1) ∈ Finset.erase b m i₁ : Finset.card (Finset.erase b m) < Finset.card b ⊢ 2 ≤ Finset.card b have i₂ := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w₂) b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b w₂ : (m.1, m.2 - 1) ∈ Finset.erase b m i₁ : Finset.card (Finset.erase b m) < Finset.card b i₂ : 0 < Finset.card (Finset.erase b m) ⊢ 2 ≤ Finset.card b exact Nat.lt_of_le_of_lt i₂ i₁ ** Qed

Subsets and Splits

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TopologicalSpace.Opens.map_eq X Y Z : TopCat f g : X ⟶ Y h : f = g ⊢ map f = map g subst h X Y Z : TopCat f : X ⟶ Y ⊢ map f = map f rfl ** Qed
TopologicalSpace.Opens.mapIso_hom_app X Y Z : TopCat f g : X ⟶ Y h : f = g U : Opens ↑Y ⊢ (map f).obj U = (map g).obj U rw [h] ** Qed
TopologicalSpace.Opens.mapIso_inv_app X Y Z : TopCat f g : X ⟶ Y h : f = g U : Opens ↑Y ⊢ (map g).obj U = (map f).obj U rw [h] ** Qed
TopologicalSpace.Opens.openEmbedding_obj_top X : TopCat U : Opens ↑X ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ⊤ = U ext1 case h X : TopCat U : Opens ↑X ⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ⊤) = ↑U exact Set.image_univ.trans Subtype.range_coe ** Qed
TopologicalSpace.Opens.inclusion_map_eq_top X : TopCat U : Opens ↑X ⊢ (map (inclusion U)).obj U = ⊤ ext1 case h X : TopCat U : Opens ↑X ⊢ ↑((map (inclusion U)).obj U) = ↑⊤ exact Subtype.coe_preimage_self _ ** Qed
TopologicalSpace.Opens.adjunction_counit_app_self X : TopCat U : Opens ↑X ⊢ (map (inclusion U) ⋙ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj U = (𝟭 (Opens ↑X)).obj U simp ** Qed
TopologicalSpace.Opens.inclusion_top_functor X : TopCat ⊢ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤)) = map (inclusionTopIso X).inv refine' CategoryTheory.Functor.ext _ _ case refine'_1 X : TopCat ⊢ ∀ (X_1 : Opens ↑((toTopCat X).obj ⊤)), (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj X_1 = (map (inclusionTopIso X).inv).obj X_1 intro U case refine'_1 X : TopCat U : Opens ↑((toTopCat X).obj ⊤) ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj U = (map (inclusionTopIso X).inv).obj U ext x case refine'_1.h.h X : TopCat U : Opens ↑((toTopCat X).obj ⊤) x : ↑X ⊢ x ∈ ↑((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj U) ↔ x ∈ ↑((map (inclusionTopIso X).inv).obj U) exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩ case refine'_2 X : TopCat ⊢ ∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y), (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).map f = eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj X_1 = (map (inclusionTopIso X).inv).obj X_1) ≫ (map (inclusionTopIso X).inv).map f ≫ eqToHom (_ : (map (inclusionTopIso X).inv).obj Y = (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj Y) intros U V f case refine'_2 X : TopCat U V : Opens ↑((toTopCat X).obj ⊤) f : U ⟶ V ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).map f = eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj U = (map (inclusionTopIso X).inv).obj U) ≫ (map (inclusionTopIso X).inv).map f ≫ eqToHom (_ : (map (inclusionTopIso X).inv).obj V = (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion ⊤))).obj V) apply Subsingleton.elim ** Qed
TopologicalSpace.Opens.functor_obj_map_obj X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y ⊢ (IsOpenMap.functor hf).obj ((map f).obj U) = (IsOpenMap.functor hf).obj ⊤ ⊓ U ext case h.h X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x✝ : ↑Y ⊢ x✝ ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) ↔ x✝ ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) constructor case h.h.mp X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x✝ : ↑Y ⊢ x✝ ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) → x✝ ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) rintro ⟨x, hx, rfl⟩ case h.h.mp.intro.intro X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x : (forget TopCat).obj X hx : x ∈ ↑((map f).obj U) ⊢ ↑f x ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) exact ⟨⟨x, trivial, rfl⟩, hx⟩ case h.h.mpr X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x✝ : ↑Y ⊢ x✝ ∈ ↑((IsOpenMap.functor hf).obj ⊤ ⊓ U) → x✝ ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) rintro ⟨⟨x, -, rfl⟩, hx⟩ case h.h.mpr.intro.intro.intro X Y : TopCat f : X ⟶ Y hf : IsOpenMap ↑f U : Opens ↑Y x : (forget TopCat).obj X hx : ↑f x ∈ ↑U ⊢ ↑f x ∈ ↑((IsOpenMap.functor hf).obj ((map f).obj U)) exact ⟨x, hx, rfl⟩ ** Qed
TopologicalSpace.Opens.functor_map_eq_inf X : TopCat U V : Opens ↑X ⊢ (IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((map (inclusion U)).obj V) = V ⊓ U ext1 case h X : TopCat U V : Opens ↑X ⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((map (inclusion U)).obj V)) = ↑(V ⊓ U) refine' Set.image_preimage_eq_inter_range.trans _ case h X : TopCat U V : Opens ↑X ⊢ (V.1 ∩ Set.range fun x => ↑(inclusion U) x) = ↑(V ⊓ U) erw [set_range_forget_map_inclusion U] case h X : TopCat U V : Opens ↑X ⊢ V.1 ∩ ↑U = ↑(V ⊓ U) rfl ** Qed
TopologicalSpace.Opens.adjunction_counit_map_functor X : TopCat U : Opens ↑X V : Opens { x // x ∈ U } ⊢ (map (inclusion U) ⋙ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) = (𝟭 (Opens ↑X)).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) dsimp X : TopCat U : Opens ↑X V : Opens { x // x ∈ U } ⊢ (IsOpenMap.functor (_ : IsOpenMap Subtype.val)).obj ((map (inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap Subtype.val)).obj V)) = (IsOpenMap.functor (_ : IsOpenMap Subtype.val)).obj V rw [map_functor_eq V] X : TopCat U : Opens ↑X V : Opens { x // x ∈ U } ⊢ (IsOpenMap.adjunction (_ : IsOpenMap ↑(inclusion U))).counit.app ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) = eqToHom (_ : (map (inclusion U) ⋙ IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V) = (𝟭 (Opens ↑X)).obj ((IsOpenMap.functor (_ : IsOpenMap ↑(inclusion U))).obj V)) apply Subsingleton.elim ** Qed
Pretrivialization.continuousOn_continuousLinearMapCoordChange ** 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₁ := (compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)).continuous 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₂ := (ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)).continuous 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₃ := continuousOn_coordChange 𝕜₁ e₁' e₁ 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) have h₄ := continuousOn_coordChange 𝕜₂ e₂ e₂' 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) refine' ((h₁.comp_continuousOn (h₄.mono _)).clm_comp (h₂.comp_continuousOn (h₃.mono _))).congr _ case refine'_1 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₂.baseSet ∩ e₂'.baseSet mfld_set_tac case refine'_2 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₁'.baseSet ∩ e₁.baseSet mfld_set_tac case refine'_3 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) ⊢ EqOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (fun x => comp ((↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) x) ((↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) x)) (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) intro b _ case refine'_3 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) b : B a✝ : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊢ continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b = (fun x => comp ((↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) x) ((↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) x)) b ext L v case refine'_3.h.h 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) b : B a✝ : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(↑(continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L) v = ↑(↑((fun x => comp ((↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) x) ((↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) ∘ fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) x)) b) L) v dsimp [continuousLinearMapCoordChange] case refine'_3.h.h 𝕜₁ : Type u_1 inst✝²² : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝²¹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝²⁰ : NormedAddCommGroup F₁ inst✝¹⁹ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹⁵ : NormedAddCommGroup F₂ inst✝¹⁴ : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹³ : (x : B) → AddCommGroup (E₂ x) inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x) inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝¹⁰ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁹ : (x : B) → TopologicalSpace (E₁ x) inst✝⁸ : FiberBundle F₁ E₁ inst✝⁷ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁶ : FiberBundle F₂ E₂ inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁ inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂ inst✝³ : MemTrivializationAtlas e₁ inst✝² : MemTrivializationAtlas e₁' inst✝¹ : MemTrivializationAtlas e₂ inst✝ : MemTrivializationAtlas e₂' h₁ : Continuous ↑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) h₂ : Continuous ↑(ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)) h₃ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet) h₄ : ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet) b : B a✝ : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (↑L (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (Trivialization.coordChangeL 𝕜₁ e₁' e₁ b))) v)) = ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (↑L (↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b) v)) rw [ContinuousLinearEquiv.symm_symm] Qed
Pretrivialization.continuousLinearMap_symm_apply' ** 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L = comp (Trivialization.symmL 𝕜₂ e₂ b) (comp L (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b)) rw [symm_apply] 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ cast (_ : Bundle.ContinuousLinearMap σ E₁ E₂ (↑(LocalEquiv.symm (continuousLinearMap σ e₁ e₂).toLocalEquiv) (b, L)).proj = Bundle.ContinuousLinearMap σ E₁ E₂ b) (↑(LocalEquiv.symm (continuousLinearMap σ e₁ e₂).toLocalEquiv) (b, L)).snd = comp (Trivialization.symmL 𝕜₂ e₂ b) (comp L (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b)) case hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ b ∈ (continuousLinearMap σ e₁ e₂).baseSet rfl case hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet L : F₁ →SL[σ] F₂ ⊢ b ∈ (continuousLinearMap σ e₁ e₂).baseSet exact hb Qed
Pretrivialization.continuousLinearMapCoordChange_apply ** 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ ⊢ ↑(continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L = (↑(continuousLinearMap σ e₁' e₂') { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }).2 ext v case h 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(↑(continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L) v = ↑(↑(continuousLinearMap σ e₁' e₂') { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }).2 v simp_rw [continuousLinearMapCoordChange, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.arrowCongrSL_apply, continuousLinearMap_apply, continuousLinearMap_symm_apply' σ e₁ e₂ hb.1, comp_apply, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.symm_symm, Trivialization.continuousLinearMapAt_apply, Trivialization.symmL_apply] case h 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (↑L (↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b) v)) = ↑(Trivialization.linearMapAt 𝕜₂ e₂' { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }.proj) (Trivialization.symm e₂ b (↑L (↑(Trivialization.linearMapAt 𝕜₁ e₁ b) (Trivialization.symm e₁' b v)))) rw [e₂.coordChangeL_apply e₂', e₁'.coordChangeL_apply e₁, e₁.coe_linearMapAt_of_mem hb.1.1, e₂'.coe_linearMapAt_of_mem hb.2.2] case h.hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ b ∈ e₁'.baseSet ∩ e₁.baseSet case h.hb 𝕜₁ : Type u_1 inst✝²⁰ : NontriviallyNormedField 𝕜₁ 𝕜₂ : Type u_2 inst✝¹⁹ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ iσ : RingHomIsometric σ B : Type u_3 F₁ : Type u_4 inst✝¹⁸ : NormedAddCommGroup F₁ inst✝¹⁷ : NormedSpace 𝕜₁ F₁ E₁ : B → Type u_5 inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x) inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x) inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_6 inst✝¹³ : NormedAddCommGroup F₂ inst✝¹² : NormedSpace 𝕜₂ F₂ E₂ : B → Type u_7 inst✝¹¹ : (x : B) → AddCommGroup (E₂ x) inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x) inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁸ : TopologicalSpace B e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝⁷ : (x : B) → TopologicalSpace (E₁ x) inst✝⁶ : FiberBundle F₁ E₁ inst✝⁵ : (x : B) → TopologicalSpace (E₂ x) ita : ∀ (x : B), TopologicalAddGroup (E₂ x) inst✝⁴ : FiberBundle F₂ E₂ inst✝³ : Trivialization.IsLinear 𝕜₁ e₁ inst✝² : Trivialization.IsLinear 𝕜₁ e₁' inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂ inst✝ : Trivialization.IsLinear 𝕜₂ e₂' b : B hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) L : F₁ →SL[σ] F₂ v : F₁ ⊢ b ∈ e₂.baseSet ∩ e₂'.baseSet exacts [⟨hb.2.1, hb.1.1⟩, ⟨hb.1.2, hb.2.2⟩] Qed
FiberBundle.totalSpaceMk_closedEmbedding ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : TopologicalSpace B inst✝⁴ : TopologicalSpace F E : B → Type u_5 inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (b : B) → TopologicalSpace (E b) inst✝¹ : FiberBundle F E inst✝ : T1Space B x : B ⊢ IsClosed (range (TotalSpace.mk x)) rw [TotalSpace.range_mk] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : TopologicalSpace B inst✝⁴ : TopologicalSpace F E : B → Type u_5 inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (b : B) → TopologicalSpace (E b) inst✝¹ : FiberBundle F E inst✝ : T1Space B x : B ⊢ IsClosed (TotalSpace.proj ⁻¹' {x}) exact isClosed_singleton.preimage <\| continuous_proj F E ** Qed
FiberBundle.exists_trivialization_Icc_subset ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ⊢ ∃ e, Icc a b ⊆ e.baseSet obtain ⟨ea, hea⟩ : ∃ ea : Trivialization F (π F E), a ∈ ea.baseSet := ⟨trivializationAt F E a, mem_baseSet_trivializationAt F E a⟩ case intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet cases' lt_or_le b a with hab hab case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b ⊢ ∃ e, Icc a b ⊆ e.baseSet set s : Set B := { x ∈ Icc a b \| ∃ e : Trivialization F (π F E), Icc a x ⊆ e.baseSet } case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ⊢ ∃ e, Icc a b ⊆ e.baseSet have ha : a ∈ s := ⟨left_mem_Icc.2 hab, ea, by simp [hea]⟩ case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s ⊢ ∃ e, Icc a b ⊆ e.baseSet have sne : s.Nonempty := ⟨a, ha⟩ case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s ⊢ ∃ e, Icc a b ⊆ e.baseSet have hsb : b ∈ upperBounds s := fun x hx => hx.1.2 case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s ⊢ ∃ e, Icc a b ⊆ e.baseSet have sbd : BddAbove s := ⟨b, hsb⟩ case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s ⊢ ∃ e, Icc a b ⊆ e.baseSet set c := sSup s case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s ⊢ ∃ e, Icc a b ⊆ e.baseSet have hsc : IsLUB s c := isLUB_csSup sne sbd case intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c ⊢ ∃ e, Icc a b ⊆ e.baseSet have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩ case intro.inr.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet cases' hc.2.eq_or_lt with heq hlt case intro.inr.intro.intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b ⊢ ∃ e, Icc a b ⊆ e.baseSet suffices : ∃ d ∈ Ioc c b, ∃ e : Trivialization F (π F E), Icc a d ⊆ e.baseSet case this ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet obtain ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, Ico c d ⊆ ec.baseSet := (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1 (mem_nhdsWithin_of_mem_nhds <\| IsOpen.mem_nhds ec.open_baseSet (hec ⟨hc.1, le_rfl⟩)) case this.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet have had : Ico a d ⊆ ec.baseSet := Ico_subset_Icc_union_Ico.trans (union_subset hec hd) case this.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet by_cases he : Disjoint (Iio d) (Ioi c) case intro.inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : b < a ⊢ ∃ e, Icc a b ⊆ e.baseSet exact ⟨ea, by simp []⟩ * ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : b < a ⊢ Icc a b ⊆ ea.baseSet ** simp [] * ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ⊢ Icc a a ⊆ ea.baseSet simp [hea] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ⊢ c ∈ s cases' hc.1.eq_or_lt with heq hlt case inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b hlt : a < c ⊢ c ∈ s refine ⟨hc, ?_⟩ case inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b hlt : a < c ⊢ ∃ e, Icc a c ⊆ e.baseSet obtain ⟨ec, hc⟩ : ∃ ec : Trivialization F (π F E), c ∈ ec.baseSet := ⟨trivializationAt F E c, mem_baseSet_trivializationAt F E c⟩ case inr.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet ⊢ ∃ e, Icc a c ⊆ e.baseSet obtain ⟨c', hc', hc'e⟩ : ∃ c' ∈ Ico a c, Ioc c' c ⊆ ec.baseSet := (mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset hlt).1 (mem_nhdsWithin_of_mem_nhds <\| IsOpen.mem_nhds ec.open_baseSet hc) case inr.intro.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet c' : B hc' : c' ∈ Ico a c hc'e : Ioc c' c ⊆ ec.baseSet ⊢ ∃ e, Icc a c ⊆ e.baseSet obtain ⟨d, ⟨hdab, ead, had⟩, hd⟩ : ∃ d ∈ s, d ∈ Ioc c' c := hsc.exists_between hc'.2 case inr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet c' : B hc' : c' ∈ Ico a c hc'e : Ioc c' c ⊆ ec.baseSet d : B hd : d ∈ Ioc c' c hdab : d ∈ Icc a b ead : Trivialization F TotalSpace.proj had : Icc a d ⊆ ead.baseSet ⊢ ∃ e, Icc a c ⊆ e.baseSet refine' ⟨ead.piecewiseLe ec d (had ⟨hdab.1, le_rfl⟩) (hc'e hd), subset_ite.2 _⟩ case inr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc✝ : c ∈ Icc a b hlt : a < c ec : Trivialization F TotalSpace.proj hc : c ∈ ec.baseSet c' : B hc' : c' ∈ Ico a c hc'e : Ioc c' c ⊆ ec.baseSet d : B hd : d ∈ Ioc c' c hdab : d ∈ Icc a b ead : Trivialization F TotalSpace.proj had : Icc a d ⊆ ead.baseSet ⊢ Icc a c ∩ Iic d ⊆ ead.baseSet ∧ Icc a c \ Iic d ⊆ (Trivialization.transFiberHomeomorph ec (Trivialization.coordChangeHomeomorph ec ead (_ : d ∈ ec.baseSet) (_ : d ∈ ead.baseSet))).baseSet refine' ⟨fun x hx => had ⟨hx.1.1, hx.2⟩, fun x hx => hc'e ⟨hd.1.trans (not_le.1 hx.2), hx.1.2⟩⟩ case inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b heq : a = c ⊢ c ∈ s rwa [← heq] case intro.inr.intro.intro.inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet heq : c = b ⊢ ∃ e, Icc a b ⊆ e.baseSet exact ⟨ec, heq ▸ hec⟩ case intro.inr.intro.intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b this : ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet rcases this with ⟨d, hdcb, hd⟩ case intro.inr.intro.intro.inr.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : ∃ e, Icc a d ⊆ e.baseSet ⊢ ∃ e, Icc a b ⊆ e.baseSet exact ((hsc.1 ⟨⟨hc.1.trans hdcb.1.le, hdcb.2⟩, hd⟩).not_lt hdcb.1).elim case pos ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet obtain ⟨ed, hed⟩ : ∃ ed : Trivialization F (π F E), d ∈ ed.baseSet := ⟨trivializationAt F E d, mem_baseSet_trivializationAt F E d⟩ case pos.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ed : Trivialization F TotalSpace.proj hed : d ∈ ed.baseSet ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet refine' ⟨d, hdcb, (ec.restrOpen (Iio d) isOpen_Iio).disjointUnion (ed.restrOpen (Ioi c) isOpen_Ioi) (he.mono (inter_subset_right _ _) (inter_subset_right _ _)), fun x hx => _⟩ case pos.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ed : Trivialization F TotalSpace.proj hed : d ∈ ed.baseSet x : B hx : x ∈ Icc a d ⊢ x ∈ (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))) (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))) (_ : Disjoint (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))).baseSet (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet rcases hx.2.eq_or_lt with (rfl \| hxd) case pos.intro.inl ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b ed : Trivialization F TotalSpace.proj x : B hdcb : x ∈ Ioc c b hd : Ico c x ⊆ ec.baseSet had : Ico a x ⊆ ec.baseSet he : Disjoint (Iio x) (Ioi c) hed : x ∈ ed.baseSet hx : x ∈ Icc a x ⊢ x ∈ (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x))) (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))) (_ : Disjoint (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x))).baseSet (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet case pos.intro.inr ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Disjoint (Iio d) (Ioi c) ed : Trivialization F TotalSpace.proj hed : d ∈ ed.baseSet x : B hx : x ∈ Icc a d hxd : x < d ⊢ x ∈ (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))) (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))) (_ : Disjoint (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))).baseSet (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet exacts [Or.inr ⟨hed, hdcb.1⟩, Or.inl ⟨had ⟨hx.1, hxd⟩, hxd⟩] case neg ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : ¬Disjoint (Iio d) (Ioi c) ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet rw [disjoint_left] at he case neg ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : ¬∀ ⦃a : B⦄, a ∈ Iio d → ¬a ∈ Ioi c ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet push_neg at he case neg ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet he : Exists fun ⦃a⦄ => a ∈ Iio d ∧ a ∈ Ioi c ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet rcases he with ⟨d', hdd' : d' < d, hd'c⟩ case neg.intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝⁷ : TopologicalSpace X inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace F E : B → Type u_5 inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (b : B) → TopologicalSpace (E b) inst✝² : ConditionallyCompleteLinearOrder B inst✝¹ : OrderTopology B inst✝ : FiberBundle F E a b : B ea : Trivialization F TotalSpace.proj hea : a ∈ ea.baseSet hab : a ≤ b s : Set B := {x \| x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet} ha : a ∈ s sne : Set.Nonempty s hsb : b ∈ upperBounds s sbd : BddAbove s c : B := sSup s hsc : IsLUB s c hc : c ∈ Icc a b ec : Trivialization F TotalSpace.proj hec : Icc a c ⊆ ec.baseSet hlt : c < b d : B hdcb : d ∈ Ioc c b hd : Ico c d ⊆ ec.baseSet had : Ico a d ⊆ ec.baseSet d' : B hdd' : d' < d hd'c : d' ∈ Ioi c ⊢ ∃ d, d ∈ Ioc c b ∧ ∃ e, Icc a d ⊆ e.baseSet exact ⟨d', ⟨hd'c, hdd'.le.trans hdcb.2⟩, ec, (Icc_subset_Ico_right hdd').trans had⟩ Qed
FiberBundleCore.mem_trivChange_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i j : ι p : B × F ⊢ p ∈ (trivChange Z i j).toLocalEquiv.source ↔ p.1 ∈ baseSet Z i ∩ baseSet Z j erw [mem_prod] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i j : ι p : B × F ⊢ p.1 ∈ baseSet Z i ∩ baseSet Z j ∧ p.2 ∈ univ ↔ p.1 ∈ baseSet Z i ∩ baseSet Z j simp ** Qed
FiberBundleCore.mem_localTrivAsLocalEquiv_target ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι p : B × F ⊢ p ∈ (localTrivAsLocalEquiv Z i).target ↔ p.1 ∈ baseSet Z i erw [mem_prod] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι p : B × F ⊢ p.1 ∈ baseSet Z i ∧ p.2 ∈ univ ↔ p.1 ∈ baseSet Z i simp only [and_true_iff, mem_univ] ** Qed
FiberBundleCore.localTrivAsLocalEquiv_trans ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι ⊢ LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j) ≈ (trivChange Z i j).toLocalEquiv constructor case left ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι ⊢ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source = (trivChange Z i j).toLocalEquiv.source ext x case left.h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι x : B × F ⊢ x ∈ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source ↔ x ∈ (trivChange Z i j).toLocalEquiv.source simp only [mem_localTrivAsLocalEquiv_target, mfld_simps] case left.h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι x : B × F ⊢ x.1 ∈ baseSet Z i ∧ ↑(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) x ∈ (localTrivAsLocalEquiv Z j).source ↔ x.1 ∈ baseSet Z i ∧ x.1 ∈ baseSet Z j rfl case right ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι ⊢ EqOn (↑(LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j))) (↑(trivChange Z i j).toLocalEquiv) (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source rintro ⟨x, v⟩ hx case right.mk ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ i j : ι x : B v : F hx : x ∈ baseSet Z i ∧ x ∈ baseSet Z j ⊢ coordChange Z (indexAt Z x) j x (coordChange Z i (indexAt Z x) x v) = coordChange Z i j x v simp only [Z.coordChange_comp, hx, mem_inter_iff, and_self_iff, mem_baseSet_at] ** Qed
FiberBundleCore.open_source' ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ IsOpen (localTrivAsLocalEquiv Z i).source apply TopologicalSpace.GenerateOpen.basic case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ (localTrivAsLocalEquiv Z i).source ∈ ⋃ i, ⋃ s, ⋃ (_ : IsOpen s), {(localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' s} simp only [exists_prop, mem_iUnion, mem_singleton_iff] case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ ∃ i_1 i_2, IsOpen i_2 ∧ (localTrivAsLocalEquiv Z i).source = (localTrivAsLocalEquiv Z i_1).source ∩ ↑(localTrivAsLocalEquiv Z i_1) ⁻¹' i_2 refine ⟨i, Z.baseSet i ×ˢ univ, (Z.isOpen_baseSet i).prod isOpen_univ, ?_⟩ case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι ⊢ (localTrivAsLocalEquiv Z i).source = (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ ext p case a.h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i✝ : ι b : B a : F i : ι p : TotalSpace Z ⊢ p ∈ (localTrivAsLocalEquiv Z i).source ↔ p ∈ (localTrivAsLocalEquiv Z i).source ∩ ↑(localTrivAsLocalEquiv Z i) ⁻¹' baseSet Z i ×ˢ univ simp only [localTrivAsLocalEquiv_apply, prod_mk_mem_set_prod_eq, mem_inter_iff, and_self_iff, mem_localTrivAsLocalEquiv_source, and_true, mem_univ, mem_preimage] ** Qed
FiberBundleCore.continuous_const_section ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v ⊢ Continuous (let_fun this := fun x => { proj := x, snd := v }; this) refine continuous_iff_continuousAt.2 fun x => ?_ ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B ⊢ ContinuousAt (let_fun this := fun x => { proj := x, snd := v }; this) x have A : Z.baseSet (Z.indexAt x) ∈ 𝓝 x := IsOpen.mem_nhds (Z.isOpen_baseSet (Z.indexAt x)) (Z.mem_baseSet_at x) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ ContinuousAt (let_fun this := fun x => { proj := x, snd := v }; this) x refine ((Z.localTrivAt x).toLocalHomeomorph.continuousAt_iff_continuousAt_comp_left ?_).2 ?_ case refine_1 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ (let_fun this := fun x => { proj := x, snd := v }; this) ⁻¹' (localTrivAt Z x).toLocalHomeomorph.toLocalEquiv.source ∈ 𝓝 x exact A case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ ContinuousAt (↑(localTrivAt Z x).toLocalHomeomorph ∘ let_fun this := fun x => { proj := x, snd := v }; this) x apply continuousAt_id.prod case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x ⊢ ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x have : ContinuousOn (fun _ : B => v) (Z.baseSet (Z.indexAt x)) := continuousOn_const case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x this : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x)) ⊢ ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x refine (this.congr fun y hy ↦ ?_).continuousAt A case refine_2 ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a v : F h : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v x : B A : baseSet Z (indexAt Z x) ∈ 𝓝 x this : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x)) y : B hy : y ∈ baseSet Z (indexAt Z x) ⊢ coordChange Z (indexAt Z y) (indexAt Z x) y v = v exact h _ _ _ ⟨mem_baseSet_at _ _, hy⟩ ** Qed
FiberBundleCore.localTrivAt_apply ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a : F p : TotalSpace Z ⊢ ↑(localTrivAt Z p.proj) p = (p.proj, p.snd) rw [localTrivAt, localTriv_apply, coordChange_self] case a ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a : F p : TotalSpace Z ⊢ p.proj ∈ baseSet Z (indexAt Z p.proj) exact Z.mem_baseSet_at p.1 ** Qed
FiberBundleCore.mem_localTrivAt_baseSet ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b✝ : B a : F b : B ⊢ b ∈ (localTrivAt Z b).baseSet rw [localTrivAt, ← baseSet_at] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b✝ : B a : F b : B ⊢ b ∈ baseSet Z (indexAt Z b) exact Z.mem_baseSet_at b ** Qed
FiberBundleCore.mk_mem_localTrivAt_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F Z : FiberBundleCore ι B F i : ι b : B a : F ⊢ { proj := b, snd := a } ∈ (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source simp only [mfld_simps] ** Qed
FiberPrebundle.continuous_symm_of_mem_pretrivializationAtlas ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas ⊢ ContinuousOn (↑(LocalEquiv.symm e.toLocalEquiv)) e.target refine' fun z H U h => preimage_nhdsWithin_coinduced' H (le_def.1 (nhds_mono _) U h) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas z : B × F H : z ∈ e.target U : Set (TotalSpace F E) h : U ∈ 𝓝 (↑(LocalEquiv.symm e.toLocalEquiv) z) ⊢ coinduced (fun x => ↑(LocalEquiv.symm e.toLocalEquiv) ↑x) inferInstance ≤ totalSpaceTopology a exact le_iSup₂ (α := TopologicalSpace (TotalSpace F E)) e he ** Qed
FiberPrebundle.isOpen_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e : Pretrivialization F TotalSpace.proj ⊢ IsOpen e.source refine isOpen_iSup_iff.mpr fun e' => isOpen_iSup_iff.mpr fun _ => ?_ ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj x✝ : e' ∈ a.pretrivializationAtlas ⊢ IsOpen e.source refine' isOpen_coinduced.mpr (isOpen_induced_iff.mpr ⟨e.target, e.open_target, _⟩) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj x✝ : e' ∈ a.pretrivializationAtlas ⊢ Subtype.val ⁻¹' e.target = Pretrivialization.setSymm e' ⁻¹' e.source ext ⟨x, hx⟩ case h.mk ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj x✝ : e' ∈ a.pretrivializationAtlas x : B × F hx : x ∈ e'.target ⊢ { val := x, property := hx } ∈ Subtype.val ⁻¹' e.target ↔ { val := x, property := hx } ∈ Pretrivialization.setSymm e' ⁻¹' e.source simp only [mem_preimage, Pretrivialization.setSymm, restrict, e.mem_target, e.mem_source, e'.proj_symm_apply hx] ** Qed
FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas ⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source) obtain ⟨u, hu1, hu2⟩ := continuousOn_iff'.mp (a.continuous_symm_of_mem_pretrivializationAtlas he') e.source (a.isOpen_source e) case intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a u : Set (B × F) hu1 : IsOpen u hu2 : ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source ∩ e'.target = u ∩ e'.target ⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source) rw [inter_comm, hu2] case intro.intro ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ e e' : Pretrivialization F TotalSpace.proj he' : e' ∈ a.pretrivializationAtlas this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a u : Set (B × F) hu1 : IsOpen u hu2 : ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source ∩ e'.target = u ∩ e'.target ⊢ IsOpen (u ∩ e'.target) exact hu1.inter e'.open_target ** Qed
FiberPrebundle.mem_pretrivializationAt_source ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B x : E b ⊢ { proj := b, snd := x } ∈ (pretrivializationAt a b).source simp only [(a.pretrivializationAt b).source_eq, mem_preimage, TotalSpace.proj] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B x : E b ⊢ b ∈ (pretrivializationAt a b).baseSet exact a.mem_base_pretrivializationAt b ** Qed
FiberPrebundle.continuous_totalSpaceMk ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B ⊢ Continuous (TotalSpace.mk b) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Continuous (TotalSpace.mk b) let e := a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas b) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj b : B this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a b ∈ a.pretrivializationAtlas) ⊢ Continuous (TotalSpace.mk b) rw [e.toLocalHomeomorph.continuous_iff_continuous_comp_left (a.totalSpaceMk_preimage_source b)] ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj b : B this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a b ∈ a.pretrivializationAtlas) ⊢ Continuous (↑e.toLocalHomeomorph ∘ TotalSpace.mk b) exact continuous_iff_le_induced.mpr (le_antisymm_iff.mp (a.totalSpaceMk_inducing b).induced).1 ** Qed
FiberPrebundle.inducing_totalSpaceMk_of_inducing_comp ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (↑(pretrivializationAt a b) ∘ TotalSpace.mk b) ⊢ Inducing (TotalSpace.mk b) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (↑(pretrivializationAt a b) ∘ TotalSpace.mk b) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Inducing (TotalSpace.mk b) rw [← restrict_comp_codRestrict (a.mem_pretrivializationAt_source b)] at h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (restrict (pretrivializationAt a b).source ↑(pretrivializationAt a b) ∘ codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Inducing (TotalSpace.mk b) apply Inducing.of_codRestrict (a.mem_pretrivializationAt_source b) ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (restrict (pretrivializationAt a b).source ↑(pretrivializationAt a b) ∘ codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Inducing (codRestrict (fun a => { proj := b, snd := a }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) refine inducing_of_inducing_compose ?_ (continuousOn_iff_continuous_restrict.mp (a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas b)).continuous_toFun) h ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj b : B h : Inducing (restrict (pretrivializationAt a b).source ↑(pretrivializationAt a b) ∘ codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Continuous (codRestrict (fun a => { proj := b, snd := a }) (pretrivializationAt a b).source (_ : ∀ (x : E b), { proj := b, snd := x } ∈ (pretrivializationAt a b).source)) exact (a.continuous_totalSpaceMk b).codRestrict (a.mem_pretrivializationAt_source b) ** Qed
FiberPrebundle.continuous_proj ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj ⊢ Continuous TotalSpace.proj letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ Continuous TotalSpace.proj letI := a.toFiberBundle ι : Type u_1 B : Type u_2 F : Type u_3 X : Type u_4 inst✝³ : TopologicalSpace X E : B → Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F inst✝ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a ⊢ Continuous TotalSpace.proj exact FiberBundle.continuous_proj F E ** Qed
FiberPrebundle.continuousOn_of_comp_right ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) ⊢ ContinuousOn f (TotalSpace.proj ⁻¹' s) letI := a.totalSpaceTopology ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a ⊢ ContinuousOn f (TotalSpace.proj ⁻¹' s) intro z hz ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s ⊢ ContinuousWithinAt f (TotalSpace.proj ⁻¹' s) z let e : Trivialization F (π F E) := a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas z.proj) ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ ContinuousWithinAt f (TotalSpace.proj ⁻¹' s) z refine' (e.continuousAt_of_comp_right _ ((hf z.proj hz).continuousAt (IsOpen.mem_nhds _ _))).continuousWithinAt case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ ↑e z ∈ (s ∩ (pretrivializationAt a z.proj).baseSet) ×ˢ univ refine' ⟨_, mem_univ _⟩ case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ (↑e z).1 ∈ s ∩ (pretrivializationAt a z.proj).baseSet rw [e.coe_fst] case refine'_1 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z.proj ∈ e.baseSet exact a.mem_base_pretrivializationAt z.proj case refine'_2 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ IsOpen ((s ∩ (pretrivializationAt a z.proj).baseSet) ×ˢ univ) exact (hs.inter (a.pretrivializationAt z.proj).open_baseSet).prod isOpen_univ case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z.proj ∈ s ∩ (pretrivializationAt a z.proj).baseSet exact ⟨hz, a.mem_base_pretrivializationAt z.proj⟩ case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z ∈ e.source rw [e.mem_source] case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 X✝ : Type u_4 inst✝⁴ : TopologicalSpace X✝ E : B → Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F inst✝¹ : (x : B) → TopologicalSpace (E x) a : FiberPrebundle F E e✝ : Pretrivialization F TotalSpace.proj X : Type u_6 inst✝ : TopologicalSpace X f : TotalSpace F E → X s : Set B hs : IsOpen s hf : ∀ (b : B), b ∈ s → ContinuousOn (f ∘ ↑(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv)) ((s ∩ (pretrivializationAt a b).baseSet) ×ˢ univ) this : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a z : TotalSpace F E hz : z ∈ TotalSpace.proj ⁻¹' s e : Trivialization F TotalSpace.proj := trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj ∈ a.pretrivializationAtlas) ⊢ z.proj ∈ e.baseSet exact a.mem_base_pretrivializationAt z.proj ** Qed
TopCat.Presheaf.toTypes_isSheaf X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf ⊢ ∃! s, IsGluing (presheafToTypes X T) U sf s choose index index_spec using fun x : ↑(iSup U) => Opens.mem_iSup.mp x.2 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) ⊢ ∃! s, IsGluing (presheafToTypes X T) U sf s let s : ∀ x : ↑(iSup U), T x := fun x => sf (index x) ⟨x.1, index_spec x⟩ X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } ⊢ ∃! s, IsGluing (presheafToTypes X T) U sf s refine' ⟨s, _, _⟩ case refine'_1 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } ⊢ (fun s => IsGluing (presheafToTypes X T) U sf s) s intro i case refine'_1 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } i : ι ⊢ ↑((presheafToTypes X T).map (leSupr U i).op) s = sf i funext x case refine'_1.h X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } i : ι x : { x // x ∈ (Opposite.op (U i)).unop } ⊢ ↑((presheafToTypes X T).map (leSupr U i).op) s x = sf i x exact congr_fun (hsf (index ⟨x, _⟩) i) ⟨x, ⟨index_spec ⟨x.1, _⟩, x.2⟩⟩ case refine'_2 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } ⊢ ∀ (y : (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (iSup U)))), (fun s => IsGluing (presheafToTypes X T) U sf s) y → y = s intro t ht case refine'_2 X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } t : (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (iSup U))) ht : IsGluing (presheafToTypes X T) U sf t ⊢ t = s funext x case refine'_2.h X : TopCat T : ↑X → Type u ι : Type u U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (U i))) hsf : IsCompatible (presheafToTypes X T) U sf index : { x // x ∈ iSup U } → ι index_spec : ∀ (x : { x // x ∈ iSup U }), ↑x ∈ U (index x) s : (x : { x // x ∈ iSup U }) → T ↑x := fun x => sf (index x) { val := ↑x, property := (_ : ↑x ∈ U (index x)) } t : (forget (Type u)).obj ((presheafToTypes X T).obj (Opposite.op (iSup U))) ht : IsGluing (presheafToTypes X T) U sf t x : { x // x ∈ (Opposite.op (iSup U)).unop } ⊢ t x = s x exact congr_fun (ht (index x)) ⟨x.1, index_spec x⟩ ** Qed
TopCat.PrelocalPredicate.sheafifyOf X : TopCat T✝ T : ↑X → Type v P : PrelocalPredicate T U : Opens ↑X f : (x : { x // x ∈ U }) → T ↑x h : pred P f x : { x // x ∈ U } ⊢ pred P fun x => f ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U) }) x) convert h ** Qed
TopCat.subpresheafToTypes.isSheaf X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s let sf' : ∀ i : ι, (presheafToTypes X T).obj (op (U i)) := fun i => (sf i).val X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s have sf'_comp : (presheafToTypes X T).IsCompatible U sf' := fun i j => congr_arg Subtype.val (sf_comp i j) X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s obtain ⟨gl, gl_spec, gl_uniq⟩ := (sheafToTypes X T).existsUnique_gluing U sf' sf'_comp case intro.intro X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ ∃! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s refine' ⟨⟨gl, _⟩, _, _⟩ case intro.intro.refine'_1 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ PrelocalPredicate.pred P.toPrelocalPredicate gl apply P.locality case intro.intro.refine'_1.x X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ ∀ (x : { x // x ∈ (op (iSup U)).unop }), ∃ V x i, PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) rintro ⟨x, mem⟩ case intro.intro.refine'_1.x.mk X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop ⊢ ∃ V x i, PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) choose i hi using Opens.mem_iSup.mp mem case intro.intro.refine'_1.x.mk X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop i : ι hi : x ∈ U i ⊢ ∃ V x i, PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) use U i, hi, Opens.leSupr U i case h X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop i : ι hi : x ∈ U i ⊢ PrelocalPredicate.pred P.toPrelocalPredicate fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x) convert (sf i).property using 1 case h.e'_5.h X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl x : ↑X mem : x ∈ (op (iSup U)).unop i : ι hi : x ∈ U i e_4✝ : U i = (op (U i)).unop ⊢ (fun x => gl ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑(op (iSup U)).unop) }) x)) = ↑(sf i) exact gl_spec i case intro.intro.refine'_2 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ (fun s => IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s) { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } exact fun i => Subtype.ext (gl_spec i) case intro.intro.refine'_3 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl ⊢ ∀ (y : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U)))), (fun s => IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s) y → y = { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } intro gl' hgl' case intro.intro.refine'_3 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl gl' : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U))) hgl' : IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf gl' ⊢ gl' = { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } refine Subtype.ext ?_ case intro.intro.refine'_3 X : TopCat T : ↑X → Type v P✝ : PrelocalPredicate T P : LocalPredicate T ι : Type v U : ι → Opens ↑X sf : (i : ι) → (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i))) sf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf sf' : (i : ι) → (presheafToTypes X T).obj (op (U i)) := fun i => ↑(sf i) sf'_comp : IsCompatible (presheafToTypes X T) U sf' gl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U))) gl_spec : IsGluing (sheafToTypes X T).val U sf' gl gl_uniq : ∀ (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))), (fun s => IsGluing (sheafToTypes X T).val U sf' s) y → y = gl gl' : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U))) hgl' : IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf gl' ⊢ ↑gl' = ↑{ val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) } exact gl_uniq gl'.1 fun i => congr_arg Subtype.val (hgl' i) ** Qed
TopCat.stalkToFiber_germ X : TopCat T : ↑X → Type v P : LocalPredicate T U : Opens ↑X x : { x // x ∈ U } f : (Sheaf.presheaf (subsheafToTypes P)).obj (op U) ⊢ stalkToFiber P (↑x) (Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) x f) = ↑f x dsimp [Presheaf.germ, stalkToFiber] X : TopCat T : ↑X → Type v P : LocalPredicate T U : Opens ↑X x : { x // x ∈ U } f : (Sheaf.presheaf (subsheafToTypes P)).obj (op U) ⊢ colimit.desc ((OpenNhds.inclusion ↑x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) { pt := T ↑x, ι := NatTrans.mk fun U_1 f => ↑f { val := ↑x, property := (_ : ↑x ∈ U_1.unop.obj) } } (colimit.ι ((OpenNhds.inclusion ↑x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) (op { obj := U, property := (_ : ↑x ∈ U) }) f) = ↑f x cases x case mk X : TopCat T : ↑X → Type v P : LocalPredicate T U : Opens ↑X f : (Sheaf.presheaf (subsheafToTypes P)).obj (op U) val✝ : ↑X property✝ : val✝ ∈ U ⊢ colimit.desc ((OpenNhds.inclusion ↑{ val := val✝, property := property✝ }).op ⋙ subpresheafToTypes P.toPrelocalPredicate) { pt := T ↑{ val := val✝, property := property✝ }, ι := NatTrans.mk fun U_1 f => ↑f { val := ↑{ val := val✝, property := property✝ }, property := (_ : ↑{ val := val✝, property := property✝ } ∈ U_1.unop.obj) } } (colimit.ι ((OpenNhds.inclusion ↑{ val := val✝, property := property✝ }).op ⋙ subpresheafToTypes P.toPrelocalPredicate) (op { obj := U, property := (_ : ↑{ val := val✝, property := property✝ } ∈ U) }) f) = ↑f { val := val✝, property := property✝ } simp ** Qed
TopCat.stalkToFiber_surjective X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t t : T x ⊢ ∃ a, stalkToFiber P x a = t rcases w t with ⟨U, f, h, rfl⟩ case intro.intro.intro X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t U : OpenNhds x f : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y h : PrelocalPredicate.pred P.toPrelocalPredicate f ⊢ ∃ a, stalkToFiber P x a = f { val := x, property := (_ : x ∈ U.obj) } fconstructor case intro.intro.intro.w X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t U : OpenNhds x f : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y h : PrelocalPredicate.pred P.toPrelocalPredicate f ⊢ Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x exact (subsheafToTypes P).presheaf.germ ⟨x, U.2⟩ ⟨f, h⟩ case intro.intro.intro.h X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (t : T x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t U : OpenNhds x f : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y h : PrelocalPredicate.pred P.toPrelocalPredicate f ⊢ stalkToFiber P x (Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ U.obj) } { val := f, property := h }) = f { val := x, property := (_ : x ∈ U.obj) } exact stalkToFiber_germ _ U.1 ⟨x, U.2⟩ ⟨f, h⟩ ** Qed
TopCat.stalkToFiber_injective X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV ⊢ tU = tV let Q : ∃ (W : (OpenNhds x)ᵒᵖ) (s : ∀ w : (unop W).1, T w) (hW : P.pred s), tU = (subsheafToTypes P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ ∧ tV = (subsheafToTypes P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ := ?_ case refine_1 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV ⊢ ∃ W s hW, tU = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } obtain ⟨U, ⟨fU, hU⟩, rfl⟩ := jointly_surjective'.{v, v} tU case refine_1.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU }) = stalkToFiber P x tV ⊢ ∃ W s hW, colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU } = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } obtain ⟨V, ⟨fV, hV⟩, rfl⟩ := jointly_surjective'.{v, v} tV case refine_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV Q : ∃ W s hW, tU = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } := ?refine_1 ⊢ tU = tV choose W s hW e using Q case refine_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) tU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x h : stalkToFiber P x tU = stalkToFiber P x tV W : (OpenNhds x)ᵒᵖ s : (w : { x_1 // x_1 ∈ W.unop.obj }) → T ↑w hW : PrelocalPredicate.pred P.toPrelocalPredicate s e : tU = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ tV = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ⊢ tU = tV exact e.1.trans e.2.symm case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU }) = stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) V { val := fV, property := hV }) ⊢ ∃ W s hW, colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU } = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) V { val := fV, property := hV } = Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } dsimp case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) U { val := fU, property := hU }) = stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (Sheaf.presheaf (subsheafToTypes P))) V { val := fV, property := hV }) ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } simp only [stalkToFiber, Types.Colimit.ι_desc_apply'] at h case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X w : ∀ (U V : OpenNhds x) (fU : (y : { x_1 // x_1 ∈ U.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fU → ∀ (fV : (y : { x_2 // x_2 ∈ V.obj }) → T ↑y), PrelocalPredicate.pred P.toPrelocalPredicate fV → fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } → ∃ W iU iV, ∀ (w : { x_4 // x_4 ∈ W.obj }), fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) = fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w) U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } specialize w (unop U) (unop V) fU hU fV hV h case refine_1.intro.intro.mk.intro.intro.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } w : ∃ W iU iV, ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } rcases w with ⟨W, iU, iV, w⟩ case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ ∃ W s hW, colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) } { val := s, property := hW } refine' ⟨op W, fun w => fU (iU w : (unop U).1), P.res _ _ hU, _⟩ case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_1 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ (op W).unop.obj ⟶ U.unop.obj case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } rcases W with ⟨W, m⟩ case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_1.mk X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : Opens ↑X m : x ∈ W iU : { obj := W, property := m } ⟶ U.unop iV : { obj := W, property := m } ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ { obj := W, property := m }.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ (op { obj := W, property := m }).unop.obj ⟶ U.unop.obj exact iU case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_2 X : TopCat T : ↑X → Type v P : LocalPredicate T x : ↑X U : (OpenNhds x)ᵒᵖ fU : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj U).unop }) → T ↑x_1 hU : PrelocalPredicate.pred P.toPrelocalPredicate fU V : (OpenNhds x)ᵒᵖ fV : (x_1 : { x_1 // x_1 ∈ ((OpenNhds.inclusion x).op.obj V).unop }) → T ↑x_1 hV : PrelocalPredicate.pred P.toPrelocalPredicate fV h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) } W : OpenNhds x iU : W ⟶ U.unop iV : W ⟶ V.unop w : ∀ (w : { x_1 // x_1 ∈ W.obj }), fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) = fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w) ⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } = Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) } { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w), property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } exact ⟨colimit_sound iU.op (Subtype.eq rfl), colimit_sound iV.op (Subtype.eq (funext w).symm)⟩ ** Qed
TopCat.epi_iff_surjective X Y : TopCat f : X ⟶ Y ⊢ Epi f ↔ Function.Surjective ↑f suffices Epi f ↔ Epi ((forget TopCat).map f) by rw [this, CategoryTheory.epi_iff_surjective] rfl X Y : TopCat f : X ⟶ Y ⊢ Epi f ↔ Epi ((forget TopCat).map f) constructor X Y : TopCat f : X ⟶ Y this : Epi f ↔ Epi ((forget TopCat).map f) ⊢ Epi f ↔ Function.Surjective ↑f rw [this, CategoryTheory.epi_iff_surjective] X Y : TopCat f : X ⟶ Y this : Epi f ↔ Epi ((forget TopCat).map f) ⊢ Function.Surjective ((forget TopCat).map f) ↔ Function.Surjective ↑f rfl case mp X Y : TopCat f : X ⟶ Y ⊢ Epi f → Epi ((forget TopCat).map f) intro case mp X Y : TopCat f : X ⟶ Y a✝ : Epi f ⊢ Epi ((forget TopCat).map f) infer_instance case mpr X Y : TopCat f : X ⟶ Y ⊢ Epi ((forget TopCat).map f) → Epi f apply Functor.epi_of_epi_map ** Qed
TopCat.mono_iff_injective X Y : TopCat f : X ⟶ Y ⊢ Mono f ↔ Function.Injective ↑f suffices Mono f ↔ Mono ((forget TopCat).map f) by rw [this, CategoryTheory.mono_iff_injective] rfl X Y : TopCat f : X ⟶ Y ⊢ Mono f ↔ Mono ((forget TopCat).map f) constructor X Y : TopCat f : X ⟶ Y this : Mono f ↔ Mono ((forget TopCat).map f) ⊢ Mono f ↔ Function.Injective ↑f rw [this, CategoryTheory.mono_iff_injective] X Y : TopCat f : X ⟶ Y this : Mono f ↔ Mono ((forget TopCat).map f) ⊢ Function.Injective ((forget TopCat).map f) ↔ Function.Injective ↑f rfl case mp X Y : TopCat f : X ⟶ Y ⊢ Mono f → Mono ((forget TopCat).map f) intro case mp X Y : TopCat f : X ⟶ Y a✝ : Mono f ⊢ Mono ((forget TopCat).map f) infer_instance case mpr X Y : TopCat f : X ⟶ Y ⊢ Mono ((forget TopCat).map f) → Mono f apply Functor.mono_of_mono_map ** Qed
TopCat.isSheaf_of_isLimit C : Type u inst✝² : Category.{v, u} C J : Type v inst✝¹ : SmallCategory J inst✝ : HasLimits C X : TopCat F : J ⥤ Presheaf C X H : ∀ (j : J), Presheaf.IsSheaf (F.obj j) c : Cone F hc : IsLimit c ⊢ Presheaf.IsSheaf c.pt let F' : J ⥤ Sheaf C X := { obj := fun j => ⟨F.obj j, H j⟩ map := fun f => ⟨F.map f⟩ } C : Type u inst✝² : Category.{v, u} C J : Type v inst✝¹ : SmallCategory J inst✝ : HasLimits C X : TopCat F : J ⥤ Presheaf C X H : ∀ (j : J), Presheaf.IsSheaf (F.obj j) c : Cone F hc : IsLimit c F' : J ⥤ Sheaf C X := CategoryTheory.Functor.mk { obj := fun j => { val := F.obj j, cond := (_ : Presheaf.IsSheaf (F.obj j)) }, map := fun {X_1 Y} f => { val := F.map f } } ⊢ Presheaf.IsSheaf c.pt let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _ C : Type u inst✝² : Category.{v, u} C J : Type v inst✝¹ : SmallCategory J inst✝ : HasLimits C X : TopCat F : J ⥤ Presheaf C X H : ∀ (j : J), Presheaf.IsSheaf (F.obj j) c : Cone F hc : IsLimit c F' : J ⥤ Sheaf C X := CategoryTheory.Functor.mk { obj := fun j => { val := F.obj j, cond := (_ : Presheaf.IsSheaf (F.obj j)) }, map := fun {X_1 Y} f => { val := F.map f } } e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun x => Iso.refl ((F' ⋙ Sheaf.forget C X).obj x) ⊢ Presheaf.IsSheaf c.pt exact Presheaf.isSheaf_of_iso ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e) (limit F').2 ** Qed
BoundedContinuousFunction.dist_set_exists F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩ case intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C✝ C : ℝ hC : ∀ ⦃x : β⦄, x ∈ range ↑f ∪ range ↑g → ∀ ⦃y : β⦄, y ∈ range ↑f ∪ range ↑g → dist x y ≤ C ⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C refine' ⟨max 0 C, le_max_left _ _, fun x => (hC _ _).trans (le_max_right _ _)⟩ <;> [left; right] <;> apply mem_range_self ** Qed
BoundedContinuousFunction.dist_lt_of_nonempty_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C ⊢ dist f g < C have c : Continuous fun x => dist (f x) (g x) := by continuity F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C c : Continuous fun x => dist (↑f x) (↑g x) ⊢ dist f g < C obtain ⟨x, -, le⟩ := IsCompact.exists_isMaxOn isCompact_univ Set.univ_nonempty (Continuous.continuousOn c) case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x✝ : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C c : Continuous fun x => dist (↑f x) (↑g x) x : α le : IsMaxOn (fun x => dist (↑f x) (↑g x)) univ x ⊢ dist f g < C exact lt_of_le_of_lt (dist_le_iff_of_nonempty.mpr fun y => le trivial) (w x) F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α w : ∀ (x : α), dist (↑f x) (↑g x) < C ⊢ Continuous fun x => dist (↑f x) (↑g x) continuity ** Qed
BoundedContinuousFunction.dist_lt_iff_of_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C ⊢ dist f g < C ↔ ∀ (x : α), dist (↑f x) (↑g x) < C fconstructor case mp F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C ⊢ dist f g < C → ∀ (x : α), dist (↑f x) (↑g x) < C intro w x case mp F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C w : dist f g < C x : α ⊢ dist (↑f x) (↑g x) < C exact lt_of_le_of_lt (dist_coe_le_dist x) w case mpr F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C ⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C by_cases h : Nonempty α case pos F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : Nonempty α ⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C exact dist_lt_of_nonempty_compact case neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C rintro - case neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ dist f g < C convert C0 case h.e'_3 F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ dist f g = 0 apply le_antisymm _ dist_nonneg' F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ dist f g ≤ 0 rw [dist_eq] F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α C0 : 0 < C h : ¬Nonempty α ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} ≤ 0 exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempty.intro x))⟩ ** Qed
BoundedContinuousFunction.nndist_eq F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} = ↑(sInf {C \| ∀ (x : α), nndist (↑f x) (↑g x) ≤ C}) rw [val_eq_coe, coe_sInf, coe_image] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} = sInf {x \| ∃ h, { val := x, property := h } ∈ {C \| ∀ (x : α), nndist (↑f x) (↑g x) ≤ C}} simp_rw [mem_setOf_eq, ← NNReal.coe_le_coe, coe_mk, exists_prop, coe_nndist] ** Qed
BoundedContinuousFunction.dist_zero_of_empty F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ inst✝ : IsEmpty α ⊢ dist f g = 0 rw [(ext isEmptyElim : f = g), dist_self] ** Qed
BoundedContinuousFunction.dist_eq_iSup F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ dist f g = ⨆ x, dist (↑f x) (↑g x) cases isEmpty_or_nonempty α case inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ h✝ : Nonempty α ⊢ dist f g = ⨆ x, dist (↑f x) (↑g x) refine' (dist_le_iff_of_nonempty.mpr <\| le_ciSup _).antisymm (ciSup_le dist_coe_le_dist) case inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ h✝ : Nonempty α ⊢ BddAbove (range fun x => dist (↑f x) (↑g x)) exact dist_set_exists.imp fun C hC => forall_range_iff.2 hC.2 case inl F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ h✝ : IsEmpty α ⊢ dist f g = ⨆ x, dist (↑f x) (↑g x) rw [iSup_of_empty', Real.sSup_empty, dist_zero_of_empty] ** Qed
BoundedContinuousFunction.nndist_eq_iSup F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ ⨆ x, dist (↑f x) (↑g x) = ↑(⨆ x, nndist (↑f x) (↑g x)) simp_rw [val_eq_coe, coe_iSup, coe_nndist] ** Qed
BoundedContinuousFunction.inducing_coeFn F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ Inducing (↑UniformFun.ofFun ∘ FunLike.coe) rw [inducing_iff_nhds] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f g : α →ᵇ β x : α C : ℝ ⊢ ∀ (a : α →ᵇ β), 𝓝 a = comap (↑UniformFun.ofFun ∘ FunLike.coe) (𝓝 ((↑UniformFun.ofFun ∘ FunLike.coe) a)) refine' fun f => eq_of_forall_le_iff fun l => _ F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ f : α →ᵇ β l : Filter (α →ᵇ β) ⊢ l ≤ 𝓝 f ↔ l ≤ comap (↑UniformFun.ofFun ∘ FunLike.coe) (𝓝 ((↑UniformFun.ofFun ∘ FunLike.coe) f)) rw [← tendsto_iff_comap, ← tendsto_id', tendsto_iff_tendstoUniformly, UniformFun.tendsto_iff_tendstoUniformly] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ f : α →ᵇ β l : Filter (α →ᵇ β) ⊢ TendstoUniformly (fun i => ↑(id i)) (↑f) l ↔ TendstoUniformly (↑UniformFun.ofFun ∘ FunLike.coe) ((↑UniformFun.ofFun ∘ FunLike.coe) f) l rfl ** Qed
BoundedContinuousFunction.lipschitz_comp ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g✝ : α →ᵇ β x✝ : α C✝ : ℝ G : β → γ C : ℝ≥0 H : LipschitzWith C G f g : α →ᵇ β x : α ⊢ ↑C * dist (↑f x) (↑g x) ≤ ↑C * dist f g gcongr case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : PseudoMetricSpace β inst✝ : PseudoMetricSpace γ f✝ g✝ : α →ᵇ β x✝ : α C✝ : ℝ G : β → γ C : ℝ≥0 H : LipschitzWith C G f g : α →ᵇ β x : α ⊢ dist (↑f x) (↑g x) ≤ dist f g apply dist_coe_le_dist Qed
BoundedContinuousFunction.dist_extend_extend F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β ⊢ dist (extend f g₁ h₁) (extend f g₂ h₂) = max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) refine' le_antisymm ((dist_le <\| le_max_iff.2 <\| Or.inl dist_nonneg).2 fun x => _) (max_le _ _) case refine'_1 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : δ ⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) rcases _root_.em (∃ y, f y = x) with (⟨x, rfl⟩ \| hx) case refine'_1.inl.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) simp only [extend_apply] case refine'_1.inl.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑g₁ x) (↑g₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) exact (dist_coe_le_dist x).trans (le_max_left _ _) case refine'_1.inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : δ hx : ¬∃ y, ↑f y = x ⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) simp only [extend_apply' hx] case refine'_1.inr F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : δ hx : ¬∃ y, ↑f y = x ⊢ dist (↑h₁ x) (↑h₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) lift x to ((range f)ᶜ : Set δ) using hx case refine'_1.inr.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : { x // x ∈ (range ↑f)ᶜ } ⊢ dist (↑h₁ ↑x) (↑h₂ ↑x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ)) calc dist (h₁ x) (h₂ x) = dist (h₁.restrict (range f)ᶜ x) (h₂.restrict (range f)ᶜ x) := rfl _ ≤ dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ) := (dist_coe_le_dist x) _ ≤ _ := le_max_right _ _ case refine'_2 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β ⊢ dist g₁ g₂ ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) refine' (dist_le dist_nonneg).2 fun x => _ case refine'_2 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑g₁ x) (↑g₂ x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) rw [← extend_apply f g₁ h₁, ← extend_apply f g₂ h₂] case refine'_2 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : α ⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) exact dist_coe_le_dist _ case refine'_3 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β ⊢ dist (restrict h₁ (range ↑f)ᶜ) (restrict h₂ (range ↑f)ᶜ) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) refine' (dist_le dist_nonneg).2 fun x => _ case refine'_3 F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : ↑(range ↑f)ᶜ ⊢ dist (↑(restrict h₁ (range ↑f)ᶜ) x) (↑(restrict h₂ (range ↑f)ᶜ) x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) calc dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop] _ ≤ _ := dist_coe_le_dist _ F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x✝ : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ g₁ g₂ : α →ᵇ β h₁ h₂ : δ →ᵇ β x : ↑(range ↑f)ᶜ ⊢ dist (↑h₁ ↑x) (↑h₂ ↑x) = dist (↑(extend f g₁ h₁) ↑x) (↑(extend f g₂ h₂) ↑x) rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop] ** Qed
BoundedContinuousFunction.isometry_extend F : Type u_1 α : Type u β : Type v γ : Type w inst✝⁴ : TopologicalSpace α inst✝³ : PseudoMetricSpace β inst✝² : PseudoMetricSpace γ f✝ g : α →ᵇ β x : α C : ℝ δ : Type u_2 inst✝¹ : TopologicalSpace δ inst✝ : DiscreteTopology δ f : α ↪ δ h : δ →ᵇ β g₁ g₂ : α →ᵇ β ⊢ dist (extend f g₁ h) (extend f g₂ h) = dist g₁ g₂ simp [dist_nonneg] ** Qed
BoundedContinuousFunction.arzela_ascoli₁ F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : Equicontinuous fun x => ↑↑x ⊢ IsCompact A simp_rw [Equicontinuous, Metric.equicontinuousAt_iff_pair] at H F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ⊢ IsCompact A refine' isCompact_of_totallyBounded_isClosed _ closed F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ⊢ TotallyBounded A refine' totallyBounded_of_finite_discretization fun ε ε0 => _ F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε rcases exists_between ε0 with ⟨ε₁, ε₁0, εε₁⟩ case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε let ε₂ := ε₁ / 2 / 2 case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε have ε₂0 : ε₂ > 0 := half_pos (half_pos ε₁0) case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε have : ∀ x : α, ∃ U, x ∈ U ∧ IsOpen U ∧ ∀ (y) (_ : y ∈ U) (z) (_ : z ∈ U) {f : α →ᵇ β}, f ∈ A → dist (f y) (f z) < ε₂ := fun x => let ⟨U, nhdsU, hU⟩ := H x _ ε₂0 let ⟨V, VU, openV, xV⟩ := _root_.mem_nhds_iff.1 nhdsU ⟨V, xV, openV, fun y hy z hz f hf => hU y (VU hy) z (VU hz) ⟨f, hf⟩⟩ case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 this : ∀ (x : α), ∃ U, x ∈ U ∧ IsOpen U ∧ ∀ (y : α), y ∈ U → ∀ (z : α), z ∈ U → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε choose U hU using this case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε rcases isCompact_univ.elim_finite_subcover_image (fun x _ => (hU x).2.1) fun x _ => mem_biUnion (mem_univ _) (hU x).1 with ⟨tα, _, ⟨_⟩, htα⟩ case intro.intro.intro.intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝ : Fintype ↑tα ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε rcases @finite_cover_balls_of_compact β _ _ isCompact_univ _ ε₂0 with ⟨tβ, _, ⟨_⟩, htβ⟩ case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε choose F hF using fun y => show ∃ z ∈ tβ, dist y z < ε₂ by simpa using htβ (mem_univ y) case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ ⊢ ∃ β_1 x F, ∀ (x y : ↑A), F x = F y → dist ↑x ↑y < ε refine' ⟨tα → tβ, by infer_instance, fun f a => ⟨F (f.1 a), (hF (f.1 a)).1⟩, _⟩ case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ ⊢ ∀ (x y : ↑A), (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) x = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) y → dist ↑x ↑y < ε rintro ⟨f, hf⟩ ⟨g, hg⟩ f_eq_g case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.mk F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } ⊢ dist ↑{ val := f, property := hf } ↑{ val := g, property := hg } < ε refine' lt_of_le_of_lt ((dist_le <\| le_of_lt ε₁0).2 fun x => _) εε₁ case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.mk F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x : α ⊢ dist (↑↑{ val := f, property := hf } x) (↑↑{ val := g, property := hg } x) ≤ ε₁ obtain ⟨x', x'tα, hx'⟩ := mem_iUnion₂.1 (htα (mem_univ x)) F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ y : β ⊢ ∃ z, z ∈ tβ ∧ dist y z < ε₂ simpa using htβ (mem_univ y) F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ ⊢ Fintype (↑tα → ↑tβ) infer_instance F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑f x) (↑f x') + dist (↑g x) (↑g x') + dist (↑f x') (↑g x') ≤ ε₂ + ε₂ + ε₁ / 2 refine' le_of_lt (add_lt_add (add_lt_add _ _) _) case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑f x) (↑f x') < ε₂ exact (hU x').2.2 _ hx' _ (hU x').1 hf case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑g x) (↑g x') < ε₂ exact (hU x').2.2 _ hx' _ (hU x').1 hg case refine'_3 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ dist (↑f x') (↑g x') < ε₁ / 2 have F_f_g : F (f x') = F (g x') := (congr_arg (fun f : tα → tβ => (f ⟨x', x'tα⟩ : β)) f_eq_g : _) case refine'_3 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' F_f_g : F (↑f x') = F (↑g x') ⊢ dist (↑f x') (↑g x') < ε₁ / 2 calc dist (f x') (g x') ≤ dist (f x') (F (f x')) + dist (g x') (F (f x')) := dist_triangle_right _ _ _ _ = dist (f x') (F (f x')) + dist (g x') (F (g x')) := by rw [F_f_g] _ < ε₂ + ε₂ := (add_lt_add (hF (f x')).2 (hF (g x')).2) _ = ε₁ / 2 := add_halves _ F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' F_f_g : F (↑f x') = F (↑g x') ⊢ dist (↑f x') (F (↑f x')) + dist (↑g x') (F (↑f x')) = dist (↑f x') (F (↑f x')) + dist (↑g x') (F (↑g x')) rw [F_f_g] F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x✝ : α C : ℝ inst✝ : CompactSpace β A : Set (α →ᵇ β) closed : IsClosed A H : ∀ (x₀ : α) (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : α), x ∈ U → ∀ (x' : α), x' ∈ U → ∀ (i : ↑A), dist (↑↑i x) (↑↑i x') < ε ε : ℝ ε0 : ε > 0 ε₁ : ℝ ε₁0 : 0 < ε₁ εε₁ : ε₁ < ε ε₂ : ℝ := ε₁ / 2 / 2 ε₂0 : ε₂ > 0 U : α → Set α hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ (y : α), y ∈ U x → ∀ (z : α), z ∈ U x → ∀ {f : α →ᵇ β}, f ∈ A → dist (↑f y) (↑f z) < ε₂ tα : Set α left✝¹ : tα ⊆ univ htα : univ ⊆ ⋃ i ∈ tα, U i a✝¹ : Fintype ↑tα tβ : Set β left✝ : tβ ⊆ univ htβ : univ ⊆ ⋃ x ∈ tβ, ball x ε₂ a✝ : Fintype ↑tβ F : β → β hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂ f : α →ᵇ β hf : f ∈ A g : α →ᵇ β hg : g ∈ A f_eq_g : (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := f, property := hf } = (fun f a => { val := F (↑↑f ↑a), property := (_ : F (↑↑f ↑a) ∈ tβ) }) { val := g, property := hg } x x' : α x'tα : x' ∈ tα hx' : x ∈ U x' ⊢ ε₂ + ε₂ + ε₁ / 2 = ε₁ rw [add_halves, add_halves] ** Qed
BoundedContinuousFunction.arzela_ascoli₂ F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x ⊢ IsCompact A have M : LipschitzWith 1 (↑) := LipschitzWith.subtype_val s F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val ⊢ IsCompact A let F : (α →ᵇ s) → α →ᵇ β := comp (↑) M F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M ⊢ IsCompact A refine' IsCompact.of_isClosed_subset ((_ : IsCompact (F ⁻¹' A)).image (continuous_comp M)) closed fun f hf => _ case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M ⊢ IsCompact (F ⁻¹' A) haveI : CompactSpace s := isCompact_iff_compactSpace.1 hs case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M this : CompactSpace ↑s ⊢ IsCompact (F ⁻¹' A) refine' arzela_ascoli₁ _ (continuous_iff_isClosed.1 (continuous_comp M) _ closed) _ case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M this : CompactSpace ↑s ⊢ Equicontinuous fun x => ↑↑x rw [uniformEmbedding_subtype_val.toUniformInducing.equicontinuous_iff] case refine'_1 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M this : CompactSpace ↑s ⊢ Equicontinuous ((fun x x_1 => x ∘ x_1) Subtype.val ∘ fun x => ↑↑x) exact H.comp (A.restrictPreimage F) case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf : f ∈ A ⊢ f ∈ comp Subtype.val M '' (F ⁻¹' A) let g := codRestrict s f fun x => in_s f x hf case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) ⊢ f ∈ comp Subtype.val M '' (F ⁻¹' A) rw [show f = F g by ext; rfl] at hf ⊢ case refine'_2 F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf✝ : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) hf : F g ∈ A ⊢ F g ∈ comp Subtype.val M '' (F ⁻¹' A) exact ⟨g, hf, rfl⟩ F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf✝ : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) hf : F g ∈ A ⊢ f = F g ext case h F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : CompactSpace α inst✝ : PseudoMetricSpace β f✝ g✝ : α →ᵇ β x : α C : ℝ s : Set β hs : IsCompact s A : Set (α →ᵇ β) closed : IsClosed A in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → ↑f x ∈ s H : Equicontinuous fun x => ↑↑x M : LipschitzWith 1 Subtype.val F : (α →ᵇ ↑s) → α →ᵇ β := comp Subtype.val M f : α →ᵇ β hf✝ : f ∈ A g : α →ᵇ ↑s := codRestrict s f (_ : ∀ (x : α), ↑f x ∈ s) hf : F g ∈ A x✝ : α ⊢ ↑f x✝ = ↑(F g) x✝ rfl ** Qed
BoundedContinuousFunction.coe_nsmulRec F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : AddMonoid β inst✝ : LipschitzAdd β f g : α →ᵇ β x : α C : ℝ ⊢ ↑(nsmulRec 0 f) = 0 • ↑f rw [nsmulRec, zero_smul, coe_zero] F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : AddMonoid β inst✝ : LipschitzAdd β f g : α →ᵇ β x : α C : ℝ n : ℕ ⊢ ↑(nsmulRec (n + 1) f) = (n + 1) • ↑f rw [nsmulRec, succ_nsmul, coe_add, coe_nsmulRec n] ** Qed
BoundedContinuousFunction.sum_apply F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : PseudoMetricSpace β inst✝¹ : AddCommMonoid β inst✝ : LipschitzAdd β ι : Type u_2 s : Finset ι f : ι → α →ᵇ β a : α ⊢ ↑(∑ i in s, f i) a = ∑ i in s, ↑(f i) a simp ** Qed
BoundedContinuousFunction.norm_eq F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ f : α →ᵇ β ⊢ ‖f‖ = sInf {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} simp [norm_def, BoundedContinuousFunction.dist_eq] ** Qed
BoundedContinuousFunction.norm_eq_of_nonempty F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ h : Nonempty α ⊢ ‖f‖ = sInf {C \| ∀ (x : α), ‖↑f x‖ ≤ C} obtain ⟨a⟩ := h case intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α ⊢ ‖f‖ = sInf {C \| ∀ (x : α), ‖↑f x‖ ≤ C} rw [norm_eq] case intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α ⊢ sInf {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} = sInf {C \| ∀ (x : α), ‖↑f x‖ ≤ C} congr case intro.e_a F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α ⊢ {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} = {C \| ∀ (x : α), ‖↑f x‖ ≤ C} ext case intro.e_a.h F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α x✝ : ℝ ⊢ x✝ ∈ {C \| 0 ≤ C ∧ ∀ (x : α), ‖↑f x‖ ≤ C} ↔ x✝ ∈ {C \| ∀ (x : α), ‖↑f x‖ ≤ C} simp only [mem_setOf_eq, and_iff_right_iff_imp] case intro.e_a.h F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ a : α x✝ : ℝ ⊢ (∀ (x : α), ‖↑f x‖ ≤ x✝) → 0 ≤ x✝ exact fun h' => le_trans (norm_nonneg (f a)) (h' a) ** Qed
BoundedContinuousFunction.norm_coe_le_norm F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x✝ : α C : ℝ x : α ⊢ ‖↑f x‖ = dist (↑f x) (↑0 x) simp [dist_zero_right] ** Qed
BoundedContinuousFunction.norm_le F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ C0 : 0 ≤ C ⊢ ‖f‖ ≤ C ↔ ∀ (x : α), ‖↑f x‖ ≤ C simpa using @dist_le _ _ _ _ f 0 _ C0 ** Qed
BoundedContinuousFunction.norm_le_of_nonempty F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : Nonempty α f : α →ᵇ β M : ℝ ⊢ ‖f‖ ≤ M ↔ ∀ (x : α), ‖↑f x‖ ≤ M simp_rw [norm_def, ← dist_zero_right] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : Nonempty α f : α →ᵇ β M : ℝ ⊢ dist f 0 ≤ M ↔ ∀ (x : α), dist (↑f x) 0 ≤ M exact dist_le_iff_of_nonempty ** Qed
BoundedContinuousFunction.norm_lt_iff_of_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α f : α →ᵇ β M : ℝ M0 : 0 < M ⊢ ‖f‖ < M ↔ ∀ (x : α), ‖↑f x‖ < M simp_rw [norm_def, ← dist_zero_right] F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝ : CompactSpace α f : α →ᵇ β M : ℝ M0 : 0 < M ⊢ dist f 0 < M ↔ ∀ (x : α), dist (↑f x) 0 < M exact dist_lt_iff_of_compact M0 ** Qed
BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α f : α →ᵇ β M : ℝ ⊢ ‖f‖ < M ↔ ∀ (x : α), ‖↑f x‖ < M simp_rw [norm_def, ← dist_zero_right] F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : SeminormedAddCommGroup β f✝ g : α →ᵇ β x : α C : ℝ inst✝¹ : Nonempty α inst✝ : CompactSpace α f : α →ᵇ β M : ℝ ⊢ dist f 0 < M ↔ ∀ (x : α), dist (↑f x) 0 < M exact dist_lt_iff_of_nonempty_compact ** Qed
BoundedContinuousFunction.norm_normComp F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖normComp f‖ = ‖f‖ simp only [norm_eq, coe_normComp, norm_norm, Function.comp] ** Qed
BoundedContinuousFunction.norm_eq_iSup_norm F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖f‖ = ⨆ x, ‖↑f x‖ simp_rw [norm_def, dist_eq_iSup, coe_zero, Pi.zero_apply, dist_zero_right] ** Qed
BoundedContinuousFunction.coe_zsmulRec F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ n : ℕ ⊢ ↑(zsmulRec (Int.ofNat n) f) = Int.ofNat n • ↑f rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmulRec, coe_nat_zsmul] F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ n : ℕ ⊢ ↑(zsmulRec (Int.negSucc n) f) = Int.negSucc n • ↑f rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmulRec] ** Qed
BoundedContinuousFunction.nnnorm_eq_iSup_nnnorm F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ⨆ x, ‖↑f x‖ = ↑(⨆ x, ‖↑f x‖₊) simp_rw [val_eq_coe, NNReal.coe_iSup, coe_nnnorm] ** Qed
BoundedContinuousFunction.abs_diff_coe_le_dist F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖↑f x - ↑g x‖ ≤ dist f g rw [dist_eq_norm] F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α inst✝ : SeminormedAddCommGroup β f g : α →ᵇ β x : α C : ℝ ⊢ ‖↑f x - ↑g x‖ ≤ ‖f - g‖ exact (f - g).norm_coe_le_norm x ** Qed
BoundedContinuousFunction.norm_compContinuous_le ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝² : TopologicalSpace α inst✝¹ : SeminormedAddCommGroup β f✝ g✝ : α →ᵇ β x : α C : ℝ inst✝ : TopologicalSpace γ f : α →ᵇ β g : C(γ, α) ⊢ ↑1 * dist f 0 ≤ ‖f‖ rw [NNReal.coe_one, one_mul, dist_zero_right] Qed
BoundedContinuousFunction.coe_npowRec F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α R : Type u_2 inst✝ : SeminormedRing R f : α →ᵇ R ⊢ ↑(npowRec 0 f) = ↑f ^ 0 rw [npowRec, pow_zero, coe_one] F : Type u_1 α : Type u β : Type v γ : Type w inst✝¹ : TopologicalSpace α R : Type u_2 inst✝ : SeminormedRing R f : α →ᵇ R n : ℕ ⊢ ↑(npowRec (n + 1) f) = ↑f ^ (n + 1) rw [npowRec, pow_succ, coe_mul, coe_npowRec f n] ** Qed
BoundedContinuousFunction.algebraMap_apply F : Type u_1 α : Type u β : Type v γ : Type w 𝕜 : Type u_2 inst✝⁵ : NormedField 𝕜 inst✝⁴ : TopologicalSpace α inst✝³ : SeminormedAddCommGroup β inst✝² : NormedSpace 𝕜 β inst✝¹ : NormedRing γ inst✝ : NormedAlgebra 𝕜 γ f g : α →ᵇ γ x : α c k : 𝕜 a : α ⊢ ↑(↑(algebraMap 𝕜 (α →ᵇ γ)) k) a = k • 1 rw [Algebra.algebraMap_eq_smul_one] F : Type u_1 α : Type u β : Type v γ : Type w 𝕜 : Type u_2 inst✝⁵ : NormedField 𝕜 inst✝⁴ : TopologicalSpace α inst✝³ : SeminormedAddCommGroup β inst✝² : NormedSpace 𝕜 β inst✝¹ : NormedRing γ inst✝ : NormedAlgebra 𝕜 γ f g : α →ᵇ γ x : α c k : 𝕜 a : α ⊢ ↑(k • 1) a = k • 1 rfl ** Qed
BoundedContinuousFunction.NNReal.upper_bound F : Type u_1 α✝ : Type u β : Type v γ : Type w α : Type u_2 inst✝ : TopologicalSpace α f : α →ᵇ ℝ≥0 x : α ⊢ ↑f x ≤ nndist f 0 have key : nndist (f x) ((0 : α →ᵇ ℝ≥0) x) ≤ nndist f 0 := @dist_coe_le_dist α ℝ≥0 _ _ f 0 x F : Type u_1 α✝ : Type u β : Type v γ : Type w α : Type u_2 inst✝ : TopologicalSpace α f : α →ᵇ ℝ≥0 x : α key : nndist (↑f x) (↑0 x) ≤ nndist f 0 ⊢ ↑f x ≤ nndist f 0 simp only [coe_zero, Pi.zero_apply] at key F : Type u_1 α✝ : Type u β : Type v γ : Type w α : Type u_2 inst✝ : TopologicalSpace α f : α →ᵇ ℝ≥0 x : α key : nndist (↑f x) 0 ≤ nndist f 0 ⊢ ↑f x ≤ nndist f 0 rwa [NNReal.nndist_zero_eq_val' (f x)] at key ** Qed
BoundedContinuousFunction.self_eq_nnrealPart_sub_nnrealPart_neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ ⊢ ↑f = toReal ∘ ↑(nnrealPart f) - toReal ∘ ↑(nnrealPart (-f)) funext x case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ ↑f x = (toReal ∘ ↑(nnrealPart f) - toReal ∘ ↑(nnrealPart (-f))) x dsimp case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ ↑f x = max (↑f x) 0 - max (-↑f x) 0 simp only [max_zero_sub_max_neg_zero_eq_self] ** Qed
BoundedContinuousFunction.abs_self_eq_nnrealPart_add_nnrealPart_neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ ⊢ abs ∘ ↑f = toReal ∘ ↑(nnrealPart f) + toReal ∘ ↑(nnrealPart (-f)) funext x case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ (abs ∘ ↑f) x = (toReal ∘ ↑(nnrealPart f) + toReal ∘ ↑(nnrealPart (-f))) x dsimp case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝ : TopologicalSpace α f : α →ᵇ ℝ x : α ⊢ \|↑f x\| = max (↑f x) 0 + max (-↑f x) 0 simp only [max_zero_add_max_neg_zero_eq_abs_self] ** Qed
SetTheory.PGame.impartialAux_def G : PGame ⊢ ImpartialAux G ↔ G ≈ -G ∧ (∀ (i : LeftMoves G), ImpartialAux (moveLeft G i)) ∧ ∀ (j : RightMoves G), ImpartialAux (moveRight G j) rw [ImpartialAux] ** Qed
SetTheory.PGame.impartial_def G : PGame ⊢ Impartial G ↔ G ≈ -G ∧ (∀ (i : LeftMoves G), Impartial (moveLeft G i)) ∧ ∀ (j : RightMoves G), Impartial (moveRight G j) simpa only [impartial_iff_aux] using impartialAux_def ** Qed
SetTheory.PGame.Impartial.impartial_congr G H : PGame e : G ≡r H ⊢ ∀ [inst : Impartial G], Impartial H intro h G H : PGame e : G ≡r H h : Impartial G ⊢ Impartial H exact impartial_def.2 ⟨Equiv.trans e.symm.equiv (Equiv.trans (neg_equiv_self G) (neg_equiv_neg_iff.2 e.equiv)), fun i => impartial_congr (e.moveLeftSymm i), fun j => impartial_congr (e.moveRightSymm j)⟩ G H : PGame x✝ : ∀ (y : (_ : PGame) ×' PGame), (invImage (fun a => PSigma.casesOn a fun G snd => (G, snd)) Prod.instWellFoundedRelationProd).1 y { fst := G, snd := H } → y.1 ≡r y.2 → ∀ [inst : Impartial y.1], Impartial y.2 e : G ≡r H h : Impartial G j : RightMoves H ⊢ (invImage (fun a => PSigma.casesOn a fun G snd => (G, snd)) Prod.instWellFoundedRelationProd).1 { fst := moveRight G (↑(Relabelling.rightMovesEquiv e).symm j), snd := moveRight H j } { fst := G, snd := H } pgame_wf_tac ** Qed
SetTheory.PGame.Impartial.nonpos G : PGame inst✝ : Impartial G h : 0 < G ⊢ False have h' := neg_lt_neg_iff.2 h G : PGame inst✝ : Impartial G h : 0 < G h' : -G < -0 ⊢ False rw [neg_zero, lt_congr_left (Equiv.symm (neg_equiv_self G))] at h' G : PGame inst✝ : Impartial G h : 0 < G h' : G < 0 ⊢ False exact (h.trans h').false ** Qed
SetTheory.PGame.Impartial.nonneg G : PGame inst✝ : Impartial G h : G < 0 ⊢ False have h' := neg_lt_neg_iff.2 h G : PGame inst✝ : Impartial G h : G < 0 h' : -0 < -G ⊢ False rw [neg_zero, lt_congr_right (Equiv.symm (neg_equiv_self G))] at h' G : PGame inst✝ : Impartial G h : G < 0 h' : 0 < G ⊢ False exact (h.trans h').false ** Qed
SetTheory.PGame.Impartial.equiv_or_fuzzy_zero G : PGame inst✝ : Impartial G ⊢ G ≈ 0 ∨ G ‖ 0 rcases lt_or_equiv_or_gt_or_fuzzy G 0 with (h \| h \| h \| h) case inl G : PGame inst✝ : Impartial G h : G < 0 ⊢ G ≈ 0 ∨ G ‖ 0 exact ((nonneg G) h).elim case inr.inl G : PGame inst✝ : Impartial G h : G ≈ 0 ⊢ G ≈ 0 ∨ G ‖ 0 exact Or.inl h case inr.inr.inl G : PGame inst✝ : Impartial G h : 0 < G ⊢ G ≈ 0 ∨ G ‖ 0 exact ((nonpos G) h).elim case inr.inr.inr G : PGame inst✝ : Impartial G h : G ‖ 0 ⊢ G ≈ 0 ∨ G ‖ 0 exact Or.inr h ** Qed
SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero G : PGame inst✝ : Impartial G H : PGame ⊢ H ≈ G ↔ H + G ≈ 0 rw [Game.PGame.equiv_iff_game_eq, ← @add_right_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add, Game.PGame.equiv_iff_game_eq] G : PGame inst✝ : Impartial G H : PGame ⊢ Quotient.mk setoid (H + G) = 0 ↔ Quotient.mk setoid (H + G) = Quotient.mk setoid 0 rfl ** Qed
SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero' G : PGame inst✝ : Impartial G H : PGame ⊢ G ≈ H ↔ G + H ≈ 0 rw [Game.PGame.equiv_iff_game_eq, ← @add_left_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add, Game.PGame.equiv_iff_game_eq] G : PGame inst✝ : Impartial G H : PGame ⊢ 0 = Quotient.mk setoid (G + H) ↔ Quotient.mk setoid (G + H) = Quotient.mk setoid 0 exact ⟨Eq.symm, Eq.symm⟩ ** Qed
SetTheory.PGame.Impartial.le_zero_iff G✝ : PGame inst✝¹ : Impartial G✝ G : PGame inst✝ : Impartial G ⊢ G ≤ 0 ↔ 0 ≤ G rw [← zero_le_neg_iff, le_congr_right (neg_equiv_self G)] ** Qed
SetTheory.PGame.Impartial.lf_zero_iff G✝ : PGame inst✝¹ : Impartial G✝ G : PGame inst✝ : Impartial G ⊢ G ⧏ 0 ↔ 0 ⧏ G rw [← zero_lf_neg_iff, lf_congr_right (neg_equiv_self G)] ** Qed
SetTheory.PGame.Impartial.forall_leftMoves_fuzzy_iff_equiv_zero G : PGame inst✝ : Impartial G ⊢ (∀ (i : LeftMoves G), moveLeft G i ‖ 0) ↔ G ≈ 0 refine' ⟨fun hb => _, fun hp i => _⟩ case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (i : LeftMoves G), moveLeft G i ‖ 0 ⊢ G ≈ 0 rw [equiv_zero_iff_le G, le_zero_lf] case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (i : LeftMoves G), moveLeft G i ‖ 0 ⊢ ∀ (i : LeftMoves G), moveLeft G i ⧏ 0 exact fun i => (hb i).1 case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : LeftMoves G ⊢ moveLeft G i ‖ 0 rw [fuzzy_zero_iff_lf] case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : LeftMoves G ⊢ moveLeft G i ⧏ 0 exact hp.1.moveLeft_lf i ** Qed
SetTheory.PGame.Impartial.forall_rightMoves_fuzzy_iff_equiv_zero G : PGame inst✝ : Impartial G ⊢ (∀ (j : RightMoves G), moveRight G j ‖ 0) ↔ G ≈ 0 refine' ⟨fun hb => _, fun hp i => _⟩ case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (j : RightMoves G), moveRight G j ‖ 0 ⊢ G ≈ 0 rw [equiv_zero_iff_ge G, zero_le_lf] case refine'_1 G : PGame inst✝ : Impartial G hb : ∀ (j : RightMoves G), moveRight G j ‖ 0 ⊢ ∀ (j : RightMoves G), 0 ⧏ moveRight G j exact fun i => (hb i).2 case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : RightMoves G ⊢ moveRight G i ‖ 0 rw [fuzzy_zero_iff_gf] case refine'_2 G : PGame inst✝ : Impartial G hp : G ≈ 0 i : RightMoves G ⊢ 0 ⧏ moveRight G i exact hp.2.lf_moveRight i ** Qed
SetTheory.PGame.Impartial.exists_left_move_equiv_iff_fuzzy_zero G : PGame inst✝ : Impartial G ⊢ (∃ i, moveLeft G i ≈ 0) ↔ G ‖ 0 refine' ⟨fun ⟨i, hi⟩ => (fuzzy_zero_iff_gf G).2 (lf_of_le_moveLeft hi.2), fun hn => _⟩ G : PGame inst✝ : Impartial G hn : G ‖ 0 ⊢ ∃ i, moveLeft G i ≈ 0 rw [fuzzy_zero_iff_gf G, zero_lf_le] at hn G : PGame inst✝ : Impartial G hn : ∃ i, 0 ≤ moveLeft G i ⊢ ∃ i, moveLeft G i ≈ 0 cases' hn with i hi case intro G : PGame inst✝ : Impartial G i : LeftMoves G hi : 0 ≤ moveLeft G i ⊢ ∃ i, moveLeft G i ≈ 0 exact ⟨i, (equiv_zero_iff_ge _).2 hi⟩ ** Qed
SetTheory.PGame.Impartial.exists_right_move_equiv_iff_fuzzy_zero G : PGame inst✝ : Impartial G ⊢ (∃ j, moveRight G j ≈ 0) ↔ G ‖ 0 refine' ⟨fun ⟨i, hi⟩ => (fuzzy_zero_iff_lf G).2 (lf_of_moveRight_le hi.1), fun hn => _⟩ G : PGame inst✝ : Impartial G hn : G ‖ 0 ⊢ ∃ j, moveRight G j ≈ 0 rw [fuzzy_zero_iff_lf G, lf_zero_le] at hn G : PGame inst✝ : Impartial G hn : ∃ j, moveRight G j ≤ 0 ⊢ ∃ j, moveRight G j ≈ 0 cases' hn with i hi case intro G : PGame inst✝ : Impartial G i : RightMoves G hi : moveRight G i ≤ 0 ⊢ ∃ j, moveRight G j ≈ 0 exact ⟨i, (equiv_zero_iff_le _).2 hi⟩ ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ IsSheaf F ↔ IsSheafOpensLeCover F refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _ C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ (∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) ↔ IsSheafOpensLeCover F constructor case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ (∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) → IsSheafOpensLeCover F intro h ι U case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι✝ : Type w U✝ : ι✝ → Opens ↑X Y : Opens ↑X hY : Y = iSup U✝ h : ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) ι : Type w U : ι → Opens ↑X ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone U)))) rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr] case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι✝ : Type w U✝ : ι✝ → Opens ↑X Y : Opens ↑X hY : Y = iSup U✝ h : ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) ι : Type w U : ι → Opens ↑X ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U (iSup U))).arrows)))) apply h case mp.a C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι✝ : Type w U✝ : ι✝ → Opens ↑X Y : Opens ↑X hY : Y = iSup U✝ h : ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) ι : Type w U : ι → Opens ↑X ⊢ Sieve.generate (presieveOfCoveringAux U (iSup U)) ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) (iSup U) apply presieveOfCovering.mem_grothendieckTopology case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y : Opens ↑X hY : Y = iSup U ⊢ IsSheafOpensLeCover F → ∀ ⦃X_1 : Opens ↑X⦄ (S : Sieve X_1), S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1 → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) intro h Y S case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y ⊢ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows)))) rw [← Sieve.generate_sieve S] case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y ⊢ Sieve.generate S.arrows ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y → Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate S.arrows).arrows)))) intro hS case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y hS : Sieve.generate S.arrows ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate S.arrows).arrows)))) rw [← (isLimitOpensLeEquivGenerate₂ F S.1 hS).nonempty_congr] case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ι : Type w U : ι → Opens ↑X Y✝ : Opens ↑X hY : Y✝ = iSup U h : IsSheafOpensLeCover F Y : Opens ↑X S : Sieve Y hS : Sieve.generate S.arrows ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone (coveringOfPresieve Y S.arrows))))) apply h ** Qed
TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 ⊢ IsLocallySurjective T ↔ ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) constructor <;> intro hT case mp C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T ⊢ ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) intro x g case mp C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X g : (forget C).obj ((stalkFunctor C x).obj 𝒢) ⊢ ∃ a, ↑((stalkFunctor C x).map T) a = g obtain ⟨U, hxU, t, rfl⟩ := 𝒢.germ_exist x g case mp.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) ⊢ ∃ a, ↑((stalkFunctor C x).map T) a = ↑(germ 𝒢 { val := x, property := hxU }) t rcases hT U t x hxU with ⟨V, ι, ⟨s, h_eq⟩, hxV⟩ case mp.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) V : Opens ↑X ι : V ⟶ U hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) h_eq : ↑(T.app (op V)) s = ↑(𝒢.map ι.op) t ⊢ ∃ a, ↑((stalkFunctor C x).map T) a = ↑(germ 𝒢 { val := x, property := hxU }) t use ℱ.germ ⟨x, hxV⟩ s case h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) V : Opens ↑X ι : V ⟶ U hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) h_eq : ↑(T.app (op V)) s = ↑(𝒢.map ι.op) t ⊢ ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = ↑(germ 𝒢 { val := x, property := hxU }) t convert stalkFunctor_map_germ_apply V ⟨x, hxV⟩ T s using 1 case h.e'_3.h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : IsLocallySurjective T x : ↑X U : Opens ↑X hxU : x ∈ U t : (forget C).obj (𝒢.obj (op U)) V : Opens ↑X ι : V ⟶ U hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) h_eq : ↑(T.app (op V)) s = ↑(𝒢.map ι.op) t e_1✝ : (forget C).obj ((stalkFunctor C x).obj 𝒢) = (forget C).obj (Limits.colim.obj ((OpenNhds.inclusion ↑{ val := x, property := hxV }).op ⋙ 𝒢)) ⊢ ↑(germ 𝒢 { val := x, property := hxU }) t = ↑(Limits.colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxV }).op ⋙ 𝒢) (op { obj := V, property := (_ : ↑{ val := x, property := hxV } ∈ V) })) (↑(T.app (op V)) s) simpa [h_eq] using (germ_res_apply 𝒢 ι ⟨x, hxV⟩ t).symm case mpr C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) ⊢ IsLocallySurjective T intro U t x hxU case mpr C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 set t_x := 𝒢.germ ⟨x, hxU⟩ t with ht_x case mpr C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 obtain ⟨s_x, hs_x : ((stalkFunctor C x).map T) s_x = t_x⟩ := hT x t_x case mpr.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t s_x : (forget C).obj ((stalkFunctor C x).obj ℱ) hs_x : ↑((stalkFunctor C x).map T) s_x = t_x ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 obtain ⟨V, hxV, s, rfl⟩ := ℱ.germ_exist x s_x case mpr.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 have key_W := 𝒢.germ_eq x hxV hxU (T.app _ s) t <\| by convert hs_x using 1 symm convert stalkFunctor_map_germ_apply _ _ _ s case mpr.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x key_W : ∃ W _m iU iV, ↑(𝒢.map iU.op) (↑(T.app (op V)) s) = ↑(𝒢.map iV.op) t ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 obtain ⟨W, hxW, hWV, hWU, h_eq⟩ := key_W case mpr.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x W : Opens ↑X hxW : x ∈ W hWV : W ⟶ V hWU : W ⟶ U h_eq : ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) = ↑(𝒢.map hWU.op) t ⊢ ∃ U_1 f, (imageSieve T t).arrows f ∧ x ∈ U_1 refine' ⟨W, hWU, ⟨ℱ.map hWV.op s, _⟩, hxW⟩ case mpr.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x W : Opens ↑X hxW : x ∈ W hWV : W ⟶ V hWU : W ⟶ U h_eq : ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) = ↑(𝒢.map hWU.op) t ⊢ ↑(T.app (op W)) (↑(ℱ.map hWV.op) s) = ↑(𝒢.map hWU.op) t convert h_eq using 1 case h.e'_2 C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x W : Opens ↑X hxW : x ∈ W hWV : W ⟶ V hWU : W ⟶ U h_eq : ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) = ↑(𝒢.map hWU.op) t ⊢ ↑(T.app (op W)) (↑(ℱ.map hWV.op) s) = ↑(𝒢.map hWV.op) (↑(T.app (op V)) s) simp only [← comp_apply, T.naturality] C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x ⊢ ↑(germ 𝒢 { val := x, property := hxV }) (↑(T.app (op V)) s) = ↑(germ 𝒢 { val := x, property := hxU }) t convert hs_x using 1 case h.e'_2.h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x e_1✝ : (fun x_1 => (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxV })) (↑(T.app (op V)) s) = (fun x_1 => (forget C).obj ((stalkFunctor C x).obj 𝒢)) (↑(germ ℱ { val := x, property := hxV }) s) ⊢ ↑(germ 𝒢 { val := x, property := hxV }) (↑(T.app (op V)) s) = ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) symm case h.e'_2.h C : Type u inst✝³ : Category.{v, u} C inst✝² : ConcreteCategory C X : TopCat ℱ 𝒢 : Presheaf C X inst✝¹ : Limits.HasColimits C inst✝ : Limits.PreservesFilteredColimits (forget C) T : ℱ ⟶ 𝒢 hT : ∀ (x : ↑X), Function.Surjective ↑((stalkFunctor C x).map T) U : Opens ↑X t : (forget C).obj (𝒢.obj (op U)) x : ↑X hxU : x ∈ U t_x : (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxU }) := ↑(germ 𝒢 { val := x, property := hxU }) t ht_x : t_x = ↑(germ 𝒢 { val := x, property := hxU }) t V : Opens ↑X hxV : x ∈ V s : (forget C).obj (ℱ.obj (op V)) hs_x : ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = t_x e_1✝ : (fun x_1 => (forget C).obj (stalk 𝒢 ↑{ val := x, property := hxV })) (↑(T.app (op V)) s) = (fun x_1 => (forget C).obj ((stalkFunctor C x).obj 𝒢)) (↑(germ ℱ { val := x, property := hxV }) s) ⊢ ↑((stalkFunctor C x).map T) (↑(germ ℱ { val := x, property := hxV }) s) = ↑(germ 𝒢 { val := x, property := hxV }) (↑(T.app (op V)) s) convert stalkFunctor_map_germ_apply _ _ _ s ** Qed
TopCat.Presheaf.isGluing_iff_pairwise C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) ⊢ IsGluing F U sf s ↔ ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i refine ⟨fun h ↦ ?_, fun h i ↦ h (op <\| Pairwise.single i)⟩ C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s ⊢ ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i rintro (i\|⟨i,j⟩) case mk.single C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s i : ι ⊢ (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app { unop := Pairwise.single i } s = objPairwiseOfFamily sf { unop := Pairwise.single i } exact h i case mk.pair C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s i j : ι ⊢ (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app { unop := Pairwise.pair i j } s = objPairwiseOfFamily sf { unop := Pairwise.pair i j } rw [← (F.mapCone (Pairwise.cocone U).op).w (op <\| Pairwise.Hom.left i j)] case mk.pair C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) s : (forget (Type u)).obj (F.obj (op (iSup U))) h : IsGluing F U sf s i j : ι ⊢ ((F.mapCone (Cocone.op (Pairwise.cocone U))).π.app { unop := Pairwise.single i } ≫ ((Pairwise.diagram U).op ⋙ F).map (op (Pairwise.Hom.left i j))) s = objPairwiseOfFamily sf { unop := Pairwise.pair i j } exact congr_arg _ (h i) ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing_types C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X ⊢ IsSheaf F ↔ IsSheafUniqueGluing F simp_rw [isSheaf_iff_isSheafPairwiseIntersections, IsSheafPairwiseIntersections, Types.isLimit_iff, IsSheafUniqueGluing, isGluing_iff_pairwise] C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι : Type x U : ι → Opens ↑X ⊢ (∀ ⦃ι : Type x⦄ (U : ι → Opens ↑X) (s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j), s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) → ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i) ↔ ∀ ⦃ι : Type x⦄ (U : ι → Opens ↑X) (sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i)))), IsCompatible F U sf → ∃! s, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i refine forall₂_congr fun ι U ↦ ⟨fun h sf cpt ↦ ?_, fun h s hs ↦ ?_⟩ case refine_1 C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X h : ∀ (s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j), s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) → ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i))) cpt : IsCompatible F U sf ⊢ ∃! s, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i exact h _ cpt.sectionPairwise.prop case refine_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X h : ∀ (sf : (i : ι) → (forget (Type u)).obj (F.obj (op (U i)))), IsCompatible F U sf → ∃! s, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s = objPairwiseOfFamily sf i s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) ⊢ ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i specialize h (fun i ↦ s <\| op <\| Pairwise.single i) fun i j ↦ (hs <\| op <\| Pairwise.Hom.left i j).trans (hs <\| op <\| Pairwise.Hom.right i j).symm case refine_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i ⊢ ∃! x, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i x = s i convert h case h.e'_2.h.h.h.h.e'_3.h.e C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i e_1✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt = (forget (Type u)).obj (F.obj (op (iSup U))) x✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt a✝ : (CategoryTheory.Pairwise ι)ᵒᵖ ⊢ s = objPairwiseOfFamily fun i => s (op (Pairwise.single i)) ext (i\|⟨i,j⟩) case h.e'_2.h.h.h.h.e'_3.h.e.h.mk.single C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i e_1✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt = (forget (Type u)).obj (F.obj (op (iSup U))) x✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt a✝ : (CategoryTheory.Pairwise ι)ᵒᵖ i : ι ⊢ s { unop := Pairwise.single i } = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) { unop := Pairwise.single i } rfl case h.e'_2.h.h.h.h.e'_3.h.e.h.mk.pair C : Type u inst✝¹ : Category.{v, u} C inst✝ : ConcreteCategory C X : TopCat F : Presheaf (Type u) X ι✝ : Type x U✝ : ι✝ → Opens ↑X ι : Type x U : ι → Opens ↑X s : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j hs : s ∈ Functor.sections ((Pairwise.diagram U).op ⋙ F) h : ∃! s_1, ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ), (F.mapCone (Cocone.op (Pairwise.cocone U))).π.app i s_1 = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) i e_1✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt = (forget (Type u)).obj (F.obj (op (iSup U))) x✝ : (F.mapCone (Cocone.op (Pairwise.cocone U))).pt a✝ : (CategoryTheory.Pairwise ι)ᵒᵖ i j : ι ⊢ s { unop := Pairwise.pair i j } = objPairwiseOfFamily (fun i => s (op (Pairwise.single i))) { unop := Pairwise.pair i j } exact (hs <\| op <\| Pairwise.Hom.left i j).symm ** Qed
TopCat.Sheaf.existsUnique_gluing' C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf ⊢ ∃! s, ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le) C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U ⊢ ∃! s, ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i obtain ⟨gl, gl_spec, gl_uniq⟩ := F.existsUnique_gluing U sf h case intro.intro C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ ∃! s, ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i refine' ⟨F.1.map (eqToHom V_eq_supr_U).op gl, _, _⟩ case intro.intro.refine'_1 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ (fun s => ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i) (↑(F.val.map (eqToHom V_eq_supr_U).op) gl) intro i case intro.intro.refine'_1 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map (iUV i).op) (↑(F.val.map (eqToHom V_eq_supr_U).op) gl) = sf i rw [← comp_apply, ← F.1.map_comp] case intro.intro.refine'_1 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map ((eqToHom V_eq_supr_U).op ≫ (iUV i).op)) gl = sf i exact gl_spec i case intro.intro.refine'_2 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op V))), (fun s => ∀ (i : ι), ↑(F.val.map (iUV i).op) s = sf i) y → y = ↑(F.val.map (eqToHom V_eq_supr_U).op) gl intro gl' gl'_spec case intro.intro.refine'_2 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl gl' : (CategoryTheory.forget C).obj (F.val.obj (op V)) gl'_spec : ∀ (i : ι), ↑(F.val.map (iUV i).op) gl' = sf i ⊢ gl' = ↑(F.val.map (eqToHom V_eq_supr_U).op) gl convert congr_arg _ (gl_uniq (F.1.map (eqToHom V_eq_supr_U.symm).op gl') fun i => _) <;> rw [← comp_apply, ← F.1.map_comp] case h.e'_2 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl gl' : (CategoryTheory.forget C).obj (F.val.obj (op V)) gl'_spec : ∀ (i : ι), ↑(F.val.map (iUV i).op) gl' = sf i ⊢ gl' = ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (eqToHom V_eq_supr_U).op)) gl' rw [eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id, id_apply] case intro.intro.refine'_2.convert_3 C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) h : IsCompatible F.val U sf V_eq_supr_U : V = iSup U gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_spec : IsGluing F.val U sf gl gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl gl' : (CategoryTheory.forget C).obj (F.val.obj (op V)) gl'_spec : ∀ (i : ι), ↑(F.val.map (iUV i).op) gl' = sf i i : ι ⊢ ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) gl' = sf i convert gl'_spec i ** Qed
TopCat.Sheaf.eq_of_locally_eq C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t ⊢ s = t let sf : ∀ i : ι, F.1.obj (op (U i)) := fun i => F.1.map (Opens.leSupr U i).op s C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s ⊢ s = t have sf_compatible : IsCompatible _ U sf := by intro i j simp_rw [← comp_apply, ← F.1.map_comp] rfl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf ⊢ s = t obtain ⟨gl, -, gl_uniq⟩ := F.existsUnique_gluing U sf sf_compatible case intro.intro C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ s = t trans gl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s ⊢ IsCompatible F.val U sf intro i j C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s i j : ι ⊢ ↑(F.val.map (infLELeft (U i) (U j)).op) (sf i) = ↑(F.val.map (infLERight (U i) (U j)).op) (sf j) simp_rw [← comp_apply, ← F.1.map_comp] C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s i j : ι ⊢ ↑(F.val.map ((leSupr U i).op ≫ (infLELeft (U i) (U j)).op)) s = ↑(F.val.map ((leSupr U j).op ≫ (infLERight (U i) (U j)).op)) s rfl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ s = gl apply gl_uniq case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ IsGluing F.val U sf s intro i case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map (leSupr U i).op) s = sf i rfl C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ gl = t symm C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ t = gl apply gl_uniq case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl ⊢ IsGluing F.val U sf t intro i case a C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X s t : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) h : ∀ (i : ι), ↑(F.val.map (leSupr U i).op) s = ↑(F.val.map (leSupr U i).op) t sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (op (U i))) := fun i => ↑(F.val.map (leSupr U i).op) s sf_compatible : IsCompatible F.val U sf gl : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U))) gl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl i : ι ⊢ ↑(F.val.map (leSupr U i).op) t = sf i rw [← h] ** Qed
TopCat.Sheaf.eq_of_locally_eq' C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t ⊢ s = t have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le) C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U ⊢ s = t suffices F.1.map (eqToHom V_eq_supr_U.symm).op s = F.1.map (eqToHom V_eq_supr_U.symm).op t by convert congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;> rw [← comp_apply, ← F.1.map_comp, eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id, id_apply] C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U ⊢ ↑(F.val.map (eqToHom (_ : iSup U = V)).op) s = ↑(F.val.map (eqToHom (_ : iSup U = V)).op) t apply eq_of_locally_eq case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U ⊢ ∀ (i : ι), ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) s) = ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) t) intro i case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U i : ι ⊢ ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) s) = ↑(F.val.map (leSupr U i).op) (↑(F.val.map (eqToHom (_ : iSup U = V)).op) t) rw [← comp_apply, ← comp_apply, ← F.1.map_comp] case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U i : ι ⊢ ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) s = ↑(F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) t convert h i C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X V : Opens ↑X iUV : (i : ι) → U i ⟶ V hcover : V ≤ iSup U s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h : ∀ (i : ι), ↑(F.val.map (iUV i).op) s = ↑(F.val.map (iUV i).op) t V_eq_supr_U : V = iSup U this : ↑(F.val.map (eqToHom (_ : iSup U = V)).op) s = ↑(F.val.map (eqToHom (_ : iSup U = V)).op) t ⊢ s = t convert congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;> rw [← comp_apply, ← F.1.map_comp, eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id, id_apply] ** Qed
TopCat.Sheaf.eq_of_locally_eq₂ C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ s = t fapply F.eq_of_locally_eq' fun t : ULift Bool => if t.1 then U₁ else U₂ case iUV C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ (i : ULift.{v, 0} Bool) → (if i.down = true then U₁ else U₂) ⟶ V exact fun i => if h : i.1 then eqToHom (if_pos h) ≫ i₁ else eqToHom (if_neg h) ≫ i₂ case hcover C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ V ≤ ⨆ t, if t.down = true then U₁ else U₂ refine' le_trans hcover _ case hcover C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ U₁ ⊔ U₂ ≤ ⨆ t, if t.down = true then U₁ else U₂ rw [sup_le_iff] case hcover C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ (U₁ ≤ ⨆ t, if t.down = true then U₁ else U₂) ∧ U₂ ≤ ⨆ t, if t.down = true then U₁ else U₂ constructor case hcover.left C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ U₁ ≤ ⨆ t, if t.down = true then U₁ else U₂ convert le_iSup (fun t : ULift Bool => if t.1 then U₁ else U₂) (ULift.up true) case hcover.right C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ U₂ ≤ ⨆ t, if t.down = true then U₁ else U₂ convert le_iSup (fun t : ULift Bool => if t.1 then U₁ else U₂) (ULift.up false) case h C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ∀ (i : ULift.{v, 0} Bool), ↑(F.val.map (if h : i.down = true then eqToHom (_ : (if i.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if i.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : i.down = true then eqToHom (_ : (if i.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if i.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t rintro ⟨_ \| _⟩ case h.up.false C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t case h.up.true C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t any_goals exact h₁ case h.up.false C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t any_goals exact h₂ case h.up.true C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := true }.down = true then eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := true }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t exact h₁ case h.up.false C : Type u inst✝⁴ : Category.{v, u} C inst✝³ : ConcreteCategory C inst✝² : HasLimits C inst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget inst✝ : PreservesLimits ConcreteCategory.forget X : TopCat F : Sheaf C X ι : Type v U : ι → Opens ↑X U₁ U₂ V : Opens ↑X i₁ : U₁ ⟶ V i₂ : U₂ ⟶ V hcover : V ≤ U₁ ⊔ U₂ s t : (CategoryTheory.forget C).obj (F.val.obj (op V)) h₁ : ↑(F.val.map i₁.op) s = ↑(F.val.map i₁.op) t h₂ : ↑(F.val.map i₂.op) s = ↑(F.val.map i₂.op) t ⊢ ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) s = ↑(F.val.map (if h : { down := false }.down = true then eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₁) ≫ i₁ else eqToHom (_ : (if { down := false }.down = true then U₁ else U₂) = U₂) ≫ i₂).op) t exact h₂ ** Qed
TopCat.Presheaf.SheafConditionEqualizerProducts.res_π C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X i : ι ⊢ res F U ≫ limit.π (Discrete.functor fun i => F.obj (op (U i))) { as := i } = F.map (leSupr U i).op rw [res, limit.lift_π, Fan.mk_π_app] ** Qed
TopCat.Presheaf.SheafConditionEqualizerProducts.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X ⊢ res F U ≫ leftRes F U = res F U ≫ rightRes F U dsimp [res, leftRes, rightRes] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X ⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.1 ≫ F.map (infLELeft (U p.1) (U p.2)).op) = (Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.2 ≫ F.map (infLERight (U p.1) (U p.2)).op refine limit.hom_ext (fun _ => ?_) C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ ((Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.1 ≫ F.map (infLELeft (U p.1) (U p.2)).op) ≫ limit.π (Discrete.functor fun p => F.obj (op (U p.1 ⊓ U p.2))) x✝ = ((Pi.lift fun i => F.map (leSupr U i).op) ≫ Pi.lift fun p => Pi.π (fun i => F.obj (op (U i))) p.2 ≫ F.map (infLERight (U p.1) (U p.2)).op) ≫ limit.π (Discrete.functor fun p => F.obj (op (U p.1 ⊓ U p.2))) x✝ simp only [limit.lift_π, limit.lift_π_assoc, Fan.mk_π_app, Category.assoc] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ F.map (leSupr U x✝.as.1).op ≫ F.map (infLELeft (U x✝.as.1) (U x✝.as.2)).op = F.map (leSupr U x✝.as.2).op ≫ F.map (infLERight (U x✝.as.1) (U x✝.as.2)).op rw [← F.map_comp] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ F.map ((leSupr U x✝.as.1).op ≫ (infLELeft (U x✝.as.1) (U x✝.as.2)).op) = F.map (leSupr U x✝.as.2).op ≫ F.map (infLERight (U x✝.as.1) (U x✝.as.2)).op rw [← F.map_comp] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasProducts C X : TopCat F : Presheaf C X ι : Type v' U : ι → Opens ↑X x✝ : Discrete (ι × ι) ⊢ F.map ((leSupr U x✝.as.1).op ≫ (infLELeft (U x✝.as.1) (U x✝.as.2)).op) = F.map ((leSupr U x✝.as.2).op ≫ (infLERight (U x✝.as.1) (U x✝.as.2)).op) congr 1 ** Qed
SetTheory.PGame.Domineering.fst_pred_mem_erase_of_mem_right b : Board m : ℤ × ℤ h : m ∈ right b ⊢ (m.1 - 1, m.2) ∈ Finset.erase b m rw [mem_right] at h b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1 - 1, m.2) ∈ b ⊢ (m.1 - 1, m.2) ∈ Finset.erase b m apply Finset.mem_erase_of_ne_of_mem _ h.2 b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1 - 1, m.2) ∈ b ⊢ (m.1 - 1, m.2) ≠ m exact ne_of_apply_ne Prod.fst (pred_ne_self m.1) ** Qed
SetTheory.PGame.Domineering.snd_pred_mem_erase_of_mem_left b : Board m : ℤ × ℤ h : m ∈ left b ⊢ (m.1, m.2 - 1) ∈ Finset.erase b m rw [mem_left] at h b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1, m.2 - 1) ∈ b ⊢ (m.1, m.2 - 1) ∈ Finset.erase b m apply Finset.mem_erase_of_ne_of_mem _ h.2 b : Board m : ℤ × ℤ h : m ∈ b ∧ (m.1, m.2 - 1) ∈ b ⊢ (m.1, m.2 - 1) ≠ m exact ne_of_apply_ne Prod.snd (pred_ne_self m.2) ** Qed
SetTheory.PGame.Domineering.card_of_mem_left b : Board m : ℤ × ℤ h : m ∈ left b ⊢ 2 ≤ Finset.card b have w₁ : m ∈ b := (Finset.mem_inter.1 h).1 b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b ⊢ 2 ≤ Finset.card b have w₂ : (m.1, m.2 - 1) ∈ b.erase m := snd_pred_mem_erase_of_mem_left h b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b w₂ : (m.1, m.2 - 1) ∈ Finset.erase b m ⊢ 2 ≤ Finset.card b have i₁ := Finset.card_erase_lt_of_mem w₁ b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b w₂ : (m.1, m.2 - 1) ∈ Finset.erase b m i₁ : Finset.card (Finset.erase b m) < Finset.card b ⊢ 2 ≤ Finset.card b have i₂ := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w₂) b : Board m : ℤ × ℤ h : m ∈ left b w₁ : m ∈ b w₂ : (m.1, m.2 - 1) ∈ Finset.erase b m i₁ : Finset.card (Finset.erase b m) < Finset.card b i₂ : 0 < Finset.card (Finset.erase b m) ⊢ 2 ≤ Finset.card b exact Nat.lt_of_le_of_lt i₂ i₁ ** Qed