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Profinite.IndexFunctor.surjective_π_app ** ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop ⊢ Function.Surjective ↑(π_app C J) ** intro x ** ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop x : ↑(obj C J) ⊢ ∃ a, ↑(π_app C J) a = x ** obtain ⟨y, hy⟩ := x.prop ** case intro ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop x : ↑(obj C J) y : (i : ι) → (fun i => X i) i hy : y ∈ C ∧ ↑(precomp Subtype.val) y = ↑x ⊢ ∃ a, ↑(π_app C J) a = x ** exact ⟨⟨y, hy.1⟩, Subtype.ext hy.2⟩ ** Qed
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Profinite.IndexFunctor.eq_of_forall_π_app_eq ** ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop a b : ↑C h : ∀ (J : Finset ι), ↑(π_app C fun x => x ∈ J) a = ↑(π_app C fun x => x ∈ J) b ⊢ a = b ** ext i ** case a.h ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop a b : ↑C h : ∀ (J : Finset ι), ↑(π_app C fun x => x ∈ J) a = ↑(π_app C fun x => x ∈ J) b i : ι ⊢ ↑a i = ↑b i ** specialize h ({i} : Finset ι) ** case a.h ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop a b : ↑C i : ι h : ↑(π_app C fun x => x ∈ {i}) a = ↑(π_app C fun x => x ∈ {i}) b ⊢ ↑a i = ↑b i ** rw [Subtype.ext_iff] at h ** case a.h ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop a b : ↑C i : ι h : ↑(↑(π_app C fun x => x ∈ {i}) a) = ↑(↑(π_app C fun x => x ∈ {i}) b) ⊢ ↑a i = ↑b i ** simp only [π_app, ContinuousMap.precomp, ContinuousMap.coe_mk,
Set.MapsTo.val_restrict_apply] at h ** case a.h ι : Type u X : ι → Type inst✝ : (i : ι) → TopologicalSpace (X i) C : Set ((i : ι) → X i) J K : ι → Prop a b : ↑C i : ι h : (fun j => ↑a ↑j) = fun j => ↑b ↑j ⊢ ↑a i = ↑b i ** exact congr_fun h ⟨i, Finset.mem_singleton.mpr rfl⟩ ** Qed
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TopCat.pullbackIsoProdSubtype_inv_fst ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z ⊢ (pullbackIsoProdSubtype f g).inv ≫ pullback.fst = pullbackFst f g ** simp [pullbackCone, pullbackIsoProdSubtype] ** Qed
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TopCat.pullbackIsoProdSubtype_inv_snd ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z ⊢ (pullbackIsoProdSubtype f g).inv ≫ pullback.snd = pullbackSnd f g ** simp [pullbackCone, pullbackIsoProdSubtype] ** Qed
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TopCat.pullbackIsoProdSubtype_hom_fst ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z ⊢ (pullbackIsoProdSubtype f g).hom ≫ pullbackFst f g = pullback.fst ** rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_fst] ** Qed
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TopCat.pullbackIsoProdSubtype_hom_snd ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z ⊢ (pullbackIsoProdSubtype f g).hom ≫ pullbackSnd f g = pullback.snd ** rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_snd] ** Qed
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TopCat.pullbackIsoProdSubtype_hom_apply ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : ConcreteCategory.forget.obj (pullback f g) ⊢ ↑f (↑pullback.fst x, ↑pullback.snd x).1 = ↑g (↑pullback.fst x, ↑pullback.snd x).2 ** simpa using ConcreteCategory.congr_hom pullback.condition x ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : ConcreteCategory.forget.obj (pullback f g) ⊢ ↑(pullbackIsoProdSubtype f g).hom x = { val := (↑pullback.fst x, ↑pullback.snd x), property := (_ : ↑f (↑pullback.fst x, ↑pullback.snd x).1 = ↑g (↑pullback.fst x, ↑pullback.snd x).2) } ** apply Subtype.ext ** case a J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : ConcreteCategory.forget.obj (pullback f g) ⊢ ↑(↑(pullbackIsoProdSubtype f g).hom x) = ↑{ val := (↑pullback.fst x, ↑pullback.snd x), property := (_ : ↑f (↑pullback.fst x, ↑pullback.snd x).1 = ↑g (↑pullback.fst x, ↑pullback.snd x).2) } ** apply Prod.ext ** case a.h₁ J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : ConcreteCategory.forget.obj (pullback f g) ⊢ (↑(↑(pullbackIsoProdSubtype f g).hom x)).1 = (↑{ val := (↑pullback.fst x, ↑pullback.snd x), property := (_ : ↑f (↑pullback.fst x, ↑pullback.snd x).1 = ↑g (↑pullback.fst x, ↑pullback.snd x).2) }).1 case a.h₂ J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : ConcreteCategory.forget.obj (pullback f g) ⊢ (↑(↑(pullbackIsoProdSubtype f g).hom x)).2 = (↑{ val := (↑pullback.fst x, ↑pullback.snd x), property := (_ : ↑f (↑pullback.fst x, ↑pullback.snd x).1 = ↑g (↑pullback.fst x, ↑pullback.snd x).2) }).2 ** exacts [ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_fst f g) x,
ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_snd f g) x] ** Qed
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TopCat.pullback_topology ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z ⊢ (pullback f g).str = induced (↑pullback.fst) X.str ⊓ induced (↑pullback.snd) Y.str ** let homeo := homeoOfIso (pullbackIsoProdSubtype f g) ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z homeo : ↑(pullback f g) ≃ₜ ↑(of { p // ↑f p.1 = ↑g p.2 }) := homeoOfIso (pullbackIsoProdSubtype f g) ⊢ (pullback f g).str = induced (↑pullback.fst) X.str ⊓ induced (↑pullback.snd) Y.str ** refine' homeo.inducing.induced.trans _ ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z homeo : ↑(pullback f g) ≃ₜ ↑(of { p // ↑f p.1 = ↑g p.2 }) := homeoOfIso (pullbackIsoProdSubtype f g) ⊢ induced (↑homeo) (topologicalSpace_coe (of { p // ↑f p.1 = ↑g p.2 })) = induced (↑pullback.fst) X.str ⊓ induced (↑pullback.snd) Y.str ** change induced homeo (induced _ ( (induced Prod.fst X.str) ⊓ (induced Prod.snd Y.str))) = _ ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z homeo : ↑(pullback f g) ≃ₜ ↑(of { p // ↑f p.1 = ↑g p.2 }) := homeoOfIso (pullbackIsoProdSubtype f g) ⊢ induced (↑homeo) (induced Subtype.val (induced Prod.fst X.str ⊓ induced Prod.snd Y.str)) = induced (↑pullback.fst) X.str ⊓ induced (↑pullback.snd) Y.str ** simp only [induced_compose, induced_inf] ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z homeo : ↑(pullback f g) ≃ₜ ↑(of { p // ↑f p.1 = ↑g p.2 }) := homeoOfIso (pullbackIsoProdSubtype f g) ⊢ induced ((Prod.fst ∘ Subtype.val) ∘ ↑(homeoOfIso (pullbackIsoProdSubtype f g))) X.str ⊓ induced ((Prod.snd ∘ Subtype.val) ∘ ↑(homeoOfIso (pullbackIsoProdSubtype f g))) Y.str = induced (↑pullback.fst) X.str ⊓ induced (↑pullback.snd) Y.str ** congr ** Qed
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TopCat.range_pullback_to_prod ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z ⊢ Set.range ↑(prod.lift pullback.fst pullback.snd) = {x | ↑(prod.fst ≫ f) x = ↑(prod.snd ≫ g) x} ** ext x ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : (forget TopCat).obj (X ⨯ Y) ⊢ x ∈ Set.range ↑(prod.lift pullback.fst pullback.snd) ↔ x ∈ {x | ↑(prod.fst ≫ f) x = ↑(prod.snd ≫ g) x} ** constructor ** case h.mp J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : (forget TopCat).obj (X ⨯ Y) ⊢ x ∈ Set.range ↑(prod.lift pullback.fst pullback.snd) → x ∈ {x | ↑(prod.fst ≫ f) x = ↑(prod.snd ≫ g) x} ** rintro ⟨y, rfl⟩ ** case h.mp.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z y : (forget TopCat).obj (pullback f g) ⊢ ↑(prod.lift pullback.fst pullback.snd) y ∈ {x | ↑(prod.fst ≫ f) x = ↑(prod.snd ≫ g) x} ** simp only [← comp_apply, Set.mem_setOf_eq] ** case h.mp.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z y : (forget TopCat).obj (pullback f g) ⊢ ↑(prod.lift pullback.fst pullback.snd ≫ prod.fst ≫ f) y = ↑(prod.lift pullback.fst pullback.snd ≫ prod.snd ≫ g) y ** congr 1 ** case h.mp.intro.e_a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z y : (forget TopCat).obj (pullback f g) ⊢ prod.lift pullback.fst pullback.snd ≫ prod.fst ≫ f = prod.lift pullback.fst pullback.snd ≫ prod.snd ≫ g ** simp [pullback.condition] ** case h.mpr J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : (forget TopCat).obj (X ⨯ Y) ⊢ x ∈ {x | ↑(prod.fst ≫ f) x = ↑(prod.snd ≫ g) x} → x ∈ Set.range ↑(prod.lift pullback.fst pullback.snd) ** rintro (h : f (_, _).1 = g (_, _).2) ** case h.mpr J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : (forget TopCat).obj (X ⨯ Y) h : ↑f (↑prod.fst x, ?m.55730).1 = ↑g (?m.55765, ↑prod.snd x).2 ⊢ x ∈ Set.range ↑(prod.lift pullback.fst pullback.snd) ** use (pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, h⟩ ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : (forget TopCat).obj (X ⨯ Y) h : ↑f (↑prod.fst x, ↑prod.snd x).1 = ↑g (↑prod.fst x, ↑prod.snd x).2 ⊢ ↑(prod.lift pullback.fst pullback.snd) (↑(pullbackIsoProdSubtype f g).inv { val := (↑prod.fst x, ↑prod.snd x), property := h }) = x ** apply Concrete.limit_ext ** case h.a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z x : (forget TopCat).obj (X ⨯ Y) h : ↑f (↑prod.fst x, ↑prod.snd x).1 = ↑g (↑prod.fst x, ↑prod.snd x).2 ⊢ ∀ (j : Discrete WalkingPair), ↑(limit.π (pair X Y) j) (↑(prod.lift pullback.fst pullback.snd) (↑(pullbackIsoProdSubtype f g).inv { val := (↑prod.fst x, ↑prod.snd x), property := h })) = ↑(limit.π (pair X Y) j) x ** rintro ⟨⟨⟩⟩ <;>
rw [←comp_apply, prod.comp_lift, ←comp_apply, limit.lift_π] <;>
aesop_cat ** Qed
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TopCat.inducing_pullback_to_prod ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z ⊢ topologicalSpace_forget (pullback f g) = induced (↑(prod.lift pullback.fst pullback.snd)) (topologicalSpace_forget (X ⨯ Y)) ** simp [prod_topology, pullback_topology, induced_compose, ← coe_comp] ** Qed
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TopCat.range_pullback_map ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Set.range ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) = ↑pullback.fst ⁻¹' Set.range ↑i₁ ∩ ↑pullback.snd ⁻¹' Set.range ↑i₂ ** ext ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) ⊢ x✝ ∈ Set.range ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) ↔ x✝ ∈ ↑pullback.fst ⁻¹' Set.range ↑i₁ ∩ ↑pullback.snd ⁻¹' Set.range ↑i₂ ** constructor ** case h.mpr J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) ⊢ x✝ ∈ ↑pullback.fst ⁻¹' Set.range ↑i₁ ∩ ↑pullback.snd ⁻¹' Set.range ↑i₂ → x✝ ∈ Set.range ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) ** rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩ ** case h.mpr.intro.intro.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ ⊢ x✝ ∈ Set.range ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) ** have : f₁ x₁ = f₂ x₂ := by
apply (TopCat.mono_iff_injective _).mp H₃
simp only [← comp_apply, eq₁, eq₂]
simp only [comp_apply, hx₁, hx₂]
simp only [← comp_apply, pullback.condition] ** case h.mpr.intro.intro.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ this : ↑f₁ x₁ = ↑f₂ x₂ ⊢ x✝ ∈ Set.range ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) ** use (pullbackIsoProdSubtype f₁ f₂).inv ⟨⟨x₁, x₂⟩, this⟩ ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ this : ↑f₁ x₁ = ↑f₂ x₂ ⊢ ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) (↑(pullbackIsoProdSubtype f₁ f₂).inv { val := (x₁, x₂), property := this }) = x✝ ** apply Concrete.limit_ext ** case h.a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ this : ↑f₁ x₁ = ↑f₂ x₂ ⊢ ∀ (j : WalkingCospan), ↑(limit.π (cospan g₁ g₂) j) (↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) (↑(pullbackIsoProdSubtype f₁ f₂).inv { val := (x₁, x₂), property := this })) = ↑(limit.π (cospan g₁ g₂) j) x✝ ** rintro (_ | _ | _) <;>
simp only [←comp_apply, Category.assoc, limit.lift_π, PullbackCone.mk_π_app_one] ** case h.mp J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) ⊢ x✝ ∈ Set.range ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) → x✝ ∈ ↑pullback.fst ⁻¹' Set.range ↑i₁ ∩ ↑pullback.snd ⁻¹' Set.range ↑i₂ ** rintro ⟨y, rfl⟩ ** case h.mp.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ y : (forget TopCat).obj (pullback f₁ f₂) ⊢ ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) y ∈ ↑pullback.fst ⁻¹' Set.range ↑i₁ ∩ ↑pullback.snd ⁻¹' Set.range ↑i₂ ** simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range, ←comp_apply, limit.lift_π,
PullbackCone.mk_pt, PullbackCone.mk_π_app] ** case h.mp.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ y : (forget TopCat).obj (pullback f₁ f₂) ⊢ (∃ y_1, ↑i₁ y_1 = ↑(pullback.fst ≫ i₁) y) ∧ ∃ y_1, ↑i₂ y_1 = ↑(pullback.snd ≫ i₂) y ** simp only [comp_apply, exists_apply_eq_apply, and_self] ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ ⊢ ↑f₁ x₁ = ↑f₂ x₂ ** apply (TopCat.mono_iff_injective _).mp H₃ ** case a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ ⊢ ↑i₃ (↑f₁ x₁) = ↑i₃ (↑f₂ x₂) ** simp only [← comp_apply, eq₁, eq₂] ** case a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ ⊢ ↑(i₁ ≫ g₁) x₁ = ↑(i₂ ≫ g₂) x₂ ** simp only [comp_apply, hx₁, hx₂] ** case a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ ⊢ ↑g₁ (↑pullback.fst x✝) = ↑g₂ (↑pullback.snd x✝) ** simp only [← comp_apply, pullback.condition] ** case h.a.none J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ this : ↑f₁ x₁ = ↑f₂ x₂ ⊢ ↑((pullbackIsoProdSubtype f₁ f₂).inv ≫ pullback.fst ≫ i₁ ≫ g₁) { val := (x₁, x₂), property := this } = ↑(limit.π (cospan g₁ g₂) none) x✝ ** simp only [cospan_one, pullbackIsoProdSubtype_inv_fst_assoc, comp_apply,
pullbackFst_apply, hx₁] ** case h.a.none J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ this : ↑f₁ x₁ = ↑f₂ x₂ ⊢ ↑g₁ (↑pullback.fst x✝) = ↑(limit.π (cospan g₁ g₂) none) x✝ ** rw [← limit.w _ WalkingCospan.Hom.inl, cospan_map_inl, comp_apply (g := g₁)] ** case h.a.some.left J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ this : ↑f₁ x₁ = ↑f₂ x₂ ⊢ ↑((pullbackIsoProdSubtype f₁ f₂).inv ≫ (PullbackCone.mk (pullback.fst ≫ i₁) (pullback.snd ≫ i₂) (_ : (pullback.fst ≫ i₁) ≫ g₁ = (pullback.snd ≫ i₂) ≫ g₂)).π.app (some WalkingPair.left)) { val := (x₁, x₂), property := this } = ↑(limit.π (cospan g₁ g₂) (some WalkingPair.left)) x✝ ** simp [hx₁] ** case h.a.some.right J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ : TopCat W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ x✝ : (forget TopCat).obj (pullback g₁ g₂) x₁ : (forget TopCat).obj W hx₁ : ↑i₁ x₁ = ↑pullback.fst x✝ x₂ : (forget TopCat).obj X hx₂ : ↑i₂ x₂ = ↑pullback.snd x✝ this : ↑f₁ x₁ = ↑f₂ x₂ ⊢ ↑((pullbackIsoProdSubtype f₁ f₂).inv ≫ (PullbackCone.mk (pullback.fst ≫ i₁) (pullback.snd ≫ i₂) (_ : (pullback.fst ≫ i₁) ≫ g₁ = (pullback.snd ≫ i₂) ≫ g₂)).π.app (some WalkingPair.right)) { val := (x₁, x₂), property := this } = ↑(limit.π (cospan g₁ g₂) (some WalkingPair.right)) x✝ ** simp [hx₂] ** Qed
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TopCat.pullback_fst_range ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S ⊢ Set.range ↑pullback.fst = {x | ∃ y, ↑f x = ↑g y} ** ext x ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S x : (forget TopCat).obj X ⊢ x ∈ Set.range ↑pullback.fst ↔ x ∈ {x | ∃ y, ↑f x = ↑g y} ** constructor ** case h.mp J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S x : (forget TopCat).obj X ⊢ x ∈ Set.range ↑pullback.fst → x ∈ {x | ∃ y, ↑f x = ↑g y} ** rintro ⟨y, rfl⟩ ** case h.mp.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S y : (forget TopCat).obj (pullback f g) ⊢ ↑pullback.fst y ∈ {x | ∃ y, ↑f x = ↑g y} ** use (pullback.snd : pullback f g ⟶ _) y ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S y : (forget TopCat).obj (pullback f g) ⊢ ↑f (↑pullback.fst y) = ↑g (↑pullback.snd y) ** exact ConcreteCategory.congr_hom pullback.condition y ** case h.mpr J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S x : (forget TopCat).obj X ⊢ x ∈ {x | ∃ y, ↑f x = ↑g y} → x ∈ Set.range ↑pullback.fst ** rintro ⟨y, eq⟩ ** case h.mpr.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S x : (forget TopCat).obj X y : ↑Y eq : ↑f x = ↑g y ⊢ x ∈ Set.range ↑pullback.fst ** use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩ ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S x : (forget TopCat).obj X y : ↑Y eq : ↑f x = ↑g y ⊢ ↑pullback.fst (↑(pullbackIsoProdSubtype f g).inv { val := (x, y), property := eq }) = x ** simp ** Qed
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TopCat.pullback_snd_range ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S ⊢ Set.range ↑pullback.snd = {y | ∃ x, ↑f x = ↑g y} ** ext y ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S y : (forget TopCat).obj Y ⊢ y ∈ Set.range ↑pullback.snd ↔ y ∈ {y | ∃ x, ↑f x = ↑g y} ** constructor ** case h.mp J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S y : (forget TopCat).obj Y ⊢ y ∈ Set.range ↑pullback.snd → y ∈ {y | ∃ x, ↑f x = ↑g y} ** rintro ⟨x, rfl⟩ ** case h.mp.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S x : (forget TopCat).obj (pullback f g) ⊢ ↑pullback.snd x ∈ {y | ∃ x, ↑f x = ↑g y} ** use (pullback.fst : pullback f g ⟶ _) x ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S x : (forget TopCat).obj (pullback f g) ⊢ ↑f (↑pullback.fst x) = ↑g (↑pullback.snd x) ** exact ConcreteCategory.congr_hom pullback.condition x ** case h.mpr J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S y : (forget TopCat).obj Y ⊢ y ∈ {y | ∃ x, ↑f x = ↑g y} → y ∈ Set.range ↑pullback.snd ** rintro ⟨x, eq⟩ ** case h.mpr.intro J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S y : (forget TopCat).obj Y x : ↑X eq : ↑f x = ↑g y ⊢ y ∈ Set.range ↑pullback.snd ** use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩ ** case h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S y : (forget TopCat).obj Y x : ↑X eq : ↑f x = ↑g y ⊢ ↑pullback.snd (↑(pullbackIsoProdSubtype f g).inv { val := (x, y), property := eq }) = y ** simp ** Qed
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TopCat.pullback_map_embedding_of_embeddings ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : Embedding ↑i₁ H₂ : Embedding ↑i₂ i₃ : S ⟶ T eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Embedding ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) ** refine'
embedding_of_embedding_compose (ContinuousMap.continuous_toFun _)
(show Continuous (prod.lift pullback.fst pullback.snd : pullback g₁ g₂ ⟶ Y ⨯ Z) from
ContinuousMap.continuous_toFun _)
_ ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : Embedding ↑i₁ H₂ : Embedding ↑i₂ i₃ : S ⟶ T eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Embedding (↑(prod.lift pullback.fst pullback.snd) ∘ ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)) ** suffices
Embedding (prod.lift pullback.fst pullback.snd ≫ Limits.prod.map i₁ i₂ : pullback f₁ f₂ ⟶ _) by
simpa [← coe_comp] using this ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : Embedding ↑i₁ H₂ : Embedding ↑i₂ i₃ : S ⟶ T eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Embedding ↑(prod.lift pullback.fst pullback.snd ≫ prod.map i₁ i₂) ** rw [coe_comp] ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : Embedding ↑i₁ H₂ : Embedding ↑i₂ i₃ : S ⟶ T eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Embedding (↑(prod.map i₁ i₂) ∘ ↑(prod.lift pullback.fst pullback.snd)) ** refine Embedding.comp (embedding_prod_map H₁ H₂) (embedding_pullback_to_prod _ _) ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : Embedding ↑i₁ H₂ : Embedding ↑i₂ i₃ : S ⟶ T eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ this : Embedding ↑(prod.lift pullback.fst pullback.snd ≫ prod.map i₁ i₂) ⊢ Embedding (↑(prod.lift pullback.fst pullback.snd) ∘ ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)) ** simpa [← coe_comp] using this ** Qed
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TopCat.pullback_map_openEmbedding_of_open_embeddings ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ OpenEmbedding ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) ** constructor ** case toEmbedding J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Embedding ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) ** apply
pullback_map_embedding_of_embeddings f₁ f₂ g₁ g₂ H₁.toEmbedding H₂.toEmbedding i₃ eq₁ eq₂ ** case open_range J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ IsOpen (Set.range ↑(pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)) ** rw [range_pullback_map] ** case open_range J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ IsOpen (↑pullback.fst ⁻¹' Set.range ↑i₁ ∩ ↑pullback.snd ⁻¹' Set.range ↑i₂) ** apply IsOpen.inter <;> apply Continuous.isOpen_preimage ** case open_range.h₁.self J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Continuous ↑pullback.fst ** apply ContinuousMap.continuous_toFun ** case open_range.h₁.a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ IsOpen (Set.range ↑i₁) ** exact H₁.open_range ** case open_range.h₂.self J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ Continuous ↑pullback.snd ** apply ContinuousMap.continuous_toFun ** case open_range.h₂.a J : Type v inst✝ : SmallCategory J X✝ Y✝ Z✝ W X Y Z S T : TopCat f₁ : W ⟶ S f₂ : X ⟶ S g₁ : Y ⟶ T g₂ : Z ⟶ T i₁ : W ⟶ Y i₂ : X ⟶ Z H₁ : OpenEmbedding ↑i₁ H₂ : OpenEmbedding ↑i₂ i₃ : S ⟶ T H₃ : Mono i₃ eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁ eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂ ⊢ IsOpen (Set.range ↑i₂) ** exact H₂.open_range ** Qed
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TopCat.snd_embedding_of_left_embedding ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : Embedding ↑f g : Y ⟶ S ⊢ Embedding ↑pullback.snd ** convert (homeoOfIso (asIso (pullback.snd : pullback (𝟙 S) g ⟶ _))).embedding.comp
(pullback_map_embedding_of_embeddings (i₂ := 𝟙 Y)
f g (𝟙 S) g H (homeoOfIso (Iso.refl _)).embedding (𝟙 _) rfl (by simp)) ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : Embedding ↑f g : Y ⟶ S e_2✝ : (forget TopCat).obj Y = ↑Y ⊢ ↑pullback.snd = ↑(homeoOfIso (asIso pullback.snd)) ∘ ↑(pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g)) ** erw [← coe_comp] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : Embedding ↑f g : Y ⟶ S e_2✝ : (forget TopCat).obj Y = ↑Y ⊢ ↑pullback.snd = ↑(pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g) ≫ (asIso pullback.snd).hom) ** simp ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : Embedding ↑f g : Y ⟶ S ⊢ g ≫ 𝟙 S = 𝟙 Y ≫ g ** simp ** Qed
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TopCat.fst_embedding_of_right_embedding ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : Embedding ↑g ⊢ Embedding ↑pullback.fst ** convert (homeoOfIso (asIso (pullback.fst : pullback f (𝟙 S) ⟶ _))).embedding.comp
(pullback_map_embedding_of_embeddings (i₁ := 𝟙 X)
f g f (𝟙 _) (homeoOfIso (Iso.refl _)).embedding H (𝟙 _) rfl (by simp)) ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : Embedding ↑g e_2✝ : (forget TopCat).obj X = ↑X ⊢ ↑pullback.fst = ↑(homeoOfIso (asIso pullback.fst)) ∘ ↑(pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S)) ** erw [← coe_comp] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : Embedding ↑g e_2✝ : (forget TopCat).obj X = ↑X ⊢ ↑pullback.fst = ↑(pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S) ≫ (asIso pullback.fst).hom) ** simp ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : Embedding ↑g ⊢ g ≫ 𝟙 S = g ≫ 𝟙 S ** simp ** Qed
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TopCat.embedding_of_pullback_embeddings ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : Embedding ↑f H₂ : Embedding ↑g ⊢ Embedding ↑(limit.π (cospan f g) WalkingCospan.one) ** convert H₂.comp (snd_embedding_of_left_embedding H₁ g) ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : Embedding ↑f H₂ : Embedding ↑g e_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S ⊢ ↑(limit.π (cospan f g) WalkingCospan.one) = ↑g ∘ ↑pullback.snd ** erw [← coe_comp] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : Embedding ↑f H₂ : Embedding ↑g e_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S ⊢ ↑(limit.π (cospan f g) WalkingCospan.one) = ↑(pullback.snd ≫ g) ** rw [←limit.w _ WalkingCospan.Hom.inr] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : Embedding ↑f H₂ : Embedding ↑g e_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S ⊢ ↑(limit.π (cospan f g) WalkingCospan.right ≫ (cospan f g).map WalkingCospan.Hom.inr) = ↑(pullback.snd ≫ g) ** rfl ** Qed
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TopCat.snd_openEmbedding_of_left_openEmbedding ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : OpenEmbedding ↑f g : Y ⟶ S ⊢ OpenEmbedding ↑pullback.snd ** convert (homeoOfIso (asIso (pullback.snd : pullback (𝟙 S) g ⟶ _))).openEmbedding.comp
(pullback_map_openEmbedding_of_open_embeddings (i₂ := 𝟙 Y) f g (𝟙 _) g H
(homeoOfIso (Iso.refl _)).openEmbedding (𝟙 _) rfl (by simp)) ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : OpenEmbedding ↑f g : Y ⟶ S e_2✝ : (forget TopCat).obj Y = ↑Y ⊢ ↑pullback.snd = ↑(homeoOfIso (asIso pullback.snd)) ∘ ↑(pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g)) ** erw [← coe_comp] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : OpenEmbedding ↑f g : Y ⟶ S e_2✝ : (forget TopCat).obj Y = ↑Y ⊢ ↑pullback.snd = ↑(pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g) ≫ (asIso pullback.snd).hom) ** simp ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S H : OpenEmbedding ↑f g : Y ⟶ S ⊢ g ≫ 𝟙 S = 𝟙 Y ≫ g ** simp ** Qed
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TopCat.fst_openEmbedding_of_right_openEmbedding ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : OpenEmbedding ↑g ⊢ OpenEmbedding ↑pullback.fst ** convert (homeoOfIso (asIso (pullback.fst : pullback f (𝟙 S) ⟶ _))).openEmbedding.comp
(pullback_map_openEmbedding_of_open_embeddings (i₁ := 𝟙 X) f g f (𝟙 _)
(homeoOfIso (Iso.refl _)).openEmbedding H (𝟙 _) rfl (by simp)) ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : OpenEmbedding ↑g e_2✝ : (forget TopCat).obj X = ↑X ⊢ ↑pullback.fst = ↑(homeoOfIso (asIso pullback.fst)) ∘ ↑(pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S)) ** erw [← coe_comp] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : OpenEmbedding ↑g e_2✝ : (forget TopCat).obj X = ↑X ⊢ ↑pullback.fst = ↑(pullback.map f g f (𝟙 S) (𝟙 X) g (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = g ≫ 𝟙 S) ≫ (asIso pullback.fst).hom) ** simp ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H : OpenEmbedding ↑g ⊢ g ≫ 𝟙 S = g ≫ 𝟙 S ** simp ** Qed
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TopCat.openEmbedding_of_pullback_open_embeddings ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : OpenEmbedding ↑f H₂ : OpenEmbedding ↑g ⊢ OpenEmbedding ↑(limit.π (cospan f g) WalkingCospan.one) ** convert H₂.comp (snd_openEmbedding_of_left_openEmbedding H₁ g) ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : OpenEmbedding ↑f H₂ : OpenEmbedding ↑g e_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S ⊢ ↑(limit.π (cospan f g) WalkingCospan.one) = ↑g ∘ ↑pullback.snd ** erw [← coe_comp] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : OpenEmbedding ↑f H₂ : OpenEmbedding ↑g e_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S ⊢ ↑(limit.π (cospan f g) WalkingCospan.one) = ↑(pullback.snd ≫ g) ** rw [←(limit.w _ WalkingCospan.Hom.inr)] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S H₁ : OpenEmbedding ↑f H₂ : OpenEmbedding ↑g e_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S ⊢ ↑(limit.π (cospan f g) WalkingCospan.right ≫ (cospan f g).map WalkingCospan.Hom.inr) = ↑(pullback.snd ≫ g) ** rfl ** Qed
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TopCat.fst_iso_of_right_embedding_range_subset ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S hg : Embedding ↑g H : Set.range ↑f ⊆ Set.range ↑g ⊢ IsIso pullback.fst ** let esto : (pullback f g : TopCat) ≃ₜ X :=
(Homeomorph.ofEmbedding _ (fst_embedding_of_right_embedding f hg)).trans
{ toFun := Subtype.val
invFun := fun x =>
⟨x, by
rw [pullback_fst_range]
exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩⟩
left_inv := fun ⟨_, _⟩ => rfl
right_inv := fun x => rfl } ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S hg : Embedding ↑g H : Set.range ↑f ⊆ Set.range ↑g esto : ↑(pullback f g) ≃ₜ ↑X := Homeomorph.trans (Homeomorph.ofEmbedding ↑pullback.fst (_ : Embedding ↑pullback.fst)) (Homeomorph.mk { toFun := Subtype.val, invFun := fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.fst) }, left_inv := (_ : ∀ (x : ↑(Set.range ↑pullback.fst)), (fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.fst) }) ↑x = x), right_inv := (_ : ∀ (x : ↑X), ↑((fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.fst) }) x) = ↑((fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.fst) }) x)) }) ⊢ IsIso pullback.fst ** convert IsIso.of_iso (isoOfHomeo esto) ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S hg : Embedding ↑g H : Set.range ↑f ⊆ Set.range ↑g x : ↑X ⊢ x ∈ Set.range ↑pullback.fst ** rw [pullback_fst_range] ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S g : Y ⟶ S hg : Embedding ↑g H : Set.range ↑f ⊆ Set.range ↑g x : ↑X ⊢ x ∈ {x | ∃ y, ↑f x = ↑g y} ** exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩ ** Qed
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TopCat.snd_iso_of_left_embedding_range_subset ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S hf : Embedding ↑f g : Y ⟶ S H : Set.range ↑g ⊆ Set.range ↑f ⊢ IsIso pullback.snd ** let esto : (pullback f g : TopCat) ≃ₜ Y :=
(Homeomorph.ofEmbedding _ (snd_embedding_of_left_embedding hf g)).trans
{ toFun := Subtype.val
invFun := fun x =>
⟨x, by
rw [pullback_snd_range]
exact ⟨_, (H (Set.mem_range_self x)).choose_spec⟩⟩
left_inv := fun ⟨_, _⟩ => rfl
right_inv := fun x => rfl } ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S hf : Embedding ↑f g : Y ⟶ S H : Set.range ↑g ⊆ Set.range ↑f esto : ↑(pullback f g) ≃ₜ ↑Y := Homeomorph.trans (Homeomorph.ofEmbedding ↑pullback.snd (_ : Embedding ↑pullback.snd)) (Homeomorph.mk { toFun := Subtype.val, invFun := fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.snd) }, left_inv := (_ : ∀ (x : ↑(Set.range ↑pullback.snd)), (fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.snd) }) ↑x = x), right_inv := (_ : ∀ (x : ↑Y), ↑((fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.snd) }) x) = ↑((fun x => { val := x, property := (_ : x ∈ Set.range ↑pullback.snd) }) x)) }) ⊢ IsIso pullback.snd ** convert IsIso.of_iso (isoOfHomeo esto) ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S hf : Embedding ↑f g : Y ⟶ S H : Set.range ↑g ⊆ Set.range ↑f x : ↑Y ⊢ x ∈ Set.range ↑pullback.snd ** rw [pullback_snd_range] ** J : Type v inst✝ : SmallCategory J X✝ Y✝ Z : TopCat X Y S : TopCat f : X ⟶ S hf : Embedding ↑f g : Y ⟶ S H : Set.range ↑g ⊆ Set.range ↑f x : ↑Y ⊢ x ∈ {y | ∃ x, ↑f x = ↑g y} ** exact ⟨_, (H (Set.mem_range_self x)).choose_spec⟩ ** Qed
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TopCat.pullback_snd_image_fst_preimage ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X ⊢ ↑pullback.snd '' (↑pullback.fst ⁻¹' U) = ↑g ⁻¹' (↑f '' U) ** ext x ** case h J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X x : (forget TopCat).obj Y ⊢ x ∈ ↑pullback.snd '' (↑pullback.fst ⁻¹' U) ↔ x ∈ ↑g ⁻¹' (↑f '' U) ** constructor ** case h.mp J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X x : (forget TopCat).obj Y ⊢ x ∈ ↑pullback.snd '' (↑pullback.fst ⁻¹' U) → x ∈ ↑g ⁻¹' (↑f '' U) ** rintro ⟨y, hy, rfl⟩ ** case h.mp.intro.intro J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X y : (forget TopCat).obj (pullback f g) hy : y ∈ ↑pullback.fst ⁻¹' U ⊢ ↑pullback.snd y ∈ ↑g ⁻¹' (↑f '' U) ** exact
⟨(pullback.fst : pullback f g ⟶ _) y, hy, ConcreteCategory.congr_hom pullback.condition y⟩ ** case h.mpr J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X x : (forget TopCat).obj Y ⊢ x ∈ ↑g ⁻¹' (↑f '' U) → x ∈ ↑pullback.snd '' (↑pullback.fst ⁻¹' U) ** rintro ⟨y, hy, eq⟩ ** case h.mpr.intro.intro J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X x : (forget TopCat).obj Y y : (forget TopCat).obj X hy : y ∈ U eq : ↑f y = ↑g x ⊢ x ∈ ↑pullback.snd '' (↑pullback.fst ⁻¹' U) ** exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq⟩, by simpa, by simp⟩ ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X x : (forget TopCat).obj Y y : (forget TopCat).obj X hy : y ∈ U eq : ↑f y = ↑g x ⊢ ↑(pullbackIsoProdSubtype f g).inv { val := (y, x), property := eq } ∈ ↑pullback.fst ⁻¹' U ** simpa ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑X x : (forget TopCat).obj Y y : (forget TopCat).obj X hy : y ∈ U eq : ↑f y = ↑g x ⊢ ↑pullback.snd (↑(pullbackIsoProdSubtype f g).inv { val := (y, x), property := eq }) = x ** simp ** Qed
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TopCat.pullback_fst_image_snd_preimage ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y ⊢ ↑pullback.fst '' (↑pullback.snd ⁻¹' U) = ↑f ⁻¹' (↑g '' U) ** ext x ** case h J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y x : (forget TopCat).obj X ⊢ x ∈ ↑pullback.fst '' (↑pullback.snd ⁻¹' U) ↔ x ∈ ↑f ⁻¹' (↑g '' U) ** constructor ** case h.mp J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y x : (forget TopCat).obj X ⊢ x ∈ ↑pullback.fst '' (↑pullback.snd ⁻¹' U) → x ∈ ↑f ⁻¹' (↑g '' U) ** rintro ⟨y, hy, rfl⟩ ** case h.mp.intro.intro J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y y : (forget TopCat).obj (pullback f g) hy : y ∈ ↑pullback.snd ⁻¹' U ⊢ ↑pullback.fst y ∈ ↑f ⁻¹' (↑g '' U) ** exact
⟨(pullback.snd : pullback f g ⟶ _) y, hy,
(ConcreteCategory.congr_hom pullback.condition y).symm⟩ ** case h.mpr J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y x : (forget TopCat).obj X ⊢ x ∈ ↑f ⁻¹' (↑g '' U) → x ∈ ↑pullback.fst '' (↑pullback.snd ⁻¹' U) ** rintro ⟨y, hy, eq⟩ ** case h.mpr.intro.intro J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y x : (forget TopCat).obj X y : (forget TopCat).obj Y hy : y ∈ U eq : ↑g y = ↑f x ⊢ x ∈ ↑pullback.fst '' (↑pullback.snd ⁻¹' U) ** exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq.symm⟩, by simpa, by simp⟩ ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y x : (forget TopCat).obj X y : (forget TopCat).obj Y hy : y ∈ U eq : ↑g y = ↑f x ⊢ ↑(pullbackIsoProdSubtype f g).inv { val := (x, y), property := (_ : ↑f x = ↑g y) } ∈ ↑pullback.snd ⁻¹' U ** simpa ** J : Type v inst✝ : SmallCategory J X Y Z : TopCat f : X ⟶ Z g : Y ⟶ Z U : Set ↑Y x : (forget TopCat).obj X y : (forget TopCat).obj Y hy : y ∈ U eq : ↑g y = ↑f x ⊢ ↑pullback.fst (↑(pullbackIsoProdSubtype f g).inv { val := (x, y), property := (_ : ↑f x = ↑g y) }) = x ** simp ** Qed
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TopCat.coinduced_of_isColimit ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax c : Cocone F hc : IsColimit c ⊢ c.pt.str = ⨆ j, coinduced (↑(c.ι.app j)) (F.obj j).str ** let homeo := homeoOfIso (hc.coconePointUniqueUpToIso (colimitCoconeIsColimit F)) ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax c : Cocone F hc : IsColimit c homeo : ↑c.pt ≃ₜ ↑(colimitCocone F).pt := homeoOfIso (IsColimit.coconePointUniqueUpToIso hc (colimitCoconeIsColimit F)) ⊢ c.pt.str = ⨆ j, coinduced (↑(c.ι.app j)) (F.obj j).str ** ext ** case a.h.a J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax c : Cocone F hc : IsColimit c homeo : ↑c.pt ≃ₜ ↑(colimitCocone F).pt := homeoOfIso (IsColimit.coconePointUniqueUpToIso hc (colimitCoconeIsColimit F)) x✝ : Set ↑c.pt ⊢ IsOpen x✝ ↔ IsOpen x✝ ** refine' homeo.symm.isOpen_preimage.symm.trans (Iff.trans _ isOpen_iSup_iff.symm) ** case a.h.a J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax c : Cocone F hc : IsColimit c homeo : ↑c.pt ≃ₜ ↑(colimitCocone F).pt := homeoOfIso (IsColimit.coconePointUniqueUpToIso hc (colimitCoconeIsColimit F)) x✝ : Set ↑c.pt ⊢ IsOpen (↑(Homeomorph.symm homeo) ⁻¹' x✝) ↔ ∀ (i : J), IsOpen x✝ ** exact isOpen_iSup_iff ** Qed
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TopCat.colimit_isOpen_iff ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax U : Set ↑(colimit F) ⊢ IsOpen U ↔ ∀ (j : J), IsOpen (↑(colimit.ι F j) ⁻¹' U) ** exact isOpen_iSup_iff ** Qed
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TopCat.coequalizer_isOpen_iff ** J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) ⊢ IsOpen U ↔ IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ** rw [colimit_isOpen_iff] ** J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) ⊢ (∀ (j : WalkingParallelPair), IsOpen (↑(colimit.ι F j) ⁻¹' U)) ↔ IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ** constructor ** case mp J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) ⊢ (∀ (j : WalkingParallelPair), IsOpen (↑(colimit.ι F j) ⁻¹' U)) → IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ** intro H ** case mp J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) H : ∀ (j : WalkingParallelPair), IsOpen (↑(colimit.ι F j) ⁻¹' U) ⊢ IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ** exact H _ ** case mpr J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) ⊢ IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) → ∀ (j : WalkingParallelPair), IsOpen (↑(colimit.ι F j) ⁻¹' U) ** intro H j ** case mpr J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) H : IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) j : WalkingParallelPair ⊢ IsOpen (↑(colimit.ι F j) ⁻¹' U) ** cases j ** case mpr.zero J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) H : IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ⊢ IsOpen (↑(colimit.ι F WalkingParallelPair.zero) ⁻¹' U) ** rw [← colimit.w F WalkingParallelPairHom.left] ** case mpr.zero J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) H : IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ⊢ IsOpen (↑(F.map WalkingParallelPairHom.left ≫ colimit.ι F WalkingParallelPair.one) ⁻¹' U) ** exact (F.map WalkingParallelPairHom.left).continuous_toFun.isOpen_preimage _ H ** case mpr.one J : Type v inst✝ : SmallCategory J F : WalkingParallelPair ⥤ TopCat U : Set ↑(colimit F) H : IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ⊢ IsOpen (↑(colimit.ι F WalkingParallelPair.one) ⁻¹' U) ** exact H ** Qed
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bot_topologicalSpace_eq_generateFrom_of_pred_succOrder ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α ⊢ ⊥ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ** refine' (eq_bot_of_singletons_open fun a => _).symm ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ⊢ IsOpen {a} ** rw [h_singleton_eq_inter] ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ⊢ IsOpen (Iio (succ a) ∩ Ioi (pred a)) ** apply @IsOpen.inter _ _ _ (generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }) ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α a : α ⊢ {a} = Iio (succ a) ∩ Ioi (pred a) ** suffices h_singleton_eq_inter' : {a} = Iic a ∩ Ici a ** case h_singleton_eq_inter' α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α a : α ⊢ {a} = Iic a ∩ Ici a ** rw [inter_comm, Ici_inter_Iic, Icc_self a] ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α a : α h_singleton_eq_inter' : {a} = Iic a ∩ Ici a ⊢ {a} = Iio (succ a) ∩ Ioi (pred a) ** rw [h_singleton_eq_inter', ← Ioi_pred, ← Iio_succ] ** case h₁ α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ⊢ IsOpen (Iio (succ a)) ** exact isOpen_generateFrom_of_mem ⟨succ a, Or.inr rfl⟩ ** case h₂ α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ⊢ IsOpen (Ioi (pred a)) ** exact isOpen_generateFrom_of_mem ⟨pred a, Or.inl rfl⟩ ** Qed
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discreteTopology_iff_orderTopology_of_pred_succ' ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α ⊢ DiscreteTopology α ↔ OrderTopology α ** refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩ ** case refine'_1 α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α h : DiscreteTopology α ⊢ inst✝⁵ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ** rw [h.eq_bot] ** case refine'_1 α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α h : DiscreteTopology α ⊢ ⊥ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ** exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder ** case refine'_2 α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α h : OrderTopology α ⊢ inst✝⁵ = ⊥ ** rw [h.topology_eq_generate_intervals] ** case refine'_2 α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : PartialOrder α inst✝³ : PredOrder α inst✝² : SuccOrder α inst✝¹ : NoMinOrder α inst✝ : NoMaxOrder α h : OrderTopology α ⊢ generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} = ⊥ ** exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder.symm ** Qed
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LinearOrder.bot_topologicalSpace_eq_generateFrom ** α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α ⊢ ⊥ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ** refine' (eq_bot_of_singletons_open fun a => _).symm ** α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α ⊢ IsOpen {a} ** have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_inter_Iic, Icc_self a] ** α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iic a ∩ Ici a ⊢ IsOpen {a} ** by_cases ha_top : IsTop a ** α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α ⊢ {a} = Iic a ∩ Ici a ** rw [inter_comm, Ici_inter_Iic, Icc_self a] ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iic a ∩ Ici a ha_top : IsTop a ⊢ IsOpen {a} ** rw [ha_top.Iic_eq, inter_comm, inter_univ] at h_singleton_eq_inter ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Ici a ha_top : IsTop a ⊢ IsOpen {a} ** by_cases ha_bot : IsBot a ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Ici a ha_top : IsTop a ha_bot : IsBot a ⊢ IsOpen {a} ** rw [ha_bot.Ici_eq] at h_singleton_eq_inter ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = univ ha_top : IsTop a ha_bot : IsBot a ⊢ IsOpen {a} ** rw [h_singleton_eq_inter] ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = univ ha_top : IsTop a ha_bot : IsBot a ⊢ IsOpen univ ** apply @isOpen_univ _ (generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }) ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Ici a ha_top : IsTop a ha_bot : ¬IsBot a ⊢ IsOpen {a} ** rw [isBot_iff_isMin] at ha_bot ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Ici a ha_top : IsTop a ha_bot : ¬IsMin a ⊢ IsOpen {a} ** rw [← Ioi_pred_of_not_isMin ha_bot] at h_singleton_eq_inter ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Ioi (pred a) ha_top : IsTop a ha_bot : ¬IsMin a ⊢ IsOpen {a} ** rw [h_singleton_eq_inter] ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Ioi (pred a) ha_top : IsTop a ha_bot : ¬IsMin a ⊢ IsOpen (Ioi (pred a)) ** exact isOpen_generateFrom_of_mem ⟨pred a, Or.inl rfl⟩ ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iic a ∩ Ici a ha_top : ¬IsTop a ⊢ IsOpen {a} ** rw [isTop_iff_isMax] at ha_top ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iic a ∩ Ici a ha_top : ¬IsMax a ⊢ IsOpen {a} ** rw [← Iio_succ_of_not_isMax ha_top] at h_singleton_eq_inter ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a ha_top : ¬IsMax a ⊢ IsOpen {a} ** by_cases ha_bot : IsBot a ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a ha_top : ¬IsMax a ha_bot : IsBot a ⊢ IsOpen {a} ** rw [ha_bot.Ici_eq, inter_univ] at h_singleton_eq_inter ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ha_top : ¬IsMax a ha_bot : IsBot a ⊢ IsOpen {a} ** rw [h_singleton_eq_inter] ** case pos α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ha_top : ¬IsMax a ha_bot : IsBot a ⊢ IsOpen (Iio (succ a)) ** exact isOpen_generateFrom_of_mem ⟨succ a, Or.inr rfl⟩ ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a ha_top : ¬IsMax a ha_bot : ¬IsBot a ⊢ IsOpen {a} ** rw [isBot_iff_isMin] at ha_bot ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a ha_top : ¬IsMax a ha_bot : ¬IsMin a ⊢ IsOpen {a} ** rw [← Ioi_pred_of_not_isMin ha_bot] at h_singleton_eq_inter ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ha_top : ¬IsMax a ha_bot : ¬IsMin a ⊢ IsOpen {a} ** rw [h_singleton_eq_inter] ** case neg α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ha_top : ¬IsMax a ha_bot : ¬IsMin a ⊢ IsOpen (Iio (succ a) ∩ Ioi (pred a)) ** apply @IsOpen.inter _ _ _ (generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }) ** case neg.h₁ α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ha_top : ¬IsMax a ha_bot : ¬IsMin a ⊢ IsOpen (Iio (succ a)) ** exact isOpen_generateFrom_of_mem ⟨succ a, Or.inr rfl⟩ ** case neg.h₂ α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α a : α h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) ha_top : ¬IsMax a ha_bot : ¬IsMin a ⊢ IsOpen (Ioi (pred a)) ** exact isOpen_generateFrom_of_mem ⟨pred a, Or.inl rfl⟩ ** Qed
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discreteTopology_iff_orderTopology_of_pred_succ ** α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α ⊢ DiscreteTopology α ↔ OrderTopology α ** refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩ ** case refine'_1 α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α h : DiscreteTopology α ⊢ inst✝³ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ** rw [h.eq_bot] ** case refine'_1 α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α h : DiscreteTopology α ⊢ ⊥ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ** exact LinearOrder.bot_topologicalSpace_eq_generateFrom ** case refine'_2 α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α h : OrderTopology α ⊢ inst✝³ = ⊥ ** rw [h.topology_eq_generate_intervals] ** case refine'_2 α : Type u_1 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : PredOrder α inst✝ : SuccOrder α h : OrderTopology α ⊢ generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} = ⊥ ** exact LinearOrder.bot_topologicalSpace_eq_generateFrom.symm ** Qed
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Bundle.Trivial.inducing_toProd ** B : Type u_1 F : Type u_2 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F ⊢ topologicalSpace B F = induced (↑(TotalSpace.toProd B F)) instTopologicalSpaceProd ** simp only [instTopologicalSpaceProd, induced_inf, induced_compose] ** B : Type u_1 F : Type u_2 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F ⊢ topologicalSpace B F = induced (Prod.fst ∘ ↑(TotalSpace.toProd B F)) inst✝¹ ⊓ induced (Prod.snd ∘ ↑(TotalSpace.toProd B F)) inst✝ ** rfl ** Qed
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Trivialization.Prod.continuous_to_fun ** B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj ⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** let f₁ : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=
fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂)) ** B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) ⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** let f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ ⟨e₁ p.1, e₂ p.2⟩ ** B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) ⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** let f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ ⟨p.1.1, p.1.2, p.2.2⟩ ** B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) ⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** have hf₁ : Continuous f₁ := (Prod.inducing_diag F₁ E₁ F₂ E₂).continuous ** B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ ⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** have hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) :=
e₁.toLocalHomeomorph.continuousOn.prod_map e₂.toLocalHomeomorph.continuousOn ** B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) ⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** have hf₃ : Continuous f₃ :=
(continuous_fst.comp continuous_fst).prod_mk (continuous_snd.prod_map continuous_snd) ** B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ ⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** refine' ((hf₃.comp_continuousOn hf₂).comp hf₁.continuousOn _).congr _ ** case refine'_2 B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ ⊢ EqOn (toFun' e₁ e₂) ((f₃ ∘ f₂) ∘ f₁) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ** rintro ⟨b, v₁, v₂⟩ ⟨hb₁, _⟩ ** case refine'_2.mk.mk.intro B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ b : B v₁ : E₁ b v₂ : E₂ b hb₁ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₁.baseSet right✝ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₂.baseSet ⊢ toFun' e₁ e₂ { proj := b, snd := (v₁, v₂) } = ((f₃ ∘ f₂) ∘ f₁) { proj := b, snd := (v₁, v₂) } ** simp only [Prod.toFun', Prod.mk.inj_iff, Function.comp_apply, and_true_iff] ** case refine'_2.mk.mk.intro B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ b : B v₁ : E₁ b v₂ : E₂ b hb₁ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₁.baseSet right✝ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₂.baseSet ⊢ b = (↑e₁ { proj := b, snd := v₁ }).1 ** rw [e₁.coe_fst] ** case refine'_2.mk.mk.intro B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ b : B v₁ : E₁ b v₂ : E₂ b hb₁ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₁.baseSet right✝ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₂.baseSet ⊢ { proj := b, snd := v₁ } ∈ e₁.source ** rw [e₁.source_eq, mem_preimage] ** case refine'_2.mk.mk.intro B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ b : B v₁ : E₁ b v₂ : E₂ b hb₁ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₁.baseSet right✝ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₂.baseSet ⊢ { proj := b, snd := v₁ }.proj ∈ e₁.baseSet ** exact hb₁ ** case refine'_1 B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ ⊢ MapsTo f₁ (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) (e₁.source ×ˢ e₂.source) ** rw [e₁.source_eq, e₂.source_eq] ** case refine'_1 B : Type u_1 inst✝⁴ : TopologicalSpace B F₁ : Type u_2 inst✝³ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝² : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝¹ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝ : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj f₁ : (TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ := fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 }) f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.1, ↑e₂ p.2) f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.1.1, p.1.2, p.2.2) hf₁ : Continuous f₁ hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) hf₃ : Continuous f₃ ⊢ MapsTo f₁ (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) ((TotalSpace.proj ⁻¹' e₁.baseSet) ×ˢ (TotalSpace.proj ⁻¹' e₂.baseSet)) ** exact mapsTo_preimage _ _ ** Qed
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Trivialization.Prod.left_inv ** B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x h : x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) ⊢ invFun' e₁ e₂ (toFun' e₁ e₂ x) = x ** obtain ⟨x, v₁, v₂⟩ := x ** case mk.mk B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) x : B v₁ : E₁ x v₂ : E₂ x h : { proj := x, snd := (v₁, v₂) } ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) ⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { proj := x, snd := (v₁, v₂) }) = { proj := x, snd := (v₁, v₂) } ** obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h ** case mk.mk.intro B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) x : B v₁ : E₁ x v₂ : E₂ x h₁ : x ∈ e₁.baseSet h₂ : x ∈ e₂.baseSet ⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { proj := x, snd := (v₁, v₂) }) = { proj := x, snd := (v₁, v₂) } ** simp only [Prod.toFun', Prod.invFun', symm_apply_apply_mk, h₁, h₂] ** Qed
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Trivialization.Prod.right_inv ** B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) x : B × F₁ × F₂ h : x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ ⊢ toFun' e₁ e₂ (invFun' e₁ e₂ x) = x ** obtain ⟨x, w₁, w₂⟩ := x ** case mk.mk B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) x : B w₁ : F₁ w₂ : F₂ h : (x, w₁, w₂) ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ ⊢ toFun' e₁ e₂ (invFun' e₁ e₂ (x, w₁, w₂)) = (x, w₁, w₂) ** obtain ⟨⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩, -⟩ := h ** case mk.mk.intro.intro B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) x : B w₁ : F₁ w₂ : F₂ h₁ : x ∈ e₁.baseSet h₂ : x ∈ e₂.baseSet ⊢ toFun' e₁ e₂ (invFun' e₁ e₂ (x, w₁, w₂)) = (x, w₁, w₂) ** simp only [Prod.toFun', Prod.invFun', apply_mk_symm, h₁, h₂] ** Qed
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Trivialization.Prod.continuous_inv_fun ** B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) ⊢ ContinuousOn (invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ) ** rw [(Prod.inducing_diag F₁ E₁ F₂ E₂).continuousOn_iff] ** B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) ⊢ ContinuousOn ((fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })) ∘ invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ) ** have H₁ : Continuous fun p : B × F₁ × F₂ ↦ ((p.1, p.2.1), (p.1, p.2.2)) :=
(continuous_id.prod_map continuous_fst).prod_mk (continuous_id.prod_map continuous_snd) ** B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) H₁ : Continuous fun p => ((p.1, p.2.1), p.1, p.2.2) ⊢ ContinuousOn ((fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })) ∘ invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ) ** refine' (e₁.continuousOn_symm.prod_map e₂.continuousOn_symm).comp H₁.continuousOn _ ** B : Type u_1 inst✝⁶ : TopologicalSpace B F₁ : Type u_2 inst✝⁵ : TopologicalSpace F₁ E₁ : B → Type u_3 inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_4 inst✝³ : TopologicalSpace F₂ E₂ : B → Type u_5 inst✝² : TopologicalSpace (TotalSpace F₂ E₂) e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : (x : B) → Zero (E₁ x) inst✝ : (x : B) → Zero (E₂ x) H₁ : Continuous fun p => ((p.1, p.2.1), p.1, p.2.2) ⊢ MapsTo (fun p => ((p.1, p.2.1), p.1, p.2.2)) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ) ((e₁.baseSet ×ˢ univ) ×ˢ e₂.baseSet ×ˢ univ) ** exact fun x h ↦ ⟨⟨h.1.1, mem_univ _⟩, ⟨h.1.2, mem_univ _⟩⟩ ** Qed
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Pullback.continuous_proj ** B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝¹ : TopologicalSpace B' inst✝ : TopologicalSpace (TotalSpace F E) f : B' → B ⊢ Continuous TotalSpace.proj ** rw [continuous_iff_le_induced, Pullback.TotalSpace.topologicalSpace, pullbackTopology_def] ** B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝¹ : TopologicalSpace B' inst✝ : TopologicalSpace (TotalSpace F E) f : B' → B ⊢ induced TotalSpace.proj inst✝¹ ⊓ induced (Pullback.lift f) inst✝ ≤ induced TotalSpace.proj inst✝¹ ** exact inf_le_left ** Qed
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Pullback.continuous_lift ** B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝¹ : TopologicalSpace B' inst✝ : TopologicalSpace (TotalSpace F E) f : B' → B ⊢ Continuous (Pullback.lift f) ** rw [continuous_iff_le_induced, Pullback.TotalSpace.topologicalSpace, pullbackTopology_def] ** B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝¹ : TopologicalSpace B' inst✝ : TopologicalSpace (TotalSpace F E) f : B' → B ⊢ induced TotalSpace.proj inst✝¹ ⊓ induced (Pullback.lift f) inst✝ ≤ induced (Pullback.lift f) inst✝ ** exact inf_le_right ** Qed
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inducing_pullbackTotalSpaceEmbedding ** B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝¹ : TopologicalSpace B' inst✝ : TopologicalSpace (TotalSpace F E) f : B' → B ⊢ Inducing (pullbackTotalSpaceEmbedding f) ** constructor ** case induced B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝¹ : TopologicalSpace B' inst✝ : TopologicalSpace (TotalSpace F E) f : B' → B ⊢ Pullback.TotalSpace.topologicalSpace F E f = induced (pullbackTotalSpaceEmbedding f) instTopologicalSpaceProd ** simp_rw [instTopologicalSpaceProd, induced_inf, induced_compose,
Pullback.TotalSpace.topologicalSpace, pullbackTopology_def] ** case induced B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝¹ : TopologicalSpace B' inst✝ : TopologicalSpace (TotalSpace F E) f : B' → B ⊢ induced TotalSpace.proj inst✝¹ ⊓ induced (Pullback.lift f) inst✝ = induced (Prod.fst ∘ pullbackTotalSpaceEmbedding f) inst✝¹ ⊓ induced (Prod.snd ∘ pullbackTotalSpaceEmbedding f) inst✝ ** rfl ** Qed
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Pullback.continuous_totalSpaceMk ** B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝⁵ : TopologicalSpace B' inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : TopologicalSpace F inst✝² : TopologicalSpace B inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E f : B' → B x : B' ⊢ Continuous (TotalSpace.mk x) ** simp only [continuous_iff_le_induced, Pullback.TotalSpace.topologicalSpace, induced_compose,
induced_inf, Function.comp, induced_const, top_inf_eq, pullbackTopology_def] ** B : Type u F : Type v E : B → Type w₁ B' : Type w₂ f✝ : B' → B inst✝⁵ : TopologicalSpace B' inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : TopologicalSpace F inst✝² : TopologicalSpace B inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E f : B' → B x : B' ⊢ instForAllTopologicalSpacePullback E f x ≤ induced (fun x_1 => Pullback.lift f { proj := x, snd := x_1 }) inst✝⁴ ** exact le_of_eq (FiberBundle.totalSpaceMk_inducing F E (f x)).induced ** Qed
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ContinuousMap.Homotopy.extend_apply_of_le_zero ** F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ : C(X, Y) F : Homotopy f₀ f₁ t : ℝ ht : t ≤ 0 x : X ⊢ ↑(↑(extend F) t) x = ↑f₀ x ** rw [← F.apply_zero] ** F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ : C(X, Y) F : Homotopy f₀ f₁ t : ℝ ht : t ≤ 0 x : X ⊢ ↑(↑(extend F) t) x = ↑F (0, x) ** exact ContinuousMap.congr_fun (Set.IccExtend_of_le_left (zero_le_one' ℝ) F.curry ht) x ** Qed
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ContinuousMap.Homotopy.extend_apply_of_one_le ** F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ : C(X, Y) F : Homotopy f₀ f₁ t : ℝ ht : 1 ≤ t x : X ⊢ ↑(↑(extend F) t) x = ↑f₁ x ** rw [← F.apply_one] ** F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ : C(X, Y) F : Homotopy f₀ f₁ t : ℝ ht : 1 ≤ t x : X ⊢ ↑(↑(extend F) t) x = ↑F (1, x) ** exact ContinuousMap.congr_fun (Set.IccExtend_of_right_le (zero_le_one' ℝ) F.curry ht) x ** Qed
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ContinuousMap.Homotopy.symm_symm ** F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ : C(X, Y) F : Homotopy f₀ f₁ ⊢ symm (symm F) = F ** ext ** case h F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ : C(X, Y) F : Homotopy f₀ f₁ x✝ : ↑I × X ⊢ ↑(symm (symm F)) x✝ = ↑F x✝ ** simp ** Qed
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ContinuousMap.Homotopy.trans_apply ** case neg F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ x : ↑I × X h✝ : ¬↑x.1 ≤ 1 / 2 ⊢ ↑(↑(extend G) (2 * ↑x.1 - 1)) x.2 = ↑G ({ val := 2 * ↑x.1 - 1, property := (_ : 2 * ↑x.1 - 1 ∈ I) }, x.2) ** rw [extend, ContinuousMap.coe_IccExtend, Set.IccExtend_of_mem] ** case neg F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ x : ↑I × X h✝ : ¬↑x.1 ≤ 1 / 2 ⊢ ↑(↑(curry G) { val := 2 * ↑x.1 - 1, property := ?neg.hx✝ }) x.2 = ↑G ({ val := 2 * ↑x.1 - 1, property := (_ : 2 * ↑x.1 - 1 ∈ I) }, x.2) case neg.hx F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ x : ↑I × X h✝ : ¬↑x.1 ≤ 1 / 2 ⊢ 2 * ↑x.1 - 1 ∈ Set.Icc 0 1 ** rfl ** Qed
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ContinuousMap.Homotopy.symm_trans ** F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ ⊢ symm (trans F G) = trans (symm G) (symm F) ** ext ⟨t, _⟩ ** case h.mk F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X ⊢ ↑(symm (trans F G)) (t, snd✝) = ↑(trans (symm G) (symm F)) (t, snd✝) ** rw [trans_apply, symm_apply, trans_apply] ** case h.mk F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X ⊢ (if h : ↑(σ (t, snd✝).1, (t, snd✝).2).1 ≤ 1 / 2 then ↑F ({ val := 2 * ↑(σ (t, snd✝).1, (t, snd✝).2).1, property := (_ : 2 * ↑(σ (t, snd✝).1, (t, snd✝).2).1 ∈ I) }, (σ (t, snd✝).1, (t, snd✝).2).2) else ↑G ({ val := 2 * ↑(σ (t, snd✝).1, (t, snd✝).2).1 - 1, property := (_ : 2 * ↑(σ (t, snd✝).1, (t, snd✝).2).1 - 1 ∈ I) }, (σ (t, snd✝).1, (t, snd✝).2).2)) = if h : ↑(t, snd✝).1 ≤ 1 / 2 then ↑(symm G) ({ val := 2 * ↑(t, snd✝).1, property := (_ : 2 * ↑(t, snd✝).1 ∈ I) }, (t, snd✝).2) else ↑(symm F) ({ val := 2 * ↑(t, snd✝).1 - 1, property := (_ : 2 * ↑(t, snd✝).1 - 1 ∈ I) }, (t, snd✝).2) ** simp only [coe_symm_eq, symm_apply] ** case h.mk F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X ⊢ (if h : 1 - ↑t ≤ 1 / 2 then ↑F ({ val := 2 * (1 - ↑t), property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t))) }, snd✝) else ↑G ({ val := 2 * (1 - ↑t) - 1, property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t) - 1)) }, snd✝)) = if h : ↑t ≤ 1 / 2 then ↑G (σ { val := 2 * ↑t, property := (_ : 2 * ↑(t, snd✝).1 ∈ I) }, snd✝) else ↑F (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑(t, snd✝).1 - 1 ∈ I) }, snd✝) ** split_ifs with h₁ h₂ h₂ ** case pos F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : 1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑F ({ val := 2 * (1 - ↑t), property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t))) }, snd✝) = ↑G (σ { val := 2 * ↑t, property := (_ : 2 * ↑(t, snd✝).1 ∈ I) }, snd✝) ** have ht : (t : ℝ) = 1 / 2 := by linarith ** case pos F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : 1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ht : ↑t = 1 / 2 ⊢ ↑F ({ val := 2 * (1 - ↑t), property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t))) }, snd✝) = ↑G (σ { val := 2 * ↑t, property := (_ : 2 * ↑(t, snd✝).1 ∈ I) }, snd✝) ** norm_num [ht] ** F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : 1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑t = 1 / 2 ** linarith ** case neg F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : 1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ ↑F ({ val := 2 * (1 - ↑t), property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t))) }, snd✝) = ↑F (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑(t, snd✝).1 - 1 ∈ I) }, snd✝) ** congr 2 ** case neg.h.e_6.h.e_fst F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : 1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ { val := 2 * (1 - ↑t), property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t))) } = σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑(t, snd✝).1 - 1 ∈ I) } ** apply Subtype.ext ** case neg.h.e_6.h.e_fst.a F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : 1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ ↑{ val := 2 * (1 - ↑t), property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t))) } = ↑(σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑(t, snd✝).1 - 1 ∈ I) }) ** simp only [coe_symm_eq] ** case neg.h.e_6.h.e_fst.a F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : 1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ 2 * (1 - ↑t) = 1 - (2 * ↑t - 1) ** linarith ** case pos F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑G ({ val := 2 * (1 - ↑t) - 1, property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t) - 1)) }, snd✝) = ↑G (σ { val := 2 * ↑t, property := (_ : 2 * ↑(t, snd✝).1 ∈ I) }, snd✝) ** congr 2 ** case pos.h.e_6.h.e_fst F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ { val := 2 * (1 - ↑t) - 1, property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t) - 1)) } = σ { val := 2 * ↑t, property := (_ : 2 * ↑(t, snd✝).1 ∈ I) } ** apply Subtype.ext ** case pos.h.e_6.h.e_fst.a F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ ↑{ val := 2 * (1 - ↑t) - 1, property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t) - 1)) } = ↑(σ { val := 2 * ↑t, property := (_ : 2 * ↑(t, snd✝).1 ∈ I) }) ** simp only [coe_symm_eq] ** case pos.h.e_6.h.e_fst.a F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : ¬1 - ↑t ≤ 1 / 2 h₂ : ↑t ≤ 1 / 2 ⊢ 2 * (1 - ↑t) - 1 = 1 - 2 * ↑t ** linarith ** case neg F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : ¬1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ ↑G ({ val := 2 * (1 - ↑t) - 1, property := (_ : (fun x => x ∈ I) (2 * (1 - ↑t) - 1)) }, snd✝) = ↑F (σ { val := 2 * ↑t - 1, property := (_ : 2 * ↑(t, snd✝).1 - 1 ∈ I) }, snd✝) ** exfalso ** case neg.h F✝ : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' f₀ f₁ f₂ : C(X, Y) F : Homotopy f₀ f₁ G : Homotopy f₁ f₂ t : ↑I snd✝ : X h₁ : ¬1 - ↑t ≤ 1 / 2 h₂ : ¬↑t ≤ 1 / 2 ⊢ False ** linarith ** Qed
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ContinuousMap.Homotopic.equivalence ** F : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' ⊢ ∀ {x y : C(X, Y)}, Homotopic x y → Homotopic y x ** apply symm ** F : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' ⊢ ∀ {x y z : C(X, Y)}, Homotopic x y → Homotopic y z → Homotopic x z ** apply trans ** Qed
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ContinuousMap.HomotopicRel.equivalence ** F : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' S : Set X ⊢ ∀ {x y : C(X, Y)}, HomotopicRel x y S → HomotopicRel y x S ** apply symm ** F : Type u_1 X : Type u Y : Type v Z : Type w Z' : Type x inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace Z' S : Set X ⊢ ∀ {x y z : C(X, Y)}, HomotopicRel x y S → HomotopicRel y z S → HomotopicRel x z S ** apply trans ** Qed
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TopCat.Presheaf.isSheaf_unit ** C : Type u inst✝ : Category.{v, u} C X : TopCat F✝ : Presheaf C X ι : Type v U✝ : ι → Opens ↑X F : Presheaf (Discrete Unit) X x✝² : Discrete Unit U : Opens ↑X S : Sieve U x✝¹ : S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U x : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op x✝²)) S.arrows x✝ : Presieve.FamilyOfElements.Compatible x ⊢ (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) (eqToHom (_ : (op x✝²).unop = F.obj (op U))) ** aesop_cat ** C : Type u inst✝ : Category.{v, u} C X : TopCat F✝ : Presheaf C X ι : Type v U✝ : ι → Opens ↑X F : Presheaf (Discrete Unit) X x✝³ : Discrete Unit U : Opens ↑X S : Sieve U x✝² : S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U x : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op x✝³)) S.arrows x✝¹ : Presieve.FamilyOfElements.Compatible x x✝ : (F ⋙ coyoneda.obj (op x✝³)).obj (op U) ⊢ (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) x✝ → x✝ = eqToHom (_ : (op x✝³).unop = F.obj (op U)) ** aesop_cat ** Qed
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exists_compact_superset ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : WeaklyLocallyCompactSpace X K : Set X hK : IsCompact K ⊢ ∃ K', IsCompact K' ∧ K ⊆ interior K' ** choose s hc hmem using fun x : X ↦ exists_compact_mem_nhds x ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s✝ t : Set X inst✝ : WeaklyLocallyCompactSpace X K : Set X hK : IsCompact K s : X → Set X hc : ∀ (x : X), IsCompact (s x) hmem : ∀ (x : X), s x ∈ 𝓝 x ⊢ ∃ K', IsCompact K' ∧ K ⊆ interior K' ** rcases hK.elim_nhds_subcover _ fun x _ ↦ interior_mem_nhds.2 (hmem x) with ⟨I, -, hIK⟩ ** case intro.intro X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s✝ t : Set X inst✝ : WeaklyLocallyCompactSpace X K : Set X hK : IsCompact K s : X → Set X hc : ∀ (x : X), IsCompact (s x) hmem : ∀ (x : X), s x ∈ 𝓝 x I : Finset X hIK : K ⊆ ⋃ x ∈ I, interior (s x) ⊢ ∃ K', IsCompact K' ∧ K ⊆ interior K' ** refine ⟨⋃ x ∈ I, s x, I.isCompact_biUnion fun _ _ ↦ hc _, hIK.trans ?_⟩ ** case intro.intro X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s✝ t : Set X inst✝ : WeaklyLocallyCompactSpace X K : Set X hK : IsCompact K s : X → Set X hc : ∀ (x : X), IsCompact (s x) hmem : ∀ (x : X), s x ∈ 𝓝 x I : Finset X hIK : K ⊆ ⋃ x ∈ I, interior (s x) ⊢ ⋃ x ∈ I, interior (s x) ⊆ interior (⋃ x ∈ I, s x) ** exact iUnion₂_subset fun x hx ↦ interior_mono <| subset_iUnion₂ (s := fun x _ ↦ s x) x hx ** Qed
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compact_basis_nhds ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace X x : X ⊢ ∀ (t : Set X), t ∈ 𝓝 x → ∃ r, r ∈ 𝓝 x ∧ IsCompact r ∧ r ⊆ t ** simpa only [and_comm] using LocallyCompactSpace.local_compact_nhds x ** Qed
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exists_compact_subset ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace X x : X U : Set X hU : IsOpen U hx : x ∈ U ⊢ ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U ** rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩ ** case intro.intro.intro X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace X x : X U : Set X hU : IsOpen U hx : x ∈ U K : Set X h1K : K ∈ 𝓝 x h2K : K ⊆ U h3K : IsCompact K ⊢ ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U ** exact ⟨K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K⟩ ** Qed
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exists_compact_between ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y s t : Set X hX : LocallyCompactSpace X K U : Set X hK : IsCompact K hU : IsOpen U h_KU : K ⊆ U ⊢ ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U ** choose V hVc hxV hKV using fun x : K => exists_compact_subset hU (h_KU x.2) ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y s t : Set X hX : LocallyCompactSpace X K U : Set X hK : IsCompact K hU : IsOpen U h_KU : K ⊆ U V : ↑K → Set X hVc : ∀ (x : ↑K), IsCompact (V x) hxV : ∀ (x : ↑K), ↑x ∈ interior (V x) hKV : ∀ (x : ↑K), V x ⊆ U ⊢ ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U ** have : K ⊆ ⋃ x, interior (V x) := fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, hxV _⟩ ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y s t : Set X hX : LocallyCompactSpace X K U : Set X hK : IsCompact K hU : IsOpen U h_KU : K ⊆ U V : ↑K → Set X hVc : ∀ (x : ↑K), IsCompact (V x) hxV : ∀ (x : ↑K), ↑x ∈ interior (V x) hKV : ∀ (x : ↑K), V x ⊆ U this : K ⊆ ⋃ x, interior (V x) ⊢ ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U ** rcases hK.elim_finite_subcover _ (fun x => @isOpen_interior X _ (V x)) this with ⟨t, ht⟩ ** case intro X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y s t✝ : Set X hX : LocallyCompactSpace X K U : Set X hK : IsCompact K hU : IsOpen U h_KU : K ⊆ U V : ↑K → Set X hVc : ∀ (x : ↑K), IsCompact (V x) hxV : ∀ (x : ↑K), ↑x ∈ interior (V x) hKV : ∀ (x : ↑K), V x ⊆ U this : K ⊆ ⋃ x, interior (V x) t : Finset ↑K ht : K ⊆ ⋃ i ∈ t, interior (V i) ⊢ ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U ** refine'
⟨_, t.isCompact_biUnion fun x _ => hVc x, fun x hx => _, Set.iUnion₂_subset fun i _ => hKV i⟩ ** case intro X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y s t✝ : Set X hX : LocallyCompactSpace X K U : Set X hK : IsCompact K hU : IsOpen U h_KU : K ⊆ U V : ↑K → Set X hVc : ∀ (x : ↑K), IsCompact (V x) hxV : ∀ (x : ↑K), ↑x ∈ interior (V x) hKV : ∀ (x : ↑K), V x ⊆ U this : K ⊆ ⋃ x, interior (V x) t : Finset ↑K ht : K ⊆ ⋃ i ∈ t, interior (V i) x : X hx : x ∈ K ⊢ x ∈ interior (⋃ i ∈ t, V i) ** rcases mem_iUnion₂.1 (ht hx) with ⟨y, hyt, hy⟩ ** case intro.intro.intro X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝¹ : TopologicalSpace X inst✝ : TopologicalSpace Y s t✝ : Set X hX : LocallyCompactSpace X K U : Set X hK : IsCompact K hU : IsOpen U h_KU : K ⊆ U V : ↑K → Set X hVc : ∀ (x : ↑K), IsCompact (V x) hxV : ∀ (x : ↑K), ↑x ∈ interior (V x) hKV : ∀ (x : ↑K), V x ⊆ U this : K ⊆ ⋃ x, interior (V x) t : Finset ↑K ht : K ⊆ ⋃ i ∈ t, interior (V i) x : X hx : x ∈ K y : ↑K hyt : y ∈ t hy : x ∈ interior (V y) ⊢ x ∈ interior (⋃ i ∈ t, V i) ** exact interior_mono (subset_iUnion₂ y hyt) hy ** Qed
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ClosedEmbedding.locallyCompactSpace ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : ClosedEmbedding f ⊢ ∀ (x : X), HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (f x) ∧ IsCompact s) fun s => f ⁻¹' s ** intro x ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : ClosedEmbedding f x : X ⊢ HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (f x) ∧ IsCompact s) fun s => f ⁻¹' s ** rw [hf.toInducing.nhds_eq_comap] ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : ClosedEmbedding f x : X ⊢ HasBasis (comap f (𝓝 (f x))) (fun s => s ∈ 𝓝 (f x) ∧ IsCompact s) fun s => f ⁻¹' s ** exact (compact_basis_nhds _).comap _ ** Qed
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OpenEmbedding.locallyCompactSpace ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : OpenEmbedding f ⊢ LocallyCompactSpace X ** have : ∀ x : X, (𝓝 x).HasBasis
(fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s := by
intro x
rw [hf.toInducing.nhds_eq_comap]
exact
((compact_basis_nhds _).restrict_subset <| hf.open_range.mem_nhds <| mem_range_self _).comap _ ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : OpenEmbedding f this : ∀ (x : X), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s ⊢ LocallyCompactSpace X ** refine' locallyCompactSpace_of_hasBasis this fun x s hs => _ ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s✝ t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : OpenEmbedding f this : ∀ (x : X), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s x : X s : Set Y hs : (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f ⊢ IsCompact (f ⁻¹' s) ** rw [hf.toInducing.isCompact_iff, image_preimage_eq_of_subset hs.2] ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s✝ t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : OpenEmbedding f this : ∀ (x : X), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s x : X s : Set Y hs : (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f ⊢ IsCompact s ** exact hs.1.2 ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : OpenEmbedding f ⊢ ∀ (x : X), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s ** intro x ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : OpenEmbedding f x : X ⊢ HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s ** rw [hf.toInducing.nhds_eq_comap] ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : LocallyCompactSpace Y f : X → Y hf : OpenEmbedding f x : X ⊢ HasBasis (comap f (𝓝 (f x))) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s ** exact
((compact_basis_nhds _).restrict_subset <| hf.open_range.mem_nhds <| mem_range_self _).comap _ ** Qed
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Ultrafilter.le_nhds_lim ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : CompactSpace X F : Ultrafilter X ⊢ ↑F ≤ 𝓝 (lim F) ** rcases isCompact_univ.ultrafilter_le_nhds F (by simp) with ⟨x, -, h⟩ ** case intro.intro X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : CompactSpace X F : Ultrafilter X x : X h : ↑F ≤ 𝓝 x ⊢ ↑F ≤ 𝓝 (lim F) ** exact le_nhds_lim ⟨x, h⟩ ** X : Type u_1 Y : Type u_2 ι : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y s t : Set X inst✝ : CompactSpace X F : Ultrafilter X ⊢ ↑F ≤ 𝓟 univ ** simp ** Qed
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precise_refinement ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ ⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a ** have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
(SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe]) ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ this : ∃ β t x x, LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ (fun r => ↑r) a ⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a ** simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ this : ∃ β t, (∀ (b : β), IsOpen (t b)) ∧ (∀ (x : X), ∃ i, x ∈ t i) ∧ LocallyFinite t ∧ ∀ (b : β), ∃ i, t b ⊆ u i ⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a ** choose α t hto hXt htf ind hind using this ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) hXt : ∀ (x : X), ∃ i, x ∈ t i htf : LocallyFinite t ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) ⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a ** choose t_inv ht_inv using hXt ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) htf : LocallyFinite t ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) ⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a ** choose U hxU hU using htf ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} ⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a ** refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩ ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ ⊢ ⋃ a, (fun r => ↑r) a = univ ** rwa [← sUnion_range, Subtype.range_coe] ** case refine'_1 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} ⊢ ∀ (a : ι), IsOpen ((fun i => ⋃ a, ⋃ (_ : ind a = i), t a) a) ** exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a ** case refine'_2 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} ⊢ ⋃ i, (fun i => ⋃ a, ⋃ (_ : ind a = i), t a) i = univ ** simp only [eq_univ_iff_forall, mem_iUnion] ** case refine'_2 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} ⊢ ∀ (x : X), ∃ i i_1 i, x ∈ t i_1 ** exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩ ** case refine'_3 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} ⊢ LocallyFinite fun i => ⋃ a, ⋃ (_ : ind a = i), t a ** refine' fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset _⟩ ** case refine'_3 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} x : X ⊢ {i | Set.Nonempty ((fun i => ⋃ a, ⋃ (_ : ind a = i), t a) i ∩ U x)} ⊆ ind '' {i | Set.Nonempty (t i ∩ U x)} ** simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff] ** case refine'_3 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} x : X ⊢ ∀ (x_1 : ι), (∃ x_2, (∃ i i_1, x_2 ∈ t i) ∧ x_2 ∈ U x) → x_1 ∈ ind '' {i | ∃ x_2, x_2 ∈ t i ∧ x_2 ∈ U x} ** rintro i ⟨y, ⟨a, rfl, hya⟩, hyU⟩ ** case refine'_3.intro.intro.intro.intro ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} x y : X hyU : y ∈ U x a : α hya : y ∈ t a ⊢ ind a ∈ ind '' {i | ∃ x_1, x_1 ∈ t i ∧ x_1 ∈ U x} ** exact mem_image_of_mem _ ⟨y, hya, hyU⟩ ** case refine'_4 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} ⊢ ∀ (a : ι), (fun i => ⋃ a, ⋃ (_ : ind a = i), t a) a ⊆ u a ** simp only [subset_def, mem_iUnion] ** case refine'_4 ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} ⊢ ∀ (a : ι) (x : X), (∃ i i_1, x ∈ t i) → x ∈ u a ** rintro i x ⟨a, rfl, hxa⟩ ** case refine'_4.intro.intro ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X u : ι → Set X uo : ∀ (a : ι), IsOpen (u a) uc : ⋃ i, u i = univ α : Type v t : α → Set X hto : ∀ (b : α), IsOpen (t b) ind : α → ι hind : ∀ (b : α), t b ⊆ u (ind b) t_inv : X → α ht_inv : ∀ (x : X), x ∈ t (t_inv x) U : X → Set X hxU : ∀ (x : X), U x ∈ 𝓝 x hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)} x : X a : α hxa : x ∈ t a ⊢ x ∈ u (ind a) ** exact hind _ hxa ** Qed
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precise_refinement_set ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X s : Set X hs : IsClosed s u : ι → Set X uo : ∀ (i : ι), IsOpen (u i) us : s ⊆ ⋃ i, u i uc : ⋃ i, Option.elim' sᶜ u i = univ ⊢ ∃ v, (∀ (i : ι), IsOpen (v i)) ∧ s ⊆ ⋃ i, v i ∧ LocallyFinite v ∧ ∀ (i : ι), v i ⊆ u i ** rcases precise_refinement (Option.elim' sᶜ u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)
uc with
⟨v, vo, vc, vf, vu⟩ ** case intro.intro.intro.intro ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X s : Set X hs : IsClosed s u : ι → Set X uo : ∀ (i : ι), IsOpen (u i) us : s ⊆ ⋃ i, u i uc : ⋃ i, Option.elim' sᶜ u i = univ v : Option ι → Set X vo : ∀ (a : Option ι), IsOpen (v a) vc : ⋃ i, v i = univ vf : LocallyFinite v vu : ∀ (a : Option ι), v a ⊆ Option.elim' sᶜ u a ⊢ ∃ v, (∀ (i : ι), IsOpen (v i)) ∧ s ⊆ ⋃ i, v i ∧ LocallyFinite v ∧ ∀ (i : ι), v i ⊆ u i ** refine' ⟨v ∘ some, fun i ↦ vo _, _, vf.comp_injective (Option.some_injective _), fun i ↦ vu _⟩ ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X s : Set X hs : IsClosed s u : ι → Set X uo : ∀ (i : ι), IsOpen (u i) us : s ⊆ ⋃ i, u i ⊢ ⋃ i, Option.elim' sᶜ u i = univ ** apply Subset.antisymm (subset_univ _) ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X s : Set X hs : IsClosed s u : ι → Set X uo : ∀ (i : ι), IsOpen (u i) us : s ⊆ ⋃ i, u i ⊢ univ ⊆ ⋃ i, Option.elim' sᶜ u i ** simp_rw [← compl_union_self s, Option.elim', iUnion_option] ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X s : Set X hs : IsClosed s u : ι → Set X uo : ∀ (i : ι), IsOpen (u i) us : s ⊆ ⋃ i, u i ⊢ sᶜ ∪ s ⊆ sᶜ ∪ ⋃ a, u a ** apply union_subset_union_right sᶜ us ** case intro.intro.intro.intro ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X s : Set X hs : IsClosed s u : ι → Set X uo : ∀ (i : ι), IsOpen (u i) us : s ⊆ ⋃ i, u i uc : ⋃ i, Option.elim' sᶜ u i = univ v : Option ι → Set X vo : ∀ (a : Option ι), IsOpen (v a) vc : ⋃ i, v i = univ vf : LocallyFinite v vu : ∀ (a : Option ι), v a ⊆ Option.elim' sᶜ u a ⊢ s ⊆ ⋃ i, (v ∘ some) i ** simp only [iUnion_option, ← compl_subset_iff_union] at vc ** case intro.intro.intro.intro ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace X s : Set X hs : IsClosed s u : ι → Set X uo : ∀ (i : ι), IsOpen (u i) us : s ⊆ ⋃ i, u i uc : ⋃ i, Option.elim' sᶜ u i = univ v : Option ι → Set X vo : ∀ (a : Option ι), IsOpen (v a) vf : LocallyFinite v vu : ∀ (a : Option ι), v a ⊆ Option.elim' sᶜ u a vc : (v none)ᶜ ⊆ ⋃ i, v (some i) ⊢ s ⊆ ⋃ i, (v ∘ some) i ** exact Subset.trans (subset_compl_comm.1 <| vu Option.none) vc ** Qed
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ClosedEmbedding.paracompactSpace ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace Y e : X → Y he : ClosedEmbedding e α : Type v s : α → Set X ho : ∀ (a : α), IsOpen (s a) hu : ⋃ a, s a = univ ⊢ ∃ β t x x, LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a ** choose U hUo hU using fun a ↦ he.isOpen_iff.1 (ho a) ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace Y e : X → Y he : ClosedEmbedding e α : Type v s : α → Set X ho : ∀ (a : α), IsOpen (s a) hu : ⋃ a, s a = univ U : α → Set Y hUo : ∀ (a : α), IsOpen (U a) hU : ∀ (a : α), e ⁻¹' U a = s a ⊢ ∃ β t x x, LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a ** simp only [← hU] at hu ⊢ ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace Y e : X → Y he : ClosedEmbedding e α : Type v s : α → Set X ho : ∀ (a : α), IsOpen (s a) U : α → Set Y hUo : ∀ (a : α), IsOpen (U a) hU : ∀ (a : α), e ⁻¹' U a = s a hu : ⋃ a, e ⁻¹' U a = univ ⊢ ∃ β t h h, LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ e ⁻¹' U a ** have heU : range e ⊆ ⋃ i, U i := by
simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using hu ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace Y e : X → Y he : ClosedEmbedding e α : Type v s : α → Set X ho : ∀ (a : α), IsOpen (s a) U : α → Set Y hUo : ∀ (a : α), IsOpen (U a) hU : ∀ (a : α), e ⁻¹' U a = s a hu : ⋃ a, e ⁻¹' U a = univ heU : range e ⊆ ⋃ i, U i ⊢ ∃ β t h h, LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ e ⁻¹' U a ** rcases precise_refinement_set he.closed_range U hUo heU with ⟨V, hVo, heV, hVf, hVU⟩ ** case intro.intro.intro.intro ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace Y e : X → Y he : ClosedEmbedding e α : Type v s : α → Set X ho : ∀ (a : α), IsOpen (s a) U : α → Set Y hUo : ∀ (a : α), IsOpen (U a) hU : ∀ (a : α), e ⁻¹' U a = s a hu : ⋃ a, e ⁻¹' U a = univ heU : range e ⊆ ⋃ i, U i V : α → Set Y hVo : ∀ (i : α), IsOpen (V i) heV : range e ⊆ ⋃ i, V i hVf : LocallyFinite V hVU : ∀ (i : α), V i ⊆ U i ⊢ ∃ β t h h, LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ e ⁻¹' U a ** refine ⟨α, fun a ↦ e ⁻¹' (V a), fun a ↦ (hVo a).preimage he.continuous, ?_,
hVf.preimage_continuous he.continuous, fun a ↦ ⟨a, preimage_mono (hVU a)⟩⟩ ** case intro.intro.intro.intro ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace Y e : X → Y he : ClosedEmbedding e α : Type v s : α → Set X ho : ∀ (a : α), IsOpen (s a) U : α → Set Y hUo : ∀ (a : α), IsOpen (U a) hU : ∀ (a : α), e ⁻¹' U a = s a hu : ⋃ a, e ⁻¹' U a = univ heU : range e ⊆ ⋃ i, U i V : α → Set Y hVo : ∀ (i : α), IsOpen (V i) heV : range e ⊆ ⋃ i, V i hVf : LocallyFinite V hVU : ∀ (i : α), V i ⊆ U i ⊢ ⋃ b, (fun a => e ⁻¹' V a) b = univ ** simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using heV ** ι : Type u X : Type v Y : Type w inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : ParacompactSpace Y e : X → Y he : ClosedEmbedding e α : Type v s : α → Set X ho : ∀ (a : α), IsOpen (s a) U : α → Set Y hUo : ∀ (a : α), IsOpen (U a) hU : ∀ (a : α), e ⁻¹' U a = s a hu : ⋃ a, e ⁻¹' U a = univ ⊢ range e ⊆ ⋃ i, U i ** simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using hu ** Qed
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refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** set K' : CompactExhaustion X := CompactExhaustion.choice X ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** set K : CompactExhaustion X := K'.shiftr.shiftr ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** set Kdiff := fun n ↦ K (n + 1) \ interior (K n) ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) := fun x ↦ by
simpa only [K'.find_shiftr] using
diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x) ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) :=
fun n ↦ ((K.isCompact _).diff isOpen_interior).inter_right hs ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** have : ∀ (n) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 (x : X) :=
fun n x ↦ (K.isClosed n).compl_mem_nhds fun hx' ↦ x.2.1.2 <| K.subset_interior_succ _ hx' ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** choose! r hrp hr using fun n (x : ↑(Kdiff (n + 1) ∩ s)) ↦ (hB x x.2.2).mem_iff.1 (this n x) ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** have hxr : ∀ (n x) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x := fun n x hx ↦
(hB x hx.2).mem_of_mem (hrp _ ⟨x, hx⟩) ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** choose T hT using fun n ↦ (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n) ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** set T' : ∀ n, Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ T n ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) ⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a) ** refine' ⟨Σn, T' n, fun a ↦ a.2, fun a ↦ r a.1 a.2, _, _, _⟩ ** ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) x : X ⊢ x ∈ Kdiff (CompactExhaustion.find K' x + 1) ** simpa only [K'.find_shiftr] using
diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x) ** case refine'_1 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) ⊢ ∀ (a : (n : ℕ) × ↑(T' n)), (fun a => ↑↑a.snd) a ∈ s ∧ p ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a) ** rintro ⟨n, x, hx⟩ ** case refine'_1.mk.mk ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) n : ℕ x : ↑(Kdiff (n + 1) ∩ s) hx : x ∈ T' n ⊢ (fun a => ↑↑a.snd) { fst := n, snd := { val := x, property := hx } } ∈ s ∧ p ((fun a => ↑↑a.snd) { fst := n, snd := { val := x, property := hx } }) ((fun a => r a.fst ↑a.snd) { fst := n, snd := { val := x, property := hx } }) ** exact ⟨x.2.2, hrp _ _⟩ ** case refine'_2 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) ⊢ s ⊆ ⋃ a, B ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a) ** refine' fun x hx ↦ mem_iUnion.2 _ ** case refine'_2 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X hx : x ∈ s ⊢ ∃ i, x ∈ B ((fun a => ↑↑a.snd) i) ((fun a => r a.fst ↑a.snd) i) ** rcases mem_iUnion₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩ ** case refine'_2.intro.mk.intro ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X hx : x ∈ s c : X hc : c ∈ Kdiff (CompactExhaustion.find K' x + 1) ∩ s hcT : { val := c, property := hc } ∈ T (CompactExhaustion.find K' x) hcx : x ∈ B (↑{ val := c, property := hc }) (r (CompactExhaustion.find K' x) { val := ↑{ val := c, property := hc }, property := (_ : ↑{ val := c, property := hc } ∈ Kdiff (CompactExhaustion.find K' x + 1) ∩ s) }) ⊢ ∃ i, x ∈ B ((fun a => ↑↑a.snd) i) ((fun a => r a.fst ↑a.snd) i) ** exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩ ** case refine'_3 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) ⊢ LocallyFinite fun a => B ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a) ** intro x ** case refine'_3 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X ⊢ ∃ t, t ∈ 𝓝 x ∧ Set.Finite {i | Set.Nonempty ((fun a => B ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a)) i ∩ t)} ** refine'
⟨interior (K (K'.find x + 3)),
IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩ ** case refine'_3 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X ⊢ Set.Finite {i | Set.Nonempty ((fun a => B ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a)) i ∩ interior (↑K (CompactExhaustion.find K' x + 3)))} ** have : (⋃ k ≤ K'.find x + 2, range (Sigma.mk k) : Set (Σn, T' n)).Finite :=
(finite_le_nat _).biUnion fun k _ ↦ finite_range _ ** case refine'_3 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X this : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x + 2), range (Sigma.mk k)) ⊢ Set.Finite {i | Set.Nonempty ((fun a => B ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a)) i ∩ interior (↑K (CompactExhaustion.find K' x + 3)))} ** apply this.subset ** case refine'_3 ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X this : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x + 2), range (Sigma.mk k)) ⊢ {i | Set.Nonempty ((fun a => B ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a)) i ∩ interior (↑K (CompactExhaustion.find K' x + 3)))} ⊆ ⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x + 2), range (Sigma.mk k) ** rintro ⟨k, c, hc⟩ ** case refine'_3.mk.mk ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X this : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x + 2), range (Sigma.mk k)) k : ℕ c : ↑(Kdiff (k + 1) ∩ s) hc : c ∈ T' k ⊢ { fst := k, snd := { val := c, property := hc } } ∈ {i | Set.Nonempty ((fun a => B ((fun a => ↑↑a.snd) a) ((fun a => r a.fst ↑a.snd) a)) i ∩ interior (↑K (CompactExhaustion.find K' x + 3)))} → { fst := k, snd := { val := c, property := hc } } ∈ ⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x + 2), range (Sigma.mk k) ** simp only [mem_iUnion, mem_setOf_eq, mem_image, Subtype.coe_mk] ** case refine'_3.mk.mk ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x : X this : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x + 2), range (Sigma.mk k)) k : ℕ c : ↑(Kdiff (k + 1) ∩ s) hc : c ∈ T' k ⊢ Set.Nonempty (B (↑c) (r k c) ∩ interior (↑(CompactExhaustion.shiftr (CompactExhaustion.shiftr (CompactExhaustion.choice X))) (CompactExhaustion.find (CompactExhaustion.choice X) x + 3))) → ∃ i i_1, { fst := k, snd := { val := c, property := hc } } ∈ range (Sigma.mk i) ** rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩ ** case refine'_3.mk.mk.intro.intro ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x✝ : X this : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x✝ + 2), range (Sigma.mk k)) k : ℕ c : ↑(Kdiff (k + 1) ∩ s) hc : c ∈ T' k x : X hxB : x ∈ B (↑c) (r k c) hxK : x ∈ interior (↑(CompactExhaustion.shiftr (CompactExhaustion.shiftr (CompactExhaustion.choice X))) (CompactExhaustion.find (CompactExhaustion.choice X) x✝ + 3)) ⊢ ∃ i i_1, { fst := k, snd := { val := c, property := hc } } ∈ range (Sigma.mk i) ** refine' ⟨k, _, ⟨c, hc⟩, rfl⟩ ** case refine'_3.mk.mk.intro.intro ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x✝ : X this : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x✝ + 2), range (Sigma.mk k)) k : ℕ c : ↑(Kdiff (k + 1) ∩ s) hc : c ∈ T' k x : X hxB : x ∈ B (↑c) (r k c) hxK : x ∈ interior (↑(CompactExhaustion.shiftr (CompactExhaustion.shiftr (CompactExhaustion.choice X))) (CompactExhaustion.find (CompactExhaustion.choice X) x✝ + 3)) ⊢ k ≤ CompactExhaustion.find (CompactExhaustion.choice X) x✝ + 2 ** have := (mem_compl_iff _ _).1 (hr k c hxB) ** case refine'_3.mk.mk.intro.intro ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝¹ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x✝ : X this✝ : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x✝ + 2), range (Sigma.mk k)) k : ℕ c : ↑(Kdiff (k + 1) ∩ s) hc : c ∈ T' k x : X hxB : x ∈ B (↑c) (r k c) hxK : x ∈ interior (↑(CompactExhaustion.shiftr (CompactExhaustion.shiftr (CompactExhaustion.choice X))) (CompactExhaustion.find (CompactExhaustion.choice X) x✝ + 3)) this : ¬x ∈ ↑K k ⊢ k ≤ CompactExhaustion.find (CompactExhaustion.choice X) x✝ + 2 ** contrapose! this with hnk ** case refine'_3.mk.mk.intro.intro ι✝ : Type u X : Type v Y : Type w inst✝⁴ : TopologicalSpace X inst✝³ : TopologicalSpace Y inst✝² : WeaklyLocallyCompactSpace X inst✝¹ : SigmaCompactSpace X inst✝ : T2Space X ι : X → Type u p : (x : X) → ι x → Prop B : (x : X) → ι x → Set X s : Set X hs : IsClosed s hB : ∀ (x : X), x ∈ s → HasBasis (𝓝 x) (p x) (B x) K' : CompactExhaustion X := CompactExhaustion.choice X K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K') Kdiff : (n : ℕ) → (fun x => Set X) (n + 1) := fun n => ↑K (n + 1) \ interior (↑K n) hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1) Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s) this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (↑K n)ᶜ ∈ 𝓝 ↑x r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x) hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (↑K n)ᶜ hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s) hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) }) T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n) x✝ : X this : Set.Finite (⋃ k, ⋃ (_ : k ≤ CompactExhaustion.find K' x✝ + 2), range (Sigma.mk k)) k : ℕ c : ↑(Kdiff (k + 1) ∩ s) hc : c ∈ T' k x : X hxB : x ∈ B (↑c) (r k c) hxK : x ∈ interior (↑(CompactExhaustion.shiftr (CompactExhaustion.shiftr (CompactExhaustion.choice X))) (CompactExhaustion.find (CompactExhaustion.choice X) x✝ + 3)) hnk : CompactExhaustion.find (CompactExhaustion.choice X) x✝ + 2 < k ⊢ x ∈ ↑K k ** exact K.subset hnk (interior_subset hxK) ** Qed
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iUnion_compactCovering ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α ⊢ ⋃ n, compactCovering α n = univ ** rw [compactCovering, iUnion_accumulate] ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α ⊢ ⋃ x, Exists.choose (_ : ∃ K, (∀ (n : ℕ), IsCompact (K n)) ∧ ⋃ n, K n = univ) x = univ ** exact (Classical.choose_spec SigmaCompactSpace.exists_compact_covering).2 ** Qed
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ClosedEmbedding.sigmaCompactSpace ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α e : β → α he : ClosedEmbedding e ⊢ ⋃ n, (fun n => e ⁻¹' compactCovering α n) n = univ ** rw [← preimage_iUnion, iUnion_compactCovering, preimage_univ] ** Qed
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LocallyFinite.countable_univ ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α ι : Type u_3 f : ι → Set α hf : LocallyFinite f hne : ∀ (i : ι), Set.Nonempty (f i) ⊢ Set.Countable univ ** have := fun n => hf.finite_nonempty_inter_compact (isCompact_compactCovering α n) ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α ι : Type u_3 f : ι → Set α hf : LocallyFinite f hne : ∀ (i : ι), Set.Nonempty (f i) this : ∀ (n : ℕ), Set.Finite {i | Set.Nonempty (f i ∩ compactCovering α n)} ⊢ Set.Countable univ ** refine (countable_iUnion fun n => (this n).countable).mono fun i _ => ?_ ** α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α ι : Type u_3 f : ι → Set α hf : LocallyFinite f hne : ∀ (i : ι), Set.Nonempty (f i) this : ∀ (n : ℕ), Set.Finite {i | Set.Nonempty (f i ∩ compactCovering α n)} i : ι x✝ : i ∈ univ ⊢ i ∈ ⋃ i, {i_1 | Set.Nonempty (f i_1 ∩ compactCovering α i)} ** rcases hne i with ⟨x, hx⟩ ** case intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α ι : Type u_3 f : ι → Set α hf : LocallyFinite f hne : ∀ (i : ι), Set.Nonempty (f i) this : ∀ (n : ℕ), Set.Finite {i | Set.Nonempty (f i ∩ compactCovering α n)} i : ι x✝ : i ∈ univ x : α hx : x ∈ f i ⊢ i ∈ ⋃ i, {i_1 | Set.Nonempty (f i_1 ∩ compactCovering α i)} ** rcases iUnion_eq_univ_iff.1 (iUnion_compactCovering α) x with ⟨n, hn⟩ ** case intro.intro α : Type u β : Type v ι✝ : Type u_1 π : ι✝ → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α ι : Type u_3 f : ι → Set α hf : LocallyFinite f hne : ∀ (i : ι), Set.Nonempty (f i) this : ∀ (n : ℕ), Set.Finite {i | Set.Nonempty (f i ∩ compactCovering α n)} i : ι x✝ : i ∈ univ x : α hx : x ∈ f i n : ℕ hn : x ∈ compactCovering α n ⊢ i ∈ ⋃ i, {i_1 | Set.Nonempty (f i_1 ∩ compactCovering α i)} ** exact mem_iUnion.2 ⟨n, x, hx, hn⟩ ** Qed
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countable_cover_nhdsWithin_of_sigma_compact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : SigmaCompactSpace α f : α → Set α s : Set α hs : IsClosed s hf : ∀ (x : α), x ∈ s → f x ∈ 𝓝[s] x ⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ x ∈ t, f x ** simp only [nhdsWithin, mem_inf_principal] at hf ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : SigmaCompactSpace α f : α → Set α s : Set α hs : IsClosed s hf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x ⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ x ∈ t, f x ** choose t ht hsub using fun n =>
((isCompact_compactCovering α n).inter_right hs).elim_nhds_subcover _ fun x hx => hf x hx.right ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t✝ : Set α inst✝ : SigmaCompactSpace α f : α → Set α s : Set α hs : IsClosed s hf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x t : ℕ → Finset α ht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s hsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ x ∈ t n, {x_1 | x_1 ∈ s → x_1 ∈ f x} ⊢ ∃ t x, Set.Countable t ∧ s ⊆ ⋃ x ∈ t, f x ** refine'
⟨⋃ n, (t n : Set α), iUnion_subset fun n x hx => (ht n x hx).2,
countable_iUnion fun n => (t n).countable_toSet, fun x hx => mem_iUnion₂.2 _⟩ ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t✝ : Set α inst✝ : SigmaCompactSpace α f : α → Set α s : Set α hs : IsClosed s hf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x t : ℕ → Finset α ht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s hsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ x ∈ t n, {x_1 | x_1 ∈ s → x_1 ∈ f x} x : α hx : x ∈ s ⊢ ∃ i j, x ∈ f i ** rcases exists_mem_compactCovering x with ⟨n, hn⟩ ** case intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t✝ : Set α inst✝ : SigmaCompactSpace α f : α → Set α s : Set α hs : IsClosed s hf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x t : ℕ → Finset α ht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s hsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ x ∈ t n, {x_1 | x_1 ∈ s → x_1 ∈ f x} x : α hx : x ∈ s n : ℕ hn : x ∈ compactCovering α n ⊢ ∃ i j, x ∈ f i ** rcases mem_iUnion₂.1 (hsub n ⟨hn, hx⟩) with ⟨y, hyt : y ∈ t n, hyf : x ∈ s → x ∈ f y⟩ ** case intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t✝ : Set α inst✝ : SigmaCompactSpace α f : α → Set α s : Set α hs : IsClosed s hf : ∀ (x : α), x ∈ s → {x_1 | x_1 ∈ s → x_1 ∈ f x} ∈ 𝓝 x t : ℕ → Finset α ht : ∀ (n : ℕ) (x : α), x ∈ t n → x ∈ compactCovering α n ∩ s hsub : ∀ (n : ℕ), compactCovering α n ∩ s ⊆ ⋃ x ∈ t n, {x_1 | x_1 ∈ s → x_1 ∈ f x} x : α hx : x ∈ s n : ℕ hn : x ∈ compactCovering α n y : α hyt : y ∈ t n hyf : x ∈ s → x ∈ f y ⊢ ∃ i j, x ∈ f i ** exact ⟨y, mem_iUnion.2 ⟨n, hyt⟩, hyf hx⟩ ** Qed
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countable_cover_nhds_of_sigma_compact ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α f : α → Set α hf : ∀ (x : α), f x ∈ 𝓝 x ⊢ ∃ s, Set.Countable s ∧ ⋃ x ∈ s, f x = univ ** simp only [← nhdsWithin_univ] at hf ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α f : α → Set α hf : ∀ (x : α), f x ∈ 𝓝[univ] x ⊢ ∃ s, Set.Countable s ∧ ⋃ x ∈ s, f x = univ ** rcases countable_cover_nhdsWithin_of_sigma_compact isClosed_univ fun x _ => hf x with
⟨s, -, hsc, hsU⟩ ** case intro.intro.intro α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s✝ t : Set α inst✝ : SigmaCompactSpace α f : α → Set α hf : ∀ (x : α), f x ∈ 𝓝[univ] x s : Set α hsc : Set.Countable s hsU : univ ⊆ ⋃ x ∈ s, f x ⊢ ∃ s, Set.Countable s ∧ ⋃ x ∈ s, f x = univ ** exact ⟨s, hsc, univ_subset_iff.1 hsU⟩ ** Qed
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CompactExhaustion.mem_diff_shiftr_find ** α : Type u β : Type v ι : Type u_1 π : ι → Type u_2 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β s t : Set α inst✝ : SigmaCompactSpace α K : CompactExhaustion α x : α ⊢ ¬CompactExhaustion.find (shiftr K) x ≤ CompactExhaustion.find K x ** simp only [find_shiftr, not_le, Nat.lt_succ_self] ** Qed
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TopCat.piIsoPi_inv_π ** J : Type v inst✝ : SmallCategory J ι : Type v α : ι → TopCatMax i : ι ⊢ (piIsoPi α).inv ≫ Pi.π α i = piπ α i ** simp [piIsoPi] ** Qed
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TopCat.piIsoPi_hom_apply ** J : Type v inst✝ : SmallCategory J ι : Type v α : ι → TopCatMax i : ι x : ↑(∏ α) ⊢ ↑(piIsoPi α).hom x i = ↑(Pi.π α i) x ** have := piIsoPi_inv_π α i ** J : Type v inst✝ : SmallCategory J ι : Type v α : ι → TopCatMax i : ι x : ↑(∏ α) this : (piIsoPi α).inv ≫ Pi.π α i = piπ α i ⊢ ↑(piIsoPi α).hom x i = ↑(Pi.π α i) x ** rw [Iso.inv_comp_eq] at this ** J : Type v inst✝ : SmallCategory J ι : Type v α : ι → TopCatMax i : ι x : ↑(∏ α) this : Pi.π α i = (piIsoPi α).hom ≫ piπ α i ⊢ ↑(piIsoPi α).hom x i = ↑(Pi.π α i) x ** exact ConcreteCategory.congr_hom this x ** Qed
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TopCat.sigmaIsoSigma_hom_ι ** J : Type v inst✝ : SmallCategory J ι : Type v α : ι → TopCatMax i : ι ⊢ Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i ** simp [sigmaIsoSigma] ** Qed
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TopCat.sigmaIsoSigma_inv_apply ** J : Type v inst✝ : SmallCategory J ι : Type v α : ι → TopCatMax i : ι x : ↑(α i) ⊢ ↑(sigmaIsoSigma α).inv { fst := i, snd := x } = ↑(Sigma.ι α i) x ** rw [← sigmaIsoSigma_hom_ι_apply, ← comp_app, ←comp_app, Category.assoc, Iso.hom_inv_id,
Category.comp_id] ** Qed
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TopCat.induced_of_isLimit ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C ⊢ C.pt.str = ⨅ j, induced (↑(C.π.app j)) (F.obj j).str ** let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F)) ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)) ⊢ C.pt.str = ⨅ j, induced (↑(C.π.app j)) (F.obj j).str ** refine' homeo.inducing.induced.trans _ ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)) ⊢ induced (↑homeo) (topologicalSpace_coe (limitConeInfi F).pt) = ⨅ j, induced (↑(C.π.app j)) (F.obj j).str ** change induced homeo (⨅ j : J, _) = _ ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)) ⊢ induced (↑homeo) (⨅ j, induced ((Types.limitCone (F ⋙ forget TopCat)).π.app j) (F.obj j).str) = ⨅ j, induced (↑(C.π.app j)) (F.obj j).str ** simp [induced_iInf, induced_compose] ** J : Type v inst✝ : SmallCategory J F : J ⥤ TopCatMax C : Cone F hC : IsLimit C homeo : ↑C.pt ≃ₜ ↑(limitConeInfi F).pt := homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)) ⊢ ⨅ i, induced ((Types.limitCone (F ⋙ forget TopCat)).π.app i ∘ ↑(homeoOfIso (IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)))) (F.obj i).str = ⨅ j, induced (↑(C.π.app j)) (F.obj j).str ** rfl ** Qed
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TopCat.prodIsoProd_hom_fst ** J : Type v inst✝ : SmallCategory J X Y : TopCat ⊢ (prodIsoProd X Y).hom ≫ prodFst = prod.fst ** simp [← Iso.eq_inv_comp, prodIsoProd] ** J : Type v inst✝ : SmallCategory J X Y : TopCat ⊢ prodFst = BinaryFan.fst (prodBinaryFan X Y) ** rfl ** Qed
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TopCat.prodIsoProd_hom_snd ** J : Type v inst✝ : SmallCategory J X Y : TopCat ⊢ (prodIsoProd X Y).hom ≫ prodSnd = prod.snd ** simp [← Iso.eq_inv_comp, prodIsoProd] ** J : Type v inst✝ : SmallCategory J X Y : TopCat ⊢ prodSnd = BinaryFan.snd (prodBinaryFan X Y) ** rfl ** Qed
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TopCat.prodIsoProd_hom_apply ** J : Type v inst✝ : SmallCategory J X Y : TopCat x : ↑(X ⨯ Y) ⊢ ↑(prodIsoProd X Y).hom x = (↑prod.fst x, ↑prod.snd x) ** apply Prod.ext ** case h₁ J : Type v inst✝ : SmallCategory J X Y : TopCat x : ↑(X ⨯ Y) ⊢ (↑(prodIsoProd X Y).hom x).1 = (↑prod.fst x, ↑prod.snd x).1 ** exact ConcreteCategory.congr_hom (prodIsoProd_hom_fst X Y) x ** case h₂ J : Type v inst✝ : SmallCategory J X Y : TopCat x : ↑(X ⨯ Y) ⊢ (↑(prodIsoProd X Y).hom x).2 = (↑prod.fst x, ↑prod.snd x).2 ** exact ConcreteCategory.congr_hom (prodIsoProd_hom_snd X Y) x ** Qed
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TopCat.prodIsoProd_inv_fst ** J : Type v inst✝ : SmallCategory J X Y : TopCat ⊢ (prodIsoProd X Y).inv ≫ prod.fst = prodFst ** simp [Iso.inv_comp_eq] ** Qed
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TopCat.prodIsoProd_inv_snd ** J : Type v inst✝ : SmallCategory J X Y : TopCat ⊢ (prodIsoProd X Y).inv ≫ prod.snd = prodSnd ** simp [Iso.inv_comp_eq] ** Qed
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TopCat.prod_topology ** J : Type v inst✝ : SmallCategory J X Y : TopCat ⊢ (X ⨯ Y).str = induced (↑prod.fst) X.str ⊓ induced (↑prod.snd) Y.str ** let homeo := homeoOfIso (prodIsoProd X Y) ** J : Type v inst✝ : SmallCategory J X Y : TopCat homeo : ↑(X ⨯ Y) ≃ₜ ↑(of (↑X × ↑Y)) := homeoOfIso (prodIsoProd X Y) ⊢ (X ⨯ Y).str = induced (↑prod.fst) X.str ⊓ induced (↑prod.snd) Y.str ** refine' homeo.inducing.induced.trans _ ** J : Type v inst✝ : SmallCategory J X Y : TopCat homeo : ↑(X ⨯ Y) ≃ₜ ↑(of (↑X × ↑Y)) := homeoOfIso (prodIsoProd X Y) ⊢ induced (↑homeo) (topologicalSpace_coe (of (↑X × ↑Y))) = induced (↑prod.fst) X.str ⊓ induced (↑prod.snd) Y.str ** change induced homeo (_ ⊓ _) = _ ** J : Type v inst✝ : SmallCategory J X Y : TopCat homeo : ↑(X ⨯ Y) ≃ₜ ↑(of (↑X × ↑Y)) := homeoOfIso (prodIsoProd X Y) ⊢ induced (↑homeo) (induced Prod.fst (topologicalSpace_coe X) ⊓ induced Prod.snd (topologicalSpace_coe Y)) = induced (↑prod.fst) X.str ⊓ induced (↑prod.snd) Y.str ** simp [induced_compose] ** J : Type v inst✝ : SmallCategory J X Y : TopCat homeo : ↑(X ⨯ Y) ≃ₜ ↑(of (↑X × ↑Y)) := homeoOfIso (prodIsoProd X Y) ⊢ induced (Prod.fst ∘ ↑(homeoOfIso (prodIsoProd X Y))) (topologicalSpace_coe X) ⊓ induced (Prod.snd ∘ ↑(homeoOfIso (prodIsoProd X Y))) (topologicalSpace_coe Y) = induced (↑prod.fst) X.str ⊓ induced (↑prod.snd) Y.str ** rfl ** Qed
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TopCat.range_prod_map ** J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z ⊢ Set.range ↑(prod.map f g) = ↑prod.fst ⁻¹' Set.range ↑f ∩ ↑prod.snd ⁻¹' Set.range ↑g ** ext x ** case h J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) ⊢ x ∈ Set.range ↑(prod.map f g) ↔ x ∈ ↑prod.fst ⁻¹' Set.range ↑f ∩ ↑prod.snd ⁻¹' Set.range ↑g ** constructor ** case h.mp J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) ⊢ x ∈ Set.range ↑(prod.map f g) → x ∈ ↑prod.fst ⁻¹' Set.range ↑f ∩ ↑prod.snd ⁻¹' Set.range ↑g ** rintro ⟨y, rfl⟩ ** case h.mp.intro J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z y : (forget TopCat).obj (W ⨯ X) ⊢ ↑(prod.map f g) y ∈ ↑prod.fst ⁻¹' Set.range ↑f ∩ ↑prod.snd ⁻¹' Set.range ↑g ** simp only [Set.mem_preimage, Set.mem_range, Set.mem_inter_iff, ← comp_apply] ** case h.mp.intro J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z y : (forget TopCat).obj (W ⨯ X) ⊢ (∃ y_1, ↑f y_1 = ↑(prod.map f g ≫ prod.fst) y) ∧ ∃ y_1, ↑g y_1 = ↑(prod.map f g ≫ prod.snd) y ** simp only [Limits.prod.map_fst, Limits.prod.map_snd, exists_apply_eq_apply, comp_apply,
and_self_iff] ** case h.mpr J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) ⊢ x ∈ ↑prod.fst ⁻¹' Set.range ↑f ∩ ↑prod.snd ⁻¹' Set.range ↑g → x ∈ Set.range ↑(prod.map f g) ** rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩ ** case h.mpr.intro.intro.intro J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ x ∈ Set.range ↑(prod.map f g) ** use (prodIsoProd W X).inv (x₁, x₂) ** case h J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑(prod.map f g) (↑(prodIsoProd W X).inv (x₁, x₂)) = x ** apply Concrete.limit_ext ** case h.a J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ∀ (j : Discrete WalkingPair), ↑(limit.π (pair Y Z) j) (↑(prod.map f g) (↑(prodIsoProd W X).inv (x₁, x₂))) = ↑(limit.π (pair Y Z) j) x ** rintro ⟨⟨⟩⟩ ** case h.a.mk.left J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑(limit.π (pair Y Z) { as := WalkingPair.left }) (↑(prod.map f g) (↑(prodIsoProd W X).inv (x₁, x₂))) = ↑(limit.π (pair Y Z) { as := WalkingPair.left }) x ** simp only [← comp_apply, Category.assoc] ** case h.a.mk.left J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑((prodIsoProd W X).inv ≫ prod.map f g ≫ limit.π (pair Y Z) { as := WalkingPair.left }) (x₁, x₂) = ↑(limit.π (pair Y Z) { as := WalkingPair.left }) x ** erw [Limits.prod.map_fst] ** case h.a.mk.left J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑((prodIsoProd W X).inv ≫ prod.fst ≫ f) (x₁, x₂) = ↑(limit.π (pair Y Z) { as := WalkingPair.left }) x ** rw [TopCat.prodIsoProd_inv_fst_assoc,TopCat.comp_app] ** case h.a.mk.left J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑f (↑prodFst (x₁, x₂)) = ↑(limit.π (pair Y Z) { as := WalkingPair.left }) x ** exact hx₁ ** case h.a.mk.right J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑(limit.π (pair Y Z) { as := WalkingPair.right }) (↑(prod.map f g) (↑(prodIsoProd W X).inv (x₁, x₂))) = ↑(limit.π (pair Y Z) { as := WalkingPair.right }) x ** simp only [← comp_apply, Category.assoc] ** case h.a.mk.right J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑((prodIsoProd W X).inv ≫ prod.map f g ≫ limit.π (pair Y Z) { as := WalkingPair.right }) (x₁, x₂) = ↑(limit.π (pair Y Z) { as := WalkingPair.right }) x ** erw [Limits.prod.map_snd] ** case h.a.mk.right J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑((prodIsoProd W X).inv ≫ prod.snd ≫ g) (x₁, x₂) = ↑(limit.π (pair Y Z) { as := WalkingPair.right }) x ** rw [TopCat.prodIsoProd_inv_snd_assoc,TopCat.comp_app] ** case h.a.mk.right J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ Y g : X ⟶ Z x : (forget TopCat).obj (Y ⨯ Z) x₁ : (forget TopCat).obj W hx₁ : ↑f x₁ = ↑prod.fst x x₂ : (forget TopCat).obj X hx₂ : ↑g x₂ = ↑prod.snd x ⊢ ↑g (↑prodSnd (x₁, x₂)) = ↑(limit.π (pair Y Z) { as := WalkingPair.right }) x ** exact hx₂ ** Qed
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TopCat.inducing_prod_map ** J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ X g : Y ⟶ Z hf : Inducing ↑f hg : Inducing ↑g ⊢ Inducing ↑(prod.map f g) ** constructor ** case induced J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ X g : Y ⟶ Z hf : Inducing ↑f hg : Inducing ↑g ⊢ topologicalSpace_forget (W ⨯ Y) = induced (↑(prod.map f g)) (topologicalSpace_forget (X ⨯ Z)) ** simp only [prod_topology, induced_compose, ← coe_comp, Limits.prod.map_fst, Limits.prod.map_snd,
induced_inf] ** case induced J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ X g : Y ⟶ Z hf : Inducing ↑f hg : Inducing ↑g ⊢ induced (↑prod.fst) W.str ⊓ induced (↑prod.snd) Y.str = induced (↑(prod.fst ≫ f)) X.str ⊓ induced (↑(prod.snd ≫ g)) Z.str ** simp only [coe_comp] ** case induced J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ X g : Y ⟶ Z hf : Inducing ↑f hg : Inducing ↑g ⊢ induced (↑prod.fst) W.str ⊓ induced (↑prod.snd) Y.str = induced (↑f ∘ ↑prod.fst) X.str ⊓ induced (↑g ∘ ↑prod.snd) Z.str ** rw [← @induced_compose _ _ _ _ _ f, ← @induced_compose _ _ _ _ _ g, ← hf.induced, ← hg.induced] ** Qed
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TopCat.embedding_prod_map ** J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ X g : Y ⟶ Z hf : Embedding ↑f hg : Embedding ↑g ⊢ Function.Injective ↑(prod.map f g) ** haveI := (TopCat.mono_iff_injective _).mpr hf.inj ** J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ X g : Y ⟶ Z hf : Embedding ↑f hg : Embedding ↑g this : Mono f ⊢ Function.Injective ↑(prod.map f g) ** haveI := (TopCat.mono_iff_injective _).mpr hg.inj ** J : Type v inst✝ : SmallCategory J W X Y Z : TopCat f : W ⟶ X g : Y ⟶ Z hf : Embedding ↑f hg : Embedding ↑g this✝ : Mono f this : Mono g ⊢ Function.Injective ↑(prod.map f g) ** exact (TopCat.mono_iff_injective _).mp inferInstance ** Qed
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TopCat.binaryCofan_isColimit_iff ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y ⊢ Nonempty (IsColimit c) ↔ OpenEmbedding ↑(BinaryCofan.inl c) ∧ OpenEmbedding ↑(BinaryCofan.inr c) ∧ IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) ** constructor ** case mp J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y ⊢ Nonempty (IsColimit c) → OpenEmbedding ↑(BinaryCofan.inl c) ∧ OpenEmbedding ↑(BinaryCofan.inr c) ∧ IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) ** rintro ⟨h⟩ ** case mp.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h : IsColimit c ⊢ OpenEmbedding ↑(BinaryCofan.inl c) ∧ OpenEmbedding ↑(BinaryCofan.inr c) ∧ IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) ** rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (binaryCofanIsColimit X Y) ⟨WalkingPair.left⟩,
← show _ = c.inr from
h.comp_coconePointUniqueUpToIso_inv (binaryCofanIsColimit X Y) ⟨WalkingPair.right⟩] ** case mp.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h : IsColimit c ⊢ OpenEmbedding ↑((TopCat.binaryCofan X Y).ι.app { as := WalkingPair.left } ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv) ∧ OpenEmbedding ↑((TopCat.binaryCofan X Y).ι.app { as := WalkingPair.right } ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv) ∧ IsCompl (Set.range ↑((TopCat.binaryCofan X Y).ι.app { as := WalkingPair.left } ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv)) (Set.range ↑((TopCat.binaryCofan X Y).ι.app { as := WalkingPair.right } ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv)) ** dsimp ** case mp.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h : IsColimit c ⊢ OpenEmbedding ↑(BinaryCofan.inl (TopCat.binaryCofan X Y) ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv) ∧ OpenEmbedding ↑(BinaryCofan.inr (TopCat.binaryCofan X Y) ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv) ∧ IsCompl (Set.range ↑(BinaryCofan.inl (TopCat.binaryCofan X Y) ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv)) (Set.range ↑(BinaryCofan.inr (TopCat.binaryCofan X Y) ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv)) ** refine' ⟨(homeoOfIso <| h.coconePointUniqueUpToIso
(binaryCofanIsColimit X Y)).symm.openEmbedding.comp openEmbedding_inl,
(homeoOfIso <| h.coconePointUniqueUpToIso
(binaryCofanIsColimit X Y)).symm.openEmbedding.comp openEmbedding_inr, _⟩ ** case mp.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h : IsColimit c ⊢ IsCompl (Set.range ↑(BinaryCofan.inl (TopCat.binaryCofan X Y) ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv)) (Set.range ↑(BinaryCofan.inr (TopCat.binaryCofan X Y) ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv)) ** erw [Set.range_comp, ← eq_compl_iff_isCompl, coe_comp, coe_comp, Set.range_comp _ Sum.inr,
← Set.image_compl_eq (homeoOfIso <| h.coconePointUniqueUpToIso
(binaryCofanIsColimit X Y)).symm.bijective, Set.compl_range_inr, Set.image_comp] ** case mp.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h : IsColimit c ⊢ ↑(IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)).inv '' (↑(BinaryCofan.inl (TopCat.binaryCofan X Y)) '' Set.range fun x => x) = ↑(Homeomorph.symm (homeoOfIso (IsColimit.coconePointUniqueUpToIso h (binaryCofanIsColimit X Y)))) '' Set.range Sum.inl ** aesop ** case mpr J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y ⊢ OpenEmbedding ↑(BinaryCofan.inl c) ∧ OpenEmbedding ↑(BinaryCofan.inr c) ∧ IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) → Nonempty (IsColimit c) ** rintro ⟨h₁, h₂, h₃⟩ ** case mpr.intro.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) ⊢ Nonempty (IsColimit c) ** have : ∀ x, x ∈ Set.range c.inl ∨ x ∈ Set.range c.inr := by
rw [eq_compl_iff_isCompl.mpr h₃.symm]
exact fun _ => or_not ** case mpr.intro.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Nonempty (IsColimit c) ** refine' ⟨BinaryCofan.IsColimit.mk _ _ _ _ _⟩ ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) ⊢ ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) ** rw [eq_compl_iff_isCompl.mpr h₃.symm] ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) ⊢ ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ (Set.range ↑(BinaryCofan.inl c))ᶜ ** exact fun _ => or_not ** case mpr.intro.intro.refine'_1 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) ⊢ {T : TopCat} → (X ⟶ T) → (Y ⟶ T) → (c.pt ⟶ T) ** intro T f g ** case mpr.intro.intro.refine'_1 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ c.pt ⟶ T ** refine' ContinuousMap.mk _ _ ** case mpr.intro.intro.refine'_1.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ Continuous fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) }) ** rw [continuous_iff_continuousAt] ** case mpr.intro.intro.refine'_1.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ ∀ (x : ↑c.pt), ContinuousAt (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x ** intro x ** case mpr.intro.intro.refine'_1.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : ↑c.pt ⊢ ContinuousAt (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x ** by_cases x ∈ Set.range c.inl ** case mpr.intro.intro.refine'_1.refine'_1 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ ↑c.pt → ↑T ** exact fun x =>
if h : x ∈ Set.range c.inl then f ((Equiv.ofInjective _ h₁.inj).symm ⟨x, h⟩)
else g ((Equiv.ofInjective _ h₂.inj).symm ⟨x, (this x).resolve_left h⟩) ** case pos J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : ↑c.pt h : x ∈ Set.range ↑(BinaryCofan.inl c) ⊢ ContinuousAt (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x ** revert h x ** case pos J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ ∀ (x : ↑c.pt), x ∈ Set.range ↑(BinaryCofan.inl c) → ContinuousAt (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x ** apply (IsOpen.continuousOn_iff _).mp ** case pos J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ ContinuousOn (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) (Set.range ↑(BinaryCofan.inl c)) ** rw [continuousOn_iff_continuous_restrict] ** case pos J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ Continuous (Set.restrict (Set.range ↑(BinaryCofan.inl c)) fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) ** convert_to Continuous (f ∘ (Homeomorph.ofEmbedding _ h₁.toEmbedding).symm) ** case pos J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ Continuous (↑f ∘ ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inl c) (_ : Embedding ↑(BinaryCofan.inl c))))) ** apply Continuous.comp ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T e_2✝ : ↑T = (forget TopCat).obj T ⊢ (Set.restrict (Set.range ↑(BinaryCofan.inl c)) fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) = ↑f ∘ ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inl c) (_ : Embedding ↑(BinaryCofan.inl c)))) ** ext ⟨x, hx⟩ ** case h.e'_5.h.h.mk J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T e_2✝ : ↑T = (forget TopCat).obj T x : ↑c.pt hx : x ∈ Set.range ↑(BinaryCofan.inl c) ⊢ Set.restrict (Set.range ↑(BinaryCofan.inl c)) (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) { val := x, property := hx } = (↑f ∘ ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inl c) (_ : Embedding ↑(BinaryCofan.inl c))))) { val := x, property := hx } ** exact dif_pos hx ** case pos.hg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ Continuous ↑f ** exact f.continuous_toFun ** case pos.hf J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ Continuous ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inl c) (_ : Embedding ↑(BinaryCofan.inl c)))) ** continuity ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ IsOpen (Set.range ↑(BinaryCofan.inl c)) ** exact h₁.open_range ** case neg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : ↑c.pt h : ¬x ∈ Set.range ↑(BinaryCofan.inl c) ⊢ ContinuousAt (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x ** revert h x ** case neg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ ∀ (x : ↑c.pt), ¬x ∈ Set.range ↑(BinaryCofan.inl c) → ContinuousAt (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x ** apply (IsOpen.continuousOn_iff _).mp ** case neg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ ContinuousOn (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False ** rw [continuousOn_iff_continuous_restrict] ** case neg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ Continuous (Set.restrict (fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False) fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) ** have : ∀ a, a ∉ Set.range c.inl → a ∈ Set.range c.inr := by
rintro a (h : a ∈ (Set.range c.inl)ᶜ)
rwa [eq_compl_iff_isCompl.mpr h₃.symm] ** case neg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Continuous (Set.restrict (fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False) fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) ** convert_to Continuous
(g ∘ (Homeomorph.ofEmbedding _ h₂.toEmbedding).symm ∘ Subtype.map _ this) ** case neg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Continuous (↑g ∘ ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inr c) (_ : Embedding ↑(BinaryCofan.inr c)))) ∘ Subtype.map (fun a => a) this) ** apply Continuous.comp ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ** rintro a (h : a ∈ (Set.range c.inl)ᶜ) ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) h : a ∈ (Set.range ↑(BinaryCofan.inl c))ᶜ ⊢ a ∈ Set.range ↑(BinaryCofan.inr c) ** rwa [eq_compl_iff_isCompl.mpr h₃.symm] ** case h.e'_5.h J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) e_1✝ : (↑fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False) = { a // ¬a ∈ Set.range ↑(BinaryCofan.inl c) } e_2✝ : ↑T = (forget TopCat).obj T ⊢ (Set.restrict (fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False) fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) = ↑g ∘ ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inr c) (_ : Embedding ↑(BinaryCofan.inr c)))) ∘ Subtype.map (fun a => a) this ** ext ⟨x, hx⟩ ** case h.e'_5.h.h.mk J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) e_1✝ : (↑fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False) = { a // ¬a ∈ Set.range ↑(BinaryCofan.inl c) } e_2✝ : ↑T = (forget TopCat).obj T x : ↑c.pt hx : x ∈ fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False ⊢ Set.restrict (fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False) (fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) { val := x, property := hx } = (↑g ∘ ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inr c) (_ : Embedding ↑(BinaryCofan.inr c)))) ∘ Subtype.map (fun a => a) this) { val := x, property := hx } ** exact dif_neg hx ** case neg.hg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Continuous ↑g ** exact g.continuous_toFun ** case neg.hf J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Continuous (↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inr c) (_ : Embedding ↑(BinaryCofan.inr c)))) ∘ Subtype.map (fun a => a) this) ** apply Continuous.comp ** case neg.hf.hg J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Continuous ↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑(BinaryCofan.inr c) (_ : Embedding ↑(BinaryCofan.inr c)))) ** continuity ** case neg.hf.hf J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Continuous (Subtype.map (fun a => a) this) ** rw [embedding_subtype_val.toInducing.continuous_iff] ** case neg.hf.hf J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T this : ∀ (a : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), ¬a ∈ Set.range ↑(BinaryCofan.inl c) → a ∈ Set.range ↑(BinaryCofan.inr c) ⊢ Continuous (Subtype.val ∘ Subtype.map (fun a => a) this) ** exact continuous_subtype_val ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ IsOpen fun a => a ∈ Set.range ↑(BinaryCofan.inl c) → False ** change IsOpen (Set.range c.inl)ᶜ ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ IsOpen (Set.range ↑(BinaryCofan.inl c))ᶜ ** rw [← eq_compl_iff_isCompl.mpr h₃.symm] ** J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ IsOpen (Set.range ↑(BinaryCofan.inr c)) ** exact h₂.open_range ** case mpr.intro.intro.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) ⊢ ∀ {T : TopCat} (f : X ⟶ T) (g : Y ⟶ T), (BinaryCofan.inl c ≫ ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) = f ** intro T f g ** case mpr.intro.intro.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ (BinaryCofan.inl c ≫ ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) = f ** ext x ** case mpr.intro.intro.refine'_2.w J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.left }) ⊢ ↑(BinaryCofan.inl c ≫ ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x = ↑f x ** refine' (dif_pos _).trans _ ** case mpr.intro.intro.refine'_2.w.refine'_1 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.left }) ⊢ ↑(BinaryCofan.inl c) x ∈ Set.range ↑(BinaryCofan.inl c) ** exact ⟨x, rfl⟩ ** case mpr.intro.intro.refine'_2.w.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.left }) ⊢ (fun h => ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := ↑(BinaryCofan.inl c) x, property := h })) (_ : ∃ y, ↑(BinaryCofan.inl c) y = ↑(BinaryCofan.inl c) x) = ↑f x ** dsimp ** case mpr.intro.intro.refine'_2.w.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.left }) ⊢ ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := ↑(BinaryCofan.inl c) x, property := (_ : ∃ y, ↑(BinaryCofan.inl c) y = ↑(BinaryCofan.inl c) x) }) = ↑f x ** conv_lhs => erw [Equiv.ofInjective_symm_apply] ** case mpr.intro.intro.refine'_3 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) ⊢ ∀ {T : TopCat} (f : X ⟶ T) (g : Y ⟶ T), (BinaryCofan.inr c ≫ ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) = g ** intro T f g ** case mpr.intro.intro.refine'_3 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T ⊢ (BinaryCofan.inr c ≫ ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) = g ** ext x ** case mpr.intro.intro.refine'_3.w J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.right }) ⊢ ↑(BinaryCofan.inr c ≫ ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x = ↑g x ** refine' (dif_neg _).trans _ ** case mpr.intro.intro.refine'_3.w.refine'_1 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.right }) ⊢ ¬↑(BinaryCofan.inr c) x ∈ Set.range ↑(BinaryCofan.inl c) ** rintro ⟨y, e⟩ ** case mpr.intro.intro.refine'_3.w.refine'_1.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.right }) y : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.left }) e : ↑(BinaryCofan.inl c) y = ↑(BinaryCofan.inr c) x ⊢ False ** have : c.inr x ∈ Set.range c.inl ⊓ Set.range c.inr := ⟨⟨_, e⟩, ⟨_, rfl⟩⟩ ** case mpr.intro.intro.refine'_3.w.refine'_1.intro J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this✝ : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.right }) y : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.left }) e : ↑(BinaryCofan.inl c) y = ↑(BinaryCofan.inr c) x this : ↑(BinaryCofan.inr c) x ∈ Set.range ↑(BinaryCofan.inl c) ⊓ Set.range ↑(BinaryCofan.inr c) ⊢ False ** rwa [disjoint_iff.mp h₃.1] at this ** case mpr.intro.intro.refine'_3.w.refine'_2 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat f : X ⟶ T g : Y ⟶ T x : (forget TopCat).obj ((pair X Y).obj { as := WalkingPair.right }) ⊢ (fun h => ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := ↑(BinaryCofan.inr c) x, property := (_ : ↑(BinaryCofan.inr c) x ∈ Set.range ↑(BinaryCofan.inr c)) })) (_ : ↑(BinaryCofan.inr c) x ∈ Set.range ↑(BinaryCofan.inl c) → False) = ↑g x ** exact congr_arg g (Equiv.ofInjective_symm_apply _ _) ** case mpr.intro.intro.refine'_4 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) ⊢ ∀ {T : TopCat} (f : X ⟶ T) (g : Y ⟶ T) (m : c.pt ⟶ T), BinaryCofan.inl c ≫ m = f → BinaryCofan.inr c ≫ m = g → m = ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑f (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑g (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) }) ** rintro T _ _ m rfl rfl ** case mpr.intro.intro.refine'_4 J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat m : c.pt ⟶ T ⊢ m = ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑(BinaryCofan.inl c ≫ m) (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑(BinaryCofan.inr c ≫ m) (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) }) ** ext x ** case mpr.intro.intro.refine'_4.w J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat m : c.pt ⟶ T x : (forget TopCat).obj c.pt ⊢ ↑m x = ↑(ContinuousMap.mk fun x => if h : x ∈ Set.range ↑(BinaryCofan.inl c) then ↑(BinaryCofan.inl c ≫ m) (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h }) else ↑(BinaryCofan.inr c ≫ m) (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) x ** change m x = dite _ _ _ ** case mpr.intro.intro.refine'_4.w J : Type v inst✝ : SmallCategory J X Y : TopCat c : BinaryCofan X Y h₁ : OpenEmbedding ↑(BinaryCofan.inl c) h₂ : OpenEmbedding ↑(BinaryCofan.inr c) h₃ : IsCompl (Set.range ↑(BinaryCofan.inl c)) (Set.range ↑(BinaryCofan.inr c)) this : ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })), x ∈ Set.range ↑(BinaryCofan.inl c) ∨ x ∈ Set.range ↑(BinaryCofan.inr c) T : TopCat m : c.pt ⟶ T x : (forget TopCat).obj c.pt ⊢ ↑m x = if h : x ∈ Set.range ↑(BinaryCofan.inl c) then (fun h => ↑(BinaryCofan.inl c ≫ m) (↑(Equiv.ofInjective ↑(BinaryCofan.inl c) (_ : Function.Injective ↑(BinaryCofan.inl c))).symm { val := x, property := h })) h else (fun h => ↑(BinaryCofan.inr c ≫ m) (↑(Equiv.ofInjective ↑(BinaryCofan.inr c) (_ : Function.Injective ↑(BinaryCofan.inr c))).symm { val := x, property := (_ : x ∈ Set.range ↑(BinaryCofan.inr c)) })) h ** split_ifs <;> exact congr_arg _ (Equiv.apply_ofInjective_symm _ ⟨_, _⟩).symm ** Qed
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Path.Homotopic.pi_lift ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ : (i : ι) → Path (as i) (bs i) ⊢ (pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (γ i)) = Quotient.mk (Homotopic.setoid (fun i => as i) fun i => bs i) (Path.pi γ) ** unfold pi ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ : (i : ι) → Path (as i) (bs i) ⊢ Quotient.map Path.pi (_ : ∀ (x y : (i : ι) → Path (as i) (bs i)), x ≈ y → Nonempty (Homotopy (Path.pi x) (Path.pi y))) (Quotient.choice fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (γ i)) = Quotient.mk (Homotopic.setoid (fun i => as i) fun i => bs i) (Path.pi γ) ** simp ** Qed
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Path.Homotopic.comp_pi_eq_pi_comp ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ₀ : (i : ι) → Homotopic.Quotient (as i) (bs i) γ₁ : (i : ι) → Homotopic.Quotient (bs i) (cs i) ⊢ pi γ₀ ⬝ pi γ₁ = pi fun i => γ₀ i ⬝ γ₁ i ** apply Quotient.induction_on_pi (p := _) γ₁ ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ₀ : (i : ι) → Homotopic.Quotient (as i) (bs i) γ₁ : (i : ι) → Homotopic.Quotient (bs i) (cs i) ⊢ ∀ (a : (i : ι) → Path (bs i) (cs i)), (pi γ₀ ⬝ pi fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) = pi fun i => γ₀ i ⬝ (fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) i ** intro a ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ₀ : (i : ι) → Homotopic.Quotient (as i) (bs i) γ₁ : (i : ι) → Homotopic.Quotient (bs i) (cs i) a : (i : ι) → Path (bs i) (cs i) ⊢ (pi γ₀ ⬝ pi fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) = pi fun i => γ₀ i ⬝ (fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) i ** apply Quotient.induction_on_pi (p := _) γ₀ ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ₀ : (i : ι) → Homotopic.Quotient (as i) (bs i) γ₁ : (i : ι) → Homotopic.Quotient (bs i) (cs i) a : (i : ι) → Path (bs i) (cs i) ⊢ ∀ (a_1 : (i : ι) → Path (as i) (bs i)), ((pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a_1 i)) ⬝ pi fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) = pi fun i => (fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a_1 i)) i ⬝ (fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) i ** intros ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ₀ : (i : ι) → Homotopic.Quotient (as i) (bs i) γ₁ : (i : ι) → Homotopic.Quotient (bs i) (cs i) a : (i : ι) → Path (bs i) (cs i) a✝ : (i : ι) → Path (as i) (bs i) ⊢ ((pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i)) ⬝ pi fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) = pi fun i => (fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i)) i ⬝ (fun i => Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i)) i ** simp only [pi_lift] ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ₀ : (i : ι) → Homotopic.Quotient (as i) (bs i) γ₁ : (i : ι) → Homotopic.Quotient (bs i) (cs i) a : (i : ι) → Path (bs i) (cs i) a✝ : (i : ι) → Path (as i) (bs i) ⊢ Quotient.mk (Homotopic.setoid (fun i => as i) fun i => bs i) (Path.pi fun i => a✝ i) ⬝ Quotient.mk (Homotopic.setoid (fun i => bs i) fun i => cs i) (Path.pi fun i => a i) = pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i) ⬝ Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i) ** rw [← Path.Homotopic.comp_lift, Path.trans_pi_eq_pi_trans, ← pi_lift] ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i γ₀ : (i : ι) → Homotopic.Quotient (as i) (bs i) γ₁ : (i : ι) → Homotopic.Quotient (bs i) (cs i) a : (i : ι) → Path (bs i) (cs i) a✝ : (i : ι) → Path (as i) (bs i) ⊢ (pi fun i => Quotient.mk (Homotopic.setoid (as i) (cs i)) (Path.trans (a✝ i) (a i))) = pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i) ⬝ Quotient.mk (Homotopic.setoid (bs i) (cs i)) (a i) ** rfl ** Qed
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Path.Homotopic.proj_pi ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i i : ι paths : (i : ι) → Homotopic.Quotient (as i) (bs i) ⊢ proj i (pi paths) = paths i ** apply Quotient.induction_on_pi (p := _) paths ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i i : ι paths : (i : ι) → Homotopic.Quotient (as i) (bs i) ⊢ ∀ (a : (i : ι) → Path (as i) (bs i)), proj i (pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a i)) = (fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a i)) i ** intro ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i i : ι paths : (i : ι) → Homotopic.Quotient (as i) (bs i) a✝ : (i : ι) → Path (as i) (bs i) ⊢ proj i (pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i)) = (fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i)) i ** unfold proj ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i i : ι paths : (i : ι) → Homotopic.Quotient (as i) (bs i) a✝ : (i : ι) → Path (as i) (bs i) ⊢ Quotient.mapFn (pi fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i)) (ContinuousMap.mk fun p => p i) = (fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i)) i ** rw [pi_lift, ← Path.Homotopic.map_lift] ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i i : ι paths : (i : ι) → Homotopic.Quotient (as i) (bs i) a✝ : (i : ι) → Path (as i) (bs i) ⊢ Quotient.mk (Homotopic.setoid (↑(ContinuousMap.mk fun p => p i) fun i => as i) (↑(ContinuousMap.mk fun p => p i) fun i => bs i)) (Path.map (Path.pi fun i => a✝ i) (_ : Continuous ↑(ContinuousMap.mk fun p => p i))) = (fun i => Quotient.mk (Homotopic.setoid (as i) (bs i)) (a✝ i)) i ** congr ** Qed
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Path.Homotopic.pi_proj ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i p : Homotopic.Quotient as bs ⊢ (pi fun i => proj i p) = p ** apply Quotient.inductionOn (motive := _) p ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i p : Homotopic.Quotient as bs ⊢ ∀ (a : Path as bs), (pi fun i => proj i (Quotient.mk (Homotopic.setoid as bs) a)) = Quotient.mk (Homotopic.setoid as bs) a ** intro ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i p : Homotopic.Quotient as bs a✝ : Path as bs ⊢ (pi fun i => proj i (Quotient.mk (Homotopic.setoid as bs) a✝)) = Quotient.mk (Homotopic.setoid as bs) a✝ ** unfold proj ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i p : Homotopic.Quotient as bs a✝ : Path as bs ⊢ (pi fun i => Quotient.mapFn (Quotient.mk (Homotopic.setoid as bs) a✝) (ContinuousMap.mk fun p => p i)) = Quotient.mk (Homotopic.setoid as bs) a✝ ** simp_rw [← Path.Homotopic.map_lift] ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i p : Homotopic.Quotient as bs a✝ : Path as bs ⊢ (pi fun i => Quotient.mk (Homotopic.setoid (↑(ContinuousMap.mk fun p => p i) as) (↑(ContinuousMap.mk fun p => p i) bs)) (Path.map a✝ (_ : Continuous ↑(ContinuousMap.mk fun p => p i)))) = Quotient.mk (Homotopic.setoid as bs) a✝ ** erw [pi_lift] ** ι : Type u_1 X : ι → Type u_2 inst✝ : (i : ι) → TopologicalSpace (X i) as bs cs : (i : ι) → X i p : Homotopic.Quotient as bs a✝ : Path as bs ⊢ Quotient.mk (Homotopic.setoid (fun i => as i) fun i => bs i) (Path.pi fun i => Path.map a✝ (_ : Continuous ↑(ContinuousMap.mk fun p => p i))) = Quotient.mk (Homotopic.setoid as bs) a✝ ** congr ** Qed
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Path.Homotopic.comp_prod_eq_prod_comp ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ ⊢ prod q₁ q₂ ⬝ prod r₁ r₂ = prod (q₁ ⬝ r₁) (q₂ ⬝ r₂) ** apply Quotient.inductionOn₂ (motive := _) q₁ q₂ ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ ⊢ ∀ (a : Path a₁ a₂) (b : Path b₁ b₂), prod (Quotient.mk (Homotopic.setoid a₁ a₂) a) (Quotient.mk (Homotopic.setoid b₁ b₂) b) ⬝ prod r₁ r₂ = prod (Quotient.mk (Homotopic.setoid a₁ a₂) a ⬝ r₁) (Quotient.mk (Homotopic.setoid b₁ b₂) b ⬝ r₂) ** intro a b ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ a : Path a₁ a₂ b : Path b₁ b₂ ⊢ prod (Quotient.mk (Homotopic.setoid a₁ a₂) a) (Quotient.mk (Homotopic.setoid b₁ b₂) b) ⬝ prod r₁ r₂ = prod (Quotient.mk (Homotopic.setoid a₁ a₂) a ⬝ r₁) (Quotient.mk (Homotopic.setoid b₁ b₂) b ⬝ r₂) ** apply Quotient.inductionOn₂ (motive := _) r₁ r₂ ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ a : Path a₁ a₂ b : Path b₁ b₂ ⊢ ∀ (a_1 : Path a₂ a₃) (b_1 : Path b₂ b₃), prod (Quotient.mk (Homotopic.setoid a₁ a₂) a) (Quotient.mk (Homotopic.setoid b₁ b₂) b) ⬝ prod (Quotient.mk (Homotopic.setoid a₂ a₃) a_1) (Quotient.mk (Homotopic.setoid b₂ b₃) b_1) = prod (Quotient.mk (Homotopic.setoid a₁ a₂) a ⬝ Quotient.mk (Homotopic.setoid a₂ a₃) a_1) (Quotient.mk (Homotopic.setoid b₁ b₂) b ⬝ Quotient.mk (Homotopic.setoid b₂ b₃) b_1) ** intros ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ a : Path a₁ a₂ b : Path b₁ b₂ a✝ : Path a₂ a₃ b✝ : Path b₂ b₃ ⊢ prod (Quotient.mk (Homotopic.setoid a₁ a₂) a) (Quotient.mk (Homotopic.setoid b₁ b₂) b) ⬝ prod (Quotient.mk (Homotopic.setoid a₂ a₃) a✝) (Quotient.mk (Homotopic.setoid b₂ b₃) b✝) = prod (Quotient.mk (Homotopic.setoid a₁ a₂) a ⬝ Quotient.mk (Homotopic.setoid a₂ a₃) a✝) (Quotient.mk (Homotopic.setoid b₁ b₂) b ⬝ Quotient.mk (Homotopic.setoid b₂ b₃) b✝) ** simp only [prod_lift, ← Path.Homotopic.comp_lift, Path.trans_prod_eq_prod_trans] ** Qed
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Path.Homotopic.projLeft_prod ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β ⊢ projLeft (prod q₁ q₂) = q₁ ** apply Quotient.inductionOn₂ (motive := _) q₁ q₂ ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β ⊢ ∀ (a : Path a₁ a₂) (b : Path b₁ b₂), projLeft (prod (Quotient.mk (Homotopic.setoid a₁ a₂) a) (Quotient.mk (Homotopic.setoid b₁ b₂) b)) = Quotient.mk (Homotopic.setoid a₁ a₂) a ** intro p₁ p₂ ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁✝ p₁' : Path a₁ a₂ p₂✝ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p₁ : Path a₁ a₂ p₂ : Path b₁ b₂ ⊢ projLeft (prod (Quotient.mk (Homotopic.setoid a₁ a₂) p₁) (Quotient.mk (Homotopic.setoid b₁ b₂) p₂)) = Quotient.mk (Homotopic.setoid a₁ a₂) p₁ ** unfold projLeft ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁✝ p₁' : Path a₁ a₂ p₂✝ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p₁ : Path a₁ a₂ p₂ : Path b₁ b₂ ⊢ Quotient.mapFn (prod (Quotient.mk (Homotopic.setoid a₁ a₂) p₁) (Quotient.mk (Homotopic.setoid b₁ b₂) p₂)) (ContinuousMap.mk Prod.fst) = Quotient.mk (Homotopic.setoid a₁ a₂) p₁ ** rw [prod_lift, ← Path.Homotopic.map_lift] ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁✝ p₁' : Path a₁ a₂ p₂✝ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p₁ : Path a₁ a₂ p₂ : Path b₁ b₂ ⊢ Quotient.mk (Homotopic.setoid (↑(ContinuousMap.mk Prod.fst) (a₁, b₁)) (↑(ContinuousMap.mk Prod.fst) (a₂, b₂))) (Path.map (Path.prod p₁ p₂) (_ : Continuous ↑(ContinuousMap.mk Prod.fst))) = Quotient.mk (Homotopic.setoid a₁ a₂) p₁ ** congr ** Qed
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Path.Homotopic.projRight_prod ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β ⊢ projRight (prod q₁ q₂) = q₂ ** apply Quotient.inductionOn₂ (motive := _) q₁ q₂ ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β ⊢ ∀ (a : Path a₁ a₂) (b : Path b₁ b₂), projRight (prod (Quotient.mk (Homotopic.setoid a₁ a₂) a) (Quotient.mk (Homotopic.setoid b₁ b₂) b)) = Quotient.mk (Homotopic.setoid b₁ b₂) b ** intro p₁ p₂ ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁✝ p₁' : Path a₁ a₂ p₂✝ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p₁ : Path a₁ a₂ p₂ : Path b₁ b₂ ⊢ projRight (prod (Quotient.mk (Homotopic.setoid a₁ a₂) p₁) (Quotient.mk (Homotopic.setoid b₁ b₂) p₂)) = Quotient.mk (Homotopic.setoid b₁ b₂) p₂ ** unfold projRight ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁✝ p₁' : Path a₁ a₂ p₂✝ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p₁ : Path a₁ a₂ p₂ : Path b₁ b₂ ⊢ Quotient.mapFn (prod (Quotient.mk (Homotopic.setoid a₁ a₂) p₁) (Quotient.mk (Homotopic.setoid b₁ b₂) p₂)) (ContinuousMap.mk Prod.snd) = Quotient.mk (Homotopic.setoid b₁ b₂) p₂ ** rw [prod_lift, ← Path.Homotopic.map_lift] ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁✝ p₁' : Path a₁ a₂ p₂✝ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p₁ : Path a₁ a₂ p₂ : Path b₁ b₂ ⊢ Quotient.mk (Homotopic.setoid (↑(ContinuousMap.mk Prod.snd) (a₁, b₁)) (↑(ContinuousMap.mk Prod.snd) (a₂, b₂))) (Path.map (Path.prod p₁ p₂) (_ : Continuous ↑(ContinuousMap.mk Prod.snd))) = Quotient.mk (Homotopic.setoid b₁ b₂) p₂ ** congr ** Qed
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Path.Homotopic.prod_projLeft_projRight ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p : Homotopic.Quotient (a₁, b₁) (a₂, b₂) ⊢ prod (projLeft p) (projRight p) = p ** apply Quotient.inductionOn (motive := _) p ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p : Homotopic.Quotient (a₁, b₁) (a₂, b₂) ⊢ ∀ (a : Path (a₁, b₁) (a₂, b₂)), prod (projLeft (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) a)) (projRight (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) a)) = Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) a ** intro p' ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p : Homotopic.Quotient (a₁, b₁) (a₂, b₂) p' : Path (a₁, b₁) (a₂, b₂) ⊢ prod (projLeft (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p')) (projRight (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p')) = Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p' ** unfold projLeft ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p : Homotopic.Quotient (a₁, b₁) (a₂, b₂) p' : Path (a₁, b₁) (a₂, b₂) ⊢ prod (Quotient.mapFn (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p') (ContinuousMap.mk Prod.fst)) (projRight (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p')) = Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p' ** unfold projRight ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p : Homotopic.Quotient (a₁, b₁) (a₂, b₂) p' : Path (a₁, b₁) (a₂, b₂) ⊢ prod (Quotient.mapFn (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p') (ContinuousMap.mk Prod.fst)) (Quotient.mapFn (Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p') (ContinuousMap.mk Prod.snd)) = Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p' ** simp only [← Path.Homotopic.map_lift, prod_lift] ** α : Type u_1 β : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : TopologicalSpace β a₁ a₂ a₃ : α b₁ b₂ b₃ : β p₁ p₁' : Path a₁ a₂ p₂ p₂' : Path b₁ b₂ q₁ : Homotopic.Quotient a₁ a₂ q₂ : Homotopic.Quotient b₁ b₂ r₁ : Homotopic.Quotient a₂ a₃ r₂ : Homotopic.Quotient b₂ b₃ c₁ c₂ : α × β p : Homotopic.Quotient (a₁, b₁) (a₂, b₂) p' : Path (a₁, b₁) (a₂, b₂) ⊢ prod (Quotient.mk (Homotopic.setoid (↑(ContinuousMap.mk Prod.fst) (a₁, b₁)) (↑(ContinuousMap.mk Prod.fst) (a₂, b₂))) (Path.map p' (_ : Continuous ↑(ContinuousMap.mk Prod.fst)))) (Quotient.mk (Homotopic.setoid (↑(ContinuousMap.mk Prod.snd) (a₁, b₁)) (↑(ContinuousMap.mk Prod.snd) (a₂, b₂))) (Path.map p' (_ : Continuous ↑(ContinuousMap.mk Prod.snd)))) = Quotient.mk (Homotopic.setoid (a₁, b₁) (a₂, b₂)) p' ** congr ** Qed
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CompHaus.effectiveEpiFamily_tfae ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b] ** tfae_have 1 → 2 ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b] ** tfae_have 2 → 3 ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b] ** tfae_have 3 → 1 ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b tfae_3_to_1 : (∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π ⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b] ** tfae_finish ** case tfae_1_to_2 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B ⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) ** intro ** case tfae_1_to_2 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B ✝ : EffectiveEpiFamily X π ⊢ Epi (Sigma.desc π) ** infer_instance ** case tfae_2_to_3 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) ⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b ** intro e ** case tfae_2_to_3 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Epi (Sigma.desc π) ⊢ ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b ** rw [epi_iff_surjective] at e ** case tfae_2_to_3 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) ⊢ ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b ** let i : ∐ X ≅ finiteCoproduct X :=
(colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _) ** case tfae_2_to_3 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) ⊢ ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b ** intro b ** case tfae_2_to_3 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) b : ↑B.toTop ⊢ ∃ a x, ↑(π a) x = b ** obtain ⟨t, rfl⟩ := e b ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t ** let q := i.hom t ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t ** refine ⟨q.1,q.2,?_⟩ ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t ** have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t ** rw [this] ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t)) ** show _ = (i.inv ≫ Sigma.desc π) (i.hom t) ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) ** suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
rw [this]; rfl ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π ** rw [Iso.inv_comp_eq] ** case tfae_2_to_3.intro α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π ** apply colimit.hom_ext ** case tfae_2_to_3.intro.w α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) ⊢ ∀ (j : Discrete α), colimit.ι (Discrete.functor X) j ≫ Sigma.desc π = colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π ** rintro ⟨a⟩ ** case tfae_2_to_3.intro.w.mk α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α ⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π = colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π ** simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
colimit.comp_coconePointUniqueUpToIso_hom_assoc] ** case tfae_2_to_3.intro.w.mk α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α ⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π ** ext ** case tfae_2_to_3.intro.w.mk.w α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this : t = ↑i.inv (↑i.hom t) a : α x✝ : (forget CompHaus).obj (X a) ⊢ ↑(π a) x✝ = ↑((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝ ** rfl ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑(i.hom ≫ i.inv) t ** simp only [i.hom_inv_id] ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t ⊢ t = ↑(𝟙 (∐ X)) t ** rfl ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this✝ : t = ↑i.inv (↑i.hom t) this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) ** rw [this] ** α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) e : Function.Surjective ↑(Sigma.desc π) i : ∐ X ≅ finiteCoproduct X := IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X) t : (forget CompHaus).obj (∐ fun b => X b) q : (fun x => (forget CompHaus).obj (finiteCoproduct X)) t := ↑i.hom t this✝ : t = ↑i.inv (↑i.hom t) this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π ⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑i.hom t) ** rfl ** case tfae_3_to_1 α : Type inst✝ : Fintype α B : CompHaus X : α → CompHaus π : (a : α) → X a ⟶ B tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π) tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b ⊢ (∀ (b : ↑B.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π ** apply effectiveEpiFamily_of_jointly_surjective ** Qed
| |
TopCat.Presheaf.isSheaf_of_isTerminal_of_indiscrete ** C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C U : TopologicalSpace.Opens ↑X s : Sieve U hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U ⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj (op c)) s.arrows ** obtain rfl | hne := eq_or_ne U ⊥ ** case inl C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊥ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥ ⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj (op c)) s.arrows ** intro _ _ ** case inl C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊥ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥ x✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op c)) s.arrows a✝ : Presieve.FamilyOfElements.Compatible x✝ ⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x✝ t ** rw [@exists_unique_iff_exists _ ⟨fun _ _ => _⟩] ** case inl C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊥ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥ x✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op c)) s.arrows a✝ : Presieve.FamilyOfElements.Compatible x✝ ⊢ ∃ x, Presieve.FamilyOfElements.IsAmalgamation x✝ x ** refine' ⟨it.from _, fun U hU hs => IsTerminal.hom_ext _ _ _⟩ ** case inl C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊥ hs✝ : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥ x✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op c)) s.arrows a✝ : Presieve.FamilyOfElements.Compatible x✝ U : TopologicalSpace.Opens ↑X hU : U ⟶ ⊥ hs : s.arrows hU ⊢ IsTerminal (F.obj (op U)) ** rwa [le_bot_iff.1 hU.le] ** C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊥ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥ x✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj (op c)) s.arrows a✝ : Presieve.FamilyOfElements.Compatible x✝ ⊢ ∀ (x x_1 : (F ⋙ coyoneda.obj (op c)).obj (op ⊥)), x = x_1 ** apply it.hom_ext ** case inr C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C U : TopologicalSpace.Opens ↑X s : Sieve U hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U hne : U ≠ ⊥ ⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj (op c)) s.arrows ** convert Presieve.isSheafFor_top_sieve (F ⋙ coyoneda.obj (@op C c)) ** case h.e'_5.h.h.e'_4 C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C U : TopologicalSpace.Opens ↑X s : Sieve U hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U hne : U ≠ ⊥ ⊢ s = ⊤ ** rw [← Sieve.id_mem_iff_eq_top] ** case h.e'_5.h.h.e'_4 C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C U : TopologicalSpace.Opens ↑X s : Sieve U hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U hne : U ≠ ⊥ ⊢ s.arrows (𝟙 U) ** have := (U.eq_bot_or_top hind).resolve_left hne ** case h.e'_5.h.h.e'_4 C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C U : TopologicalSpace.Opens ↑X s : Sieve U hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U hne : U ≠ ⊥ this : U = ⊤ ⊢ s.arrows (𝟙 U) ** subst this ** case h.e'_5.h.h.e'_4 C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊤ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤ hne : ⊤ ≠ ⊥ ⊢ s.arrows (𝟙 ⊤) ** obtain he | ⟨⟨x⟩⟩ := isEmpty_or_nonempty X ** case h.e'_5.h.h.e'_4.inr.intro C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊤ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤ hne : ⊤ ≠ ⊥ x : ↑X ⊢ s.arrows (𝟙 ⊤) ** obtain ⟨U, f, hf, hm⟩ := hs x _root_.trivial ** case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊤ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤ hne : ⊤ ≠ ⊥ x : ↑X U : TopologicalSpace.Opens ↑X f : U ⟶ ⊤ hf : s.arrows f hm : x ∈ U ⊢ s.arrows (𝟙 ⊤) ** obtain rfl | rfl := U.eq_bot_or_top hind ** case h.e'_5.h.h.e'_4.inl C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊤ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤ hne : ⊤ ≠ ⊥ he : IsEmpty ↑X ⊢ s.arrows (𝟙 ⊤) ** exact (hne <| SetLike.ext'_iff.2 <| Set.univ_eq_empty_iff.2 he).elim ** case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro.inl C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊤ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤ hne : ⊤ ≠ ⊥ x : ↑X f : ⊥ ⟶ ⊤ hf : s.arrows f hm : x ∈ ⊥ ⊢ s.arrows (𝟙 ⊤) ** cases hm ** case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro.inr C : Type u inst✝ : Category.{v, u} C X : TopCat hind : X.str = ⊤ F : Presheaf C X it : IsTerminal (F.obj (op ⊥)) c : C s : Sieve ⊤ hs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤ hne : ⊤ ≠ ⊥ x : ↑X f : ⊤ ⟶ ⊤ hf : s.arrows f hm : x ∈ ⊤ ⊢ s.arrows (𝟙 ⊤) ** convert hf ** Qed
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CompHaus.pullback_fst_eq ** X Y B : CompHaus f : X ⟶ B g : Y ⟶ B ⊢ pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst ** dsimp [pullbackIsoPullback] ** X Y B : CompHaus f : X ⟶ B g : Y ⟶ B ⊢ pullback.fst f g = (Limits.IsLimit.conePointUniqueUpToIso (pullback.isLimit f g) (Limits.limit.isLimit (Limits.cospan f g))).hom ≫ Limits.pullback.fst ** simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π] ** Qed
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CompHaus.pullback_snd_eq ** X Y B : CompHaus f : X ⟶ B g : Y ⟶ B ⊢ pullback.snd f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.snd ** dsimp [pullbackIsoPullback] ** X Y B : CompHaus f : X ⟶ B g : Y ⟶ B ⊢ pullback.snd f g = (Limits.IsLimit.conePointUniqueUpToIso (pullback.isLimit f g) (Limits.limit.isLimit (Limits.cospan f g))).hom ≫ Limits.pullback.snd ** simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π] ** Qed
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CompHaus.Sigma.ι_comp_toFiniteCoproduct ** α : Type inst✝ : Fintype α X : α → CompHaus a : α ⊢ Limits.Sigma.ι X a ≫ (coproductIsoCoproduct X).inv = finiteCoproduct.ι X a ** dsimp [coproductIsoCoproduct] ** α : Type inst✝ : Fintype α X : α → CompHaus a : α ⊢ Limits.Sigma.ι X a ≫ (Limits.IsColimit.coconePointUniqueUpToIso (finiteCoproduct.isColimit X) (Limits.colimit.isColimit (Discrete.functor X))).inv = finiteCoproduct.ι X a ** simp only [Limits.colimit.comp_coconePointUniqueUpToIso_inv, finiteCoproduct.cocone_pt,
finiteCoproduct.cocone_ι, Discrete.natTrans_app] ** Qed
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Bundle.Trivial.trivialization.coordChangeL ** 𝕜 : Type u_1 B : Type u_2 F : Type u_3 inst✝³ : NontriviallyNormedField 𝕜 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B b : B ⊢ Trivialization.coordChangeL 𝕜 (trivialization B F) (trivialization B F) b = ContinuousLinearEquiv.refl 𝕜 F ** ext v ** case h.h 𝕜 : Type u_1 B : Type u_2 F : Type u_3 inst✝³ : NontriviallyNormedField 𝕜 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B b : B v : F ⊢ ↑(Trivialization.coordChangeL 𝕜 (trivialization B F) (trivialization B F) b) v = ↑(ContinuousLinearEquiv.refl 𝕜 F) v ** rw [Trivialization.coordChangeL_apply'] ** case h.h 𝕜 : Type u_1 B : Type u_2 F : Type u_3 inst✝³ : NontriviallyNormedField 𝕜 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B b : B v : F ⊢ (↑(trivialization B F) (↑(LocalHomeomorph.symm (trivialization B F).toLocalHomeomorph) (b, v))).2 = ↑(ContinuousLinearEquiv.refl 𝕜 F) v case h.h.hb 𝕜 : Type u_1 B : Type u_2 F : Type u_3 inst✝³ : NontriviallyNormedField 𝕜 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B b : B v : F ⊢ b ∈ (trivialization B F).baseSet ∩ (trivialization B F).baseSet ** exacts [rfl, ⟨mem_univ _, mem_univ _⟩] ** Qed
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Trivialization.coordChangeL_prod ** 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet ⊢ ↑(coordChangeL 𝕜 (prod e₁ e₂) (prod e₁' e₂') b) = ContinuousLinearMap.prodMap ↑(coordChangeL 𝕜 e₁ e₁' b) ↑(coordChangeL 𝕜 e₂ e₂' b) ** rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap'] ** 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet ⊢ ∀ (x : F₁ × F₂), ↑↑(coordChangeL 𝕜 (prod e₁ e₂) (prod e₁' e₂') b) x = Prod.map (↑↑(coordChangeL 𝕜 e₁ e₁' b)) (↑↑(coordChangeL 𝕜 e₂ e₂' b)) x ** rintro ⟨v₁, v₂⟩ ** case mk 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet v₁ : F₁ v₂ : F₂ ⊢ ↑↑(coordChangeL 𝕜 (prod e₁ e₂) (prod e₁' e₂') b) (v₁, v₂) = Prod.map ↑↑(coordChangeL 𝕜 e₁ e₁' b) ↑↑(coordChangeL 𝕜 e₂ e₂' b) (v₁, v₂) ** show
(e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) =
(e₁.coordChangeL 𝕜 e₁' b v₁, e₂.coordChangeL 𝕜 e₂' b v₂) ** case mk 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet v₁ : F₁ v₂ : F₂ ⊢ ↑(coordChangeL 𝕜 (prod e₁ e₂) (prod e₁' e₂') b) (v₁, v₂) = (↑(coordChangeL 𝕜 e₁ e₁' b) v₁, ↑(coordChangeL 𝕜 e₂ e₂' b) v₂) ** rw [e₁.coordChangeL_apply e₁', e₂.coordChangeL_apply e₂', (e₁.prod e₂).coordChangeL_apply'] ** case mk 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet v₁ : F₁ v₂ : F₂ ⊢ (↑(prod e₁' e₂') (↑(LocalHomeomorph.symm (prod e₁ e₂).toLocalHomeomorph) (b, v₁, v₂))).2 = ((↑e₁' { proj := b, snd := Trivialization.symm e₁ b v₁ }).2, (↑e₂' { proj := b, snd := Trivialization.symm e₂ b v₂ }).2) case mk.hb 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet v₁ : F₁ v₂ : F₂ ⊢ b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet case mk.hb 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet v₁ : F₁ v₂ : F₂ ⊢ b ∈ e₂.baseSet ∩ e₂'.baseSet case mk.hb 𝕜 : Type u_1 B : Type u_2 inst✝¹⁵ : NontriviallyNormedField 𝕜 inst✝¹⁴ : TopologicalSpace B F₁ : Type u_3 inst✝¹³ : NormedAddCommGroup F₁ inst✝¹² : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹⁰ : NormedAddCommGroup F₂ inst✝⁹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁷ : (x : B) → AddCommMonoid (E₁ x) inst✝⁶ : (x : B) → Module 𝕜 (E₁ x) inst✝⁵ : (x : B) → AddCommMonoid (E₂ x) inst✝⁴ : (x : B) → Module 𝕜 (E₂ x) e₁ e₁' : Trivialization F₁ TotalSpace.proj e₂ e₂' : Trivialization F₂ TotalSpace.proj inst✝³ : Trivialization.IsLinear 𝕜 e₁ inst✝² : Trivialization.IsLinear 𝕜 e₁' inst✝¹ : Trivialization.IsLinear 𝕜 e₂ inst✝ : Trivialization.IsLinear 𝕜 e₂' b : B hb : b ∈ (prod e₁ e₂).baseSet ∩ (prod e₁' e₂').baseSet v₁ : F₁ v₂ : F₂ ⊢ b ∈ e₁.baseSet ∩ e₁'.baseSet ** exacts [rfl, hb, ⟨hb.1.2, hb.2.2⟩, ⟨hb.1.1, hb.2.1⟩] ** Qed
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Trivialization.continuousLinearEquivAt_prod ** 𝕜 : Type u_1 B : Type u_2 inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : TopologicalSpace B F₁ : Type u_3 inst✝¹⁵ : NormedAddCommGroup F₁ inst✝¹⁴ : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹² : NormedAddCommGroup F₂ inst✝¹¹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁹ : (x : B) → AddCommMonoid (E₁ x) inst✝⁸ : (x : B) → Module 𝕜 (E₁ x) inst✝⁷ : (x : B) → AddCommMonoid (E₂ x) inst✝⁶ : (x : B) → Module 𝕜 (E₂ x) inst✝⁵ : (x : B) → TopologicalSpace (E₁ x) inst✝⁴ : (x : B) → TopologicalSpace (E₂ x) inst✝³ : FiberBundle F₁ E₁ inst✝² : FiberBundle F₂ E₂ e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : Trivialization.IsLinear 𝕜 e₁ inst✝ : Trivialization.IsLinear 𝕜 e₂ x : B hx : x ∈ (prod e₁ e₂).baseSet ⊢ continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx = ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet)) (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet)) ** ext v : 2 ** case h.h 𝕜 : Type u_1 B : Type u_2 inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : TopologicalSpace B F₁ : Type u_3 inst✝¹⁵ : NormedAddCommGroup F₁ inst✝¹⁴ : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹² : NormedAddCommGroup F₂ inst✝¹¹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁹ : (x : B) → AddCommMonoid (E₁ x) inst✝⁸ : (x : B) → Module 𝕜 (E₁ x) inst✝⁷ : (x : B) → AddCommMonoid (E₂ x) inst✝⁶ : (x : B) → Module 𝕜 (E₂ x) inst✝⁵ : (x : B) → TopologicalSpace (E₁ x) inst✝⁴ : (x : B) → TopologicalSpace (E₂ x) inst✝³ : FiberBundle F₁ E₁ inst✝² : FiberBundle F₂ E₂ e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : Trivialization.IsLinear 𝕜 e₁ inst✝ : Trivialization.IsLinear 𝕜 e₂ x : B hx : x ∈ (prod e₁ e₂).baseSet v : E₁ x × E₂ x ⊢ ↑(continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx) v = ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet)) (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet))) v ** obtain ⟨v₁, v₂⟩ := v ** case h.h.mk 𝕜 : Type u_1 B : Type u_2 inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : TopologicalSpace B F₁ : Type u_3 inst✝¹⁵ : NormedAddCommGroup F₁ inst✝¹⁴ : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹² : NormedAddCommGroup F₂ inst✝¹¹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁹ : (x : B) → AddCommMonoid (E₁ x) inst✝⁸ : (x : B) → Module 𝕜 (E₁ x) inst✝⁷ : (x : B) → AddCommMonoid (E₂ x) inst✝⁶ : (x : B) → Module 𝕜 (E₂ x) inst✝⁵ : (x : B) → TopologicalSpace (E₁ x) inst✝⁴ : (x : B) → TopologicalSpace (E₂ x) inst✝³ : FiberBundle F₁ E₁ inst✝² : FiberBundle F₂ E₂ e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : Trivialization.IsLinear 𝕜 e₁ inst✝ : Trivialization.IsLinear 𝕜 e₂ x : B hx : x ∈ (prod e₁ e₂).baseSet v₁ : E₁ x v₂ : E₂ x ⊢ ↑(continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx) (v₁, v₂) = ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet)) (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet))) (v₁, v₂) ** rw [(e₁.prod e₂).continuousLinearEquivAt_apply 𝕜, Trivialization.prod] ** case h.h.mk 𝕜 : Type u_1 B : Type u_2 inst✝¹⁷ : NontriviallyNormedField 𝕜 inst✝¹⁶ : TopologicalSpace B F₁ : Type u_3 inst✝¹⁵ : NormedAddCommGroup F₁ inst✝¹⁴ : NormedSpace 𝕜 F₁ E₁ : B → Type u_4 inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁) F₂ : Type u_5 inst✝¹² : NormedAddCommGroup F₂ inst✝¹¹ : NormedSpace 𝕜 F₂ E₂ : B → Type u_6 inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂) inst✝⁹ : (x : B) → AddCommMonoid (E₁ x) inst✝⁸ : (x : B) → Module 𝕜 (E₁ x) inst✝⁷ : (x : B) → AddCommMonoid (E₂ x) inst✝⁶ : (x : B) → Module 𝕜 (E₂ x) inst✝⁵ : (x : B) → TopologicalSpace (E₁ x) inst✝⁴ : (x : B) → TopologicalSpace (E₂ x) inst✝³ : FiberBundle F₁ E₁ inst✝² : FiberBundle F₂ E₂ e₁ : Trivialization F₁ TotalSpace.proj e₂ : Trivialization F₂ TotalSpace.proj inst✝¹ : Trivialization.IsLinear 𝕜 e₁ inst✝ : Trivialization.IsLinear 𝕜 e₂ x : B hx : x ∈ (prod e₁ e₂).baseSet v₁ : E₁ x v₂ : E₂ x ⊢ (fun y => (↑{ toLocalHomeomorph := { toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }, baseSet := e₁.baseSet ∩ e₂.baseSet, open_baseSet := (_ : IsOpen (e₁.baseSet ∩ e₂.baseSet)), source_eq := (_ : { toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source = { toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source), target_eq := (_ : { toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.target = { toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.target), proj_toFun := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ { toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source → (↑{ toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) } x).1 = (↑{ toLocalEquiv := { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }, open_source := (_ : IsOpen { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → (Prod.toFun' e₁ e₂ x).1 ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).2 ∈ univ), map_target' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.1 ∈ e₁.baseSet ∩ e₂.baseSet), left_inv' := (_ : ∀ (x : TotalSpace (F₁ × F₂) fun x => E₁ x × E₂ x), x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) → Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x), right_inv' := (_ : ∀ (x : B × F₁ × F₂), x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source), open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)), continuous_toFun := (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))), continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) } x).1) } { proj := x, snd := y }).2) (v₁, v₂) = ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet)) (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet))) (v₁, v₂) ** exact (congr_arg Prod.snd (prod_apply 𝕜 hx.1 hx.2 v₁ v₂) : _) ** Qed
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TopologicalSpace.Opens.op_map_id_obj ** X Y Z : TopCat U : (Opens ↑X)ᵒᵖ ⊢ (map (𝟙 X)).op.obj U = U ** simp ** Qed
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TopologicalSpace.Opens.map_iSup ** X Y Z : TopCat f : X ⟶ Y ι : Type u_1 U : ι → Opens ↑Y ⊢ (map f).obj (iSup U) = iSup ((map f).toPrefunctor.obj ∘ U) ** ext1 ** case h X Y Z : TopCat f : X ⟶ Y ι : Type u_1 U : ι → Opens ↑Y ⊢ ↑((map f).obj (iSup U)) = ↑(iSup ((map f).toPrefunctor.obj ∘ U)) ** rw [iSup_def, iSup_def, map_obj] ** case h X Y Z : TopCat f : X ⟶ Y ι : Type u_1 U : ι → Opens ↑Y ⊢ ↑{ carrier := ↑f ⁻¹' ⋃ i, ↑(U i), is_open' := (_ : IsOpen (↑f ⁻¹' ⋃ i, ↑(U i))) } = ↑{ carrier := ⋃ i, ↑(((map f).toPrefunctor.obj ∘ U) i), is_open' := (_ : IsOpen (⋃ i, ↑(((map f).toPrefunctor.obj ∘ U) i))) } ** dsimp ** case h X Y Z : TopCat f : X ⟶ Y ι : Type u_1 U : ι → Opens ↑Y ⊢ ↑f ⁻¹' ⋃ i, ↑(U i) = ⋃ i, ↑f ⁻¹' ↑(U i) ** rw [Set.preimage_iUnion] ** Qed
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TopologicalSpace.Opens.map_id_eq ** X Y Z : TopCat ⊢ map (𝟙 X) = 𝟭 (Opens ↑X) ** rfl ** Qed
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