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exists_polynomial_near_of_continuousOn ** a b : ℝ f : ℝ → ℝ c : ContinuousOn f (Set.Icc a b) ε : ℝ pos : 0 < ε ⊢ ∃ p, ∀ (x : ℝ), x ∈ Set.Icc a b → |Polynomial.eval x p - f x| < ε ** let f' : C(Set.Icc a b, ℝ) := ⟨fun x => f x, continuousOn_iff_continuous_restrict.mp c⟩ ** a b : ℝ f : ℝ → ℝ c : ContinuousOn f (Set.Icc a b) ε : ℝ pos : 0 < ε f' : C(↑(Set.Icc a b), ℝ) := ContinuousMap.mk fun x => f ↑x ⊢ ∃ p, ∀ (x : ℝ), x ∈ Set.Icc a b → |Polynomial.eval x p - f x| < ε ** obtain ⟨p, b⟩ := exists_polynomial_near_continuousMap a b f' ε pos ** case intro a b✝ : ℝ f : ℝ → ℝ c : ContinuousOn f (Set.Icc a b✝) ε : ℝ pos : 0 < ε f' : C(↑(Set.Icc a b✝), ℝ) := ContinuousMap.mk fun x => f ↑x p : ℝ[X] b : ‖Polynomial.toContinuousMapOn p (Set.Icc a b✝) - f'‖ < ε ⊢ ∃ p, ∀ (x : ℝ), x ∈ Set.Icc a b✝ → |Polynomial.eval x p - f x| < ε ** use p ** case h a b✝ : ℝ f : ℝ → ℝ c : ContinuousOn f (Set.Icc a b✝) ε : ℝ pos : 0 < ε f' : C(↑(Set.Icc a b✝), ℝ) := ContinuousMap.mk fun x => f ↑x p : ℝ[X] b : ‖Polynomial.toContinuousMapOn p (Set.Icc a b✝) - f'‖ < ε ⊢ ∀ (x : ℝ), x ∈ Set.Icc a b✝ → |Polynomial.eval x p - f x| < ε ** rw [norm_lt_iff _ pos] at b ** case h a b✝ : ℝ f : ℝ → ℝ c : ContinuousOn f (Set.Icc a b✝) ε : ℝ pos : 0 < ε f' : C(↑(Set.Icc a b✝), ℝ) := ContinuousMap.mk fun x => f ↑x p : ℝ[X] b : ∀ (x : ↑(Set.Icc a b✝)), ‖↑(Polynomial.toContinuousMapOn p (Set.Icc a b✝) - f') x‖ < ε ⊢ ∀ (x : ℝ), x ∈ Set.Icc a b✝ → |Polynomial.eval x p - f x| < ε ** intro x m ** case h a b✝ : ℝ f : ℝ → ℝ c : ContinuousOn f (Set.Icc a b✝) ε : ℝ pos : 0 < ε f' : C(↑(Set.Icc a b✝), ℝ) := ContinuousMap.mk fun x => f ↑x p : ℝ[X] b : ∀ (x : ↑(Set.Icc a b✝)), ‖↑(Polynomial.toContinuousMapOn p (Set.Icc a b✝) - f') x‖ < ε x : ℝ m : x ∈ Set.Icc a b✝ ⊢ |Polynomial.eval x p - f x| < ε ** exact b ⟨x, m⟩ ** Qed
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IsClosed.mk_lt_continuum ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s ⊢ #↑s < 𝔠 ** by_contra' h ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s ⊢ False ** rcases exists_countable_dense X with ⟨t, htc, htd⟩ ** case intro.intro X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t ⊢ False ** haveI := htc.to_subtype ** case intro.intro X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t this : Countable ↑t ⊢ False ** refine (Cardinal.cantor 𝔠).not_le ?_ ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t this : Countable ↑t ⊢ 2 ^ 𝔠 ≤ #C(↑s, ℝ) ** rw [(ContinuousMap.equivFnOfDiscrete _ _).cardinal_eq, mk_arrow, mk_real, lift_continuum,
lift_uzero] ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t this : Countable ↑t ⊢ 2 ^ 𝔠 ≤ 𝔠 ^ #↑s ** exact (power_le_power_left two_ne_zero h).trans (power_le_power_right (nat_lt_continuum 2).le) ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t this : Countable ↑t f : C(↑s, ℝ) → C(X, ℝ) hf : ∀ (x : C(↑s, ℝ)), ContinuousMap.restrict s (f x) = x ⊢ #C(↑s, ℝ) ≤ #C(X, ℝ) ** have hfi : Injective f := LeftInverse.injective hf ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t this : Countable ↑t f : C(↑s, ℝ) → C(X, ℝ) hf : ∀ (x : C(↑s, ℝ)), ContinuousMap.restrict s (f x) = x hfi : Injective f ⊢ #C(↑s, ℝ) ≤ #C(X, ℝ) ** exact mk_le_of_injective hfi ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t this : Countable ↑t ⊢ #(↑t → ℝ) ≤ 𝔠 ** rw [mk_arrow, mk_real, lift_uzero, lift_continuum, continuum, ← power_mul] ** X : Type u inst✝³ : TopologicalSpace X inst✝² : SeparableSpace X inst✝¹ : NormalSpace X s : Set X hs : IsClosed s inst✝ : DiscreteTopology ↑s h : 𝔠 ≤ #↑s t : Set X htc : Set.Countable t htd : Dense t this : Countable ↑t ⊢ 2 ^ (ℵ₀ * #↑t) ≤ 2 ^ ℵ₀ ** exact power_le_power_left two_ne_zero mk_le_aleph0 ** Qed
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ZeroAtInftyContinuousMap.coe_nsmulRec ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β x : α inst✝¹ : AddMonoid β inst✝ : ContinuousAdd β f g : α →C₀ β ⊢ ↑(nsmulRec 0 f) = 0 • ↑f ** rw [nsmulRec, zero_smul, coe_zero] ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β x : α inst✝¹ : AddMonoid β inst✝ : ContinuousAdd β f g : α →C₀ β n : ℕ ⊢ ↑(nsmulRec (n + 1) f) = (n + 1) • ↑f ** rw [nsmulRec, succ_nsmul, coe_add, coe_nsmulRec n] ** Qed
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ZeroAtInftyContinuousMap.coe_zsmulRec ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β x : α inst✝¹ : AddGroup β inst✝ : TopologicalAddGroup β f g : α →C₀ β n : ℕ ⊢ ↑(zsmulRec (Int.ofNat n) f) = Int.ofNat n • ↑f ** rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmulRec, coe_nat_zsmul] ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β x : α inst✝¹ : AddGroup β inst✝ : TopologicalAddGroup β f g : α →C₀ β n : ℕ ⊢ ↑(zsmulRec (Int.negSucc n) f) = Int.negSucc n • ↑f ** rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmulRec] ** Qed
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ZeroAtInftyContinuousMap.bounded ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F ⊢ ∃ C, ∀ (x y : α), dist (↑f x) (↑f y) ≤ C ** obtain ⟨K : Set α, hK₁, hK₂⟩ := mem_cocompact.mp
(tendsto_def.mp (zero_at_infty (f : F)) _ (closedBall_mem_nhds (0 : β) zero_lt_one)) ** case intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F K : Set α hK₁ : IsCompact K hK₂ : Kᶜ ⊆ ↑f ⁻¹' closedBall 0 1 ⊢ ∃ C, ∀ (x y : α), dist (↑f x) (↑f y) ≤ C ** obtain ⟨C, hC⟩ := (hK₁.image (map_continuous f)).isBounded.subset_closedBall (0 : β) ** case intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F K : Set α hK₁ : IsCompact K hK₂ : Kᶜ ⊆ ↑f ⁻¹' closedBall 0 1 C : ℝ hC : ↑f '' K ⊆ closedBall 0 C ⊢ ∃ C, ∀ (x y : α), dist (↑f x) (↑f y) ≤ C ** refine' ⟨max C 1 + max C 1, fun x y => _⟩ ** case intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F K : Set α hK₁ : IsCompact K hK₂ : Kᶜ ⊆ ↑f ⁻¹' closedBall 0 1 C : ℝ hC : ↑f '' K ⊆ closedBall 0 C x y : α this : ∀ (x : α), ↑f x ∈ closedBall 0 (max C 1) ⊢ dist (↑f x) (↑f y) ≤ max C 1 + max C 1 ** exact (dist_triangle (f x) 0 (f y)).trans
(add_le_add (mem_closedBall.mp <| this x) (mem_closedBall'.mp <| this y)) ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F K : Set α hK₁ : IsCompact K hK₂ : Kᶜ ⊆ ↑f ⁻¹' closedBall 0 1 C : ℝ hC : ↑f '' K ⊆ closedBall 0 C x y : α ⊢ ∀ (x : α), ↑f x ∈ closedBall 0 (max C 1) ** intro x ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F K : Set α hK₁ : IsCompact K hK₂ : Kᶜ ⊆ ↑f ⁻¹' closedBall 0 1 C : ℝ hC : ↑f '' K ⊆ closedBall 0 C x✝ y x : α ⊢ ↑f x ∈ closedBall 0 (max C 1) ** by_cases hx : x ∈ K ** case pos F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F K : Set α hK₁ : IsCompact K hK₂ : Kᶜ ⊆ ↑f ⁻¹' closedBall 0 1 C : ℝ hC : ↑f '' K ⊆ closedBall 0 C x✝ y x : α hx : x ∈ K ⊢ ↑f x ∈ closedBall 0 (max C 1) ** exact (mem_closedBall.mp <| hC ⟨x, hx, rfl⟩).trans (le_max_left _ _) ** case neg F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f : F K : Set α hK₁ : IsCompact K hK₂ : Kᶜ ⊆ ↑f ⁻¹' closedBall 0 1 C : ℝ hC : ↑f '' K ⊆ closedBall 0 C x✝ y x : α hx : ¬x ∈ K ⊢ ↑f x ∈ closedBall 0 (max C 1) ** exact (mem_closedBall.mp <| mem_preimage.mp (hK₂ hx)).trans (le_max_right _ _) ** Qed
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ZeroAtInftyContinuousMap.toBcf_injective ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f g : α →C₀ β h : toBcf f = toBcf g ⊢ f = g ** ext x ** case h F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β f g : α →C₀ β h : toBcf f = toBcf g x : α ⊢ ↑f x = ↑g x ** simpa only using FunLike.congr_fun h x ** Qed
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ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly ** F✝ : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F✝ α β C : ℝ f✝ g : α →C₀ β ι : Type u_2 F : ι → α →C₀ β f : α →C₀ β l : Filter ι ⊢ Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => ↑(F i)) (↑f) l ** simpa only [Metric.tendsto_nhds] using
@BoundedContinuousFunction.tendsto_iff_tendstoUniformly _ _ _ _ _ (fun i => (F i).toBcf)
f.toBcf l ** Qed
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ZeroAtInftyContinuousMap.isometry_toBcf ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f g : α →C₀ β ⊢ Isometry toBcf ** tauto ** Qed
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ZeroAtInftyContinuousMap.closed_range_toBcf ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f g : α →C₀ β ⊢ IsClosed (range toBcf) ** refine' isClosed_iff_clusterPt.mpr fun f hf => _ ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g : α →C₀ β f : α →ᵇ β hf : ClusterPt f (𝓟 (range toBcf)) ⊢ f ∈ range toBcf ** rw [clusterPt_principal_iff] at hf ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g : α →C₀ β f : α →ᵇ β hf : ∀ (U : Set (α →ᵇ β)), U ∈ 𝓝 f → Set.Nonempty (U ∩ range toBcf) ⊢ f ∈ range toBcf ** have : Tendsto f (cocompact α) (𝓝 0) := by
refine' Metric.tendsto_nhds.mpr fun ε hε => _
obtain ⟨_, hg, g, rfl⟩ := hf (ball f (ε / 2)) (ball_mem_nhds f <| half_pos hε)
refine' (Metric.tendsto_nhds.mp (zero_at_infty g) (ε / 2) (half_pos hε)).mp
(eventually_of_forall fun x hx => _)
calc
dist (f x) 0 ≤ dist (g.toBcf x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
_ < dist g.toBcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
_ < ε := by simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2) ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g : α →C₀ β f : α →ᵇ β hf : ∀ (U : Set (α →ᵇ β)), U ∈ 𝓝 f → Set.Nonempty (U ∩ range toBcf) this : Tendsto (↑f) (cocompact α) (𝓝 0) ⊢ f ∈ range toBcf ** exact ⟨⟨f.toContinuousMap, this⟩, rfl⟩ ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g : α →C₀ β f : α →ᵇ β hf : ∀ (U : Set (α →ᵇ β)), U ∈ 𝓝 f → Set.Nonempty (U ∩ range toBcf) ⊢ Tendsto (↑f) (cocompact α) (𝓝 0) ** refine' Metric.tendsto_nhds.mpr fun ε hε => _ ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g : α →C₀ β f : α →ᵇ β hf : ∀ (U : Set (α →ᵇ β)), U ∈ 𝓝 f → Set.Nonempty (U ∩ range toBcf) ε : ℝ hε : ε > 0 ⊢ ∀ᶠ (x : α) in cocompact α, dist (↑f x) 0 < ε ** obtain ⟨_, hg, g, rfl⟩ := hf (ball f (ε / 2)) (ball_mem_nhds f <| half_pos hε) ** case intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g✝ : α →C₀ β f : α →ᵇ β hf : ∀ (U : Set (α →ᵇ β)), U ∈ 𝓝 f → Set.Nonempty (U ∩ range toBcf) ε : ℝ hε : ε > 0 g : α →C₀ β hg : toBcf g ∈ ball f (ε / 2) ⊢ ∀ᶠ (x : α) in cocompact α, dist (↑f x) 0 < ε ** refine' (Metric.tendsto_nhds.mp (zero_at_infty g) (ε / 2) (half_pos hε)).mp
(eventually_of_forall fun x hx => _) ** case intro.intro.intro F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g✝ : α →C₀ β f : α →ᵇ β hf : ∀ (U : Set (α →ᵇ β)), U ∈ 𝓝 f → Set.Nonempty (U ∩ range toBcf) ε : ℝ hε : ε > 0 g : α →C₀ β hg : toBcf g ∈ ball f (ε / 2) x : α hx : dist (↑g x) 0 < ε / 2 ⊢ dist (↑f x) 0 < ε ** calc
dist (f x) 0 ≤ dist (g.toBcf x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
_ < dist g.toBcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
_ < ε := by simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2) ** F : Type u_1 α : Type u β : Type v γ : Type w inst✝³ : TopologicalSpace α inst✝² : MetricSpace β inst✝¹ : Zero β inst✝ : ZeroAtInftyContinuousMapClass F α β C : ℝ f✝ g✝ : α →C₀ β f : α →ᵇ β hf : ∀ (U : Set (α →ᵇ β)), U ∈ 𝓝 f → Set.Nonempty (U ∩ range toBcf) ε : ℝ hε : ε > 0 g : α →C₀ β hg : toBcf g ∈ ball f (ε / 2) x : α hx : dist (↑g x) 0 < ε / 2 ⊢ dist (toBcf g) f + ε / 2 < ε ** simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2) ** Qed
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exists_clopen_upper_or_lower_of_ne ** α : Type u_1 inst✝² : TopologicalSpace α inst✝¹ : PartialOrder α inst✝ : PriestleySpace α x y : α h : x ≠ y ⊢ ∃ U, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ ¬y ∈ U ** obtain h | h := h.not_le_or_not_le ** case inl α : Type u_1 inst✝² : TopologicalSpace α inst✝¹ : PartialOrder α inst✝ : PriestleySpace α x y : α h✝ : x ≠ y h : ¬x ≤ y ⊢ ∃ U, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ ¬y ∈ U ** exact (exists_clopen_upper_of_not_le h).imp fun _ ↦ And.imp_right <| And.imp_left Or.inl ** case inr α : Type u_1 inst✝² : TopologicalSpace α inst✝¹ : PartialOrder α inst✝ : PriestleySpace α x y : α h✝ : x ≠ y h : ¬y ≤ x ⊢ ∃ U, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ ¬y ∈ U ** obtain ⟨U, hU, hU', hy, hx⟩ := exists_clopen_lower_of_not_le h ** case inr.intro.intro.intro.intro α : Type u_1 inst✝² : TopologicalSpace α inst✝¹ : PartialOrder α inst✝ : PriestleySpace α x y : α h✝ : x ≠ y h : ¬y ≤ x U : Set α hU : IsClopen U hU' : IsLowerSet U hy : ¬y ∈ U hx : x ∈ U ⊢ ∃ U, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ ¬y ∈ U ** exact ⟨U, hU, Or.inr hU', hx, hy⟩ ** Qed
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TopologicalSpace.Opens.coe_iSup ** ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Sort u_5 s : ι → Opens α ⊢ ↑(⨆ i, s i) = ⋃ i, ↑(s i) ** simp [iSup] ** Qed
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TopologicalSpace.Opens.mem_iSup ** ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Sort u_5 x : α s : ι → Opens α ⊢ x ∈ iSup s ↔ ∃ i, x ∈ s i ** rw [← SetLike.mem_coe] ** ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Sort u_5 x : α s : ι → Opens α ⊢ x ∈ ↑(iSup s) ↔ ∃ i, x ∈ s i ** simp ** Qed
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TopologicalSpace.Opens.mem_sSup ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ Us : Set (Opens α) x : α ⊢ x ∈ sSup Us ↔ ∃ u, u ∈ Us ∧ x ∈ u ** simp_rw [sSup_eq_iSup, mem_iSup, exists_prop] ** Qed
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TopologicalSpace.Opens.openEmbedding_of_le ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ U V : Opens α i : U ≤ V ⊢ IsOpen (range (inclusion (_ : ↑U ⊆ ↑V))) ** rw [Set.range_inclusion i] ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ U V : Opens α i : U ≤ V ⊢ IsOpen {x | ↑x ∈ ↑U} ** exact U.isOpen.preimage continuous_subtype_val ** Qed
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TopologicalSpace.Opens.not_nonempty_iff_eq_bot ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ U : Opens α ⊢ ¬Set.Nonempty ↑U ↔ U = ⊥ ** rw [← coe_inj, coe_bot, ← Set.not_nonempty_iff_eq_empty] ** Qed
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TopologicalSpace.Opens.ne_bot_iff_nonempty ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ U : Opens α ⊢ U ≠ ⊥ ↔ Set.Nonempty ↑U ** rw [Ne.def, ← not_nonempty_iff_eq_bot, not_not] ** Qed
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TopologicalSpace.Opens.eq_bot_or_top ** ι : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α✝ inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ α : Type u_5 t : TopologicalSpace α h : t = ⊤ U : Opens α ⊢ U = ⊥ ∨ U = ⊤ ** subst h ** ι : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α✝ inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ α : Type u_5 U : Opens α ⊢ U = ⊥ ∨ U = ⊤ ** letI : TopologicalSpace α := ⊤ ** ι : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α✝ inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ α : Type u_5 U : Opens α this : TopologicalSpace α := ⊤ ⊢ U = ⊥ ∨ U = ⊤ ** rw [← coe_eq_empty, ← coe_eq_univ, ← isOpen_top_iff] ** ι : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α✝ inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ α : Type u_5 U : Opens α this : TopologicalSpace α := ⊤ ⊢ IsOpen ↑U ** exact U.2 ** Qed
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TopologicalSpace.Opens.isBasis_iff_nbhd ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) ⊢ IsBasis B ↔ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U ** constructor <;> intro h ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : IsBasis B ⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U ** rintro ⟨sU, hU⟩ x hx ** case mp.mk ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : IsBasis B sU : Set α hU : IsOpen sU x : α hx : x ∈ { carrier := sU, is_open' := hU } ⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ { carrier := sU, is_open' := hU } ** rcases h.mem_nhds_iff.mp (IsOpen.mem_nhds hU hx) with ⟨sV, ⟨⟨V, H₁, H₂⟩, hsV⟩⟩ ** case mp.mk.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : IsBasis B sU : Set α hU : IsOpen sU x : α hx : x ∈ { carrier := sU, is_open' := hU } sV : Set α hsV : x ∈ sV ∧ sV ⊆ sU V : Opens α H₁ : V ∈ B H₂ : ↑V = sV ⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ { carrier := sU, is_open' := hU } ** refine' ⟨V, H₁, _⟩ ** case mp.mk.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : IsBasis B sU : Set α hU : IsOpen sU x : α hx : x ∈ { carrier := sU, is_open' := hU } sV : Set α hsV : x ∈ sV ∧ sV ⊆ sU V : Opens α H₁ : V ∈ B H₂ : ↑V = sV ⊢ x ∈ V ∧ V ≤ { carrier := sU, is_open' := hU } ** cases V ** case mp.mk.intro.intro.intro.intro.mk ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : IsBasis B sU : Set α hU : IsOpen sU x : α hx : x ∈ { carrier := sU, is_open' := hU } sV : Set α hsV : x ∈ sV ∧ sV ⊆ sU carrier✝ : Set α is_open'✝ : IsOpen carrier✝ H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B H₂ : ↑{ carrier := carrier✝, is_open' := is_open'✝ } = sV ⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧ { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU } ** dsimp at H₂ ** case mp.mk.intro.intro.intro.intro.mk ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : IsBasis B sU : Set α hU : IsOpen sU x : α hx : x ∈ { carrier := sU, is_open' := hU } sV : Set α hsV : x ∈ sV ∧ sV ⊆ sU carrier✝ : Set α is_open'✝ : IsOpen carrier✝ H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B H₂ : carrier✝ = sV ⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧ { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU } ** subst H₂ ** case mp.mk.intro.intro.intro.intro.mk ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : IsBasis B sU : Set α hU : IsOpen sU x : α hx : x ∈ { carrier := sU, is_open' := hU } carrier✝ : Set α is_open'✝ : IsOpen carrier✝ H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B hsV : x ∈ carrier✝ ∧ carrier✝ ⊆ sU ⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧ { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU } ** exact hsV ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U ⊢ IsBasis B ** refine' isTopologicalBasis_of_open_of_nhds _ _ ** case mpr.refine'_1 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U ⊢ ∀ (u : Set α), u ∈ SetLike.coe '' B → IsOpen u ** rintro sU ⟨U, -, rfl⟩ ** case mpr.refine'_1.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U U : Opens α ⊢ IsOpen ↑U ** exact U.2 ** case mpr.refine'_2 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U ⊢ ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v, v ∈ SetLike.coe '' B ∧ a ∈ v ∧ v ⊆ u ** intro x sU hx hsU ** case mpr.refine'_2 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U x : α sU : Set α hx : x ∈ sU hsU : IsOpen sU ⊢ ∃ v, v ∈ SetLike.coe '' B ∧ x ∈ v ∧ v ⊆ sU ** rcases @h ⟨sU, hsU⟩ x hx with ⟨V, hV, H⟩ ** case mpr.refine'_2.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U x : α sU : Set α hx : x ∈ sU hsU : IsOpen sU V : Opens α hV : V ∈ B H : x ∈ V ∧ V ≤ { carrier := sU, is_open' := hsU } ⊢ ∃ v, v ∈ SetLike.coe '' B ∧ x ∈ v ∧ v ⊆ sU ** exact ⟨V, ⟨V, hV, rfl⟩, H⟩ ** Qed
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TopologicalSpace.Opens.isBasis_iff_cover ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) ⊢ IsBasis B ↔ ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us ** constructor ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) ⊢ IsBasis B → ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us ** intro hB U ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) hB : IsBasis B U : Opens α ⊢ ∃ Us, Us ⊆ B ∧ U = sSup Us ** refine ⟨{ V : Opens α | V ∈ B ∧ V ≤ U }, fun U hU => hU.left, ext ?_⟩ ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) hB : IsBasis B U : Opens α ⊢ ↑U = ↑(sSup {V | V ∈ B ∧ V ≤ U}) ** rw [coe_sSup, hB.open_eq_sUnion' U.isOpen] ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) hB : IsBasis B U : Opens α ⊢ ⋃₀ {s | s ∈ SetLike.coe '' B ∧ s ⊆ ↑U} = ⋃ i ∈ {V | V ∈ B ∧ V ≤ U}, ↑i ** simp_rw [sUnion_eq_biUnion, iUnion, mem_setOf_eq, iSup_and, iSup_image] ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) hB : IsBasis B U : Opens α ⊢ ⨆ b ∈ B, ⨆ (_ : ↑b ⊆ ↑U), ↑b = ⨆ i ∈ B, ⨆ (_ : i ≤ U), ↑i ** rfl ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) ⊢ (∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us) → IsBasis B ** intro h ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us ⊢ IsBasis B ** rw [isBasis_iff_nbhd] ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us ⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U ** intro U x hx ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us U : Opens α x : α hx : x ∈ U ⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U ** rcases h U with ⟨Us, hUs, rfl⟩ ** case mpr.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us x : α Us : Set (Opens α) hUs : Us ⊆ B hx : x ∈ sSup Us ⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ sSup Us ** rcases mem_sSup.1 hx with ⟨U, Us, xU⟩ ** case mpr.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ B : Set (Opens α) h : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us x : α Us✝ : Set (Opens α) hUs : Us✝ ⊆ B hx : x ∈ sSup Us✝ U : Opens α Us : U ∈ Us✝ xU : x ∈ U ⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ sSup Us✝ ** exact ⟨U, hUs Us, xU, le_sSup Us⟩ ** Qed
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TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion ** ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_5 b : ι → Opens α hb : IsBasis (range b) hb' : ∀ (i : ι), IsCompact ↑(b i) U : Set α ⊢ IsCompact U ∧ IsOpen U ↔ ∃ s, Set.Finite s ∧ U = ⋃ i ∈ s, ↑(b i) ** apply isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis fun i : ι => (b i).1 ** case hb ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_5 b : ι → Opens α hb : IsBasis (range b) hb' : ∀ (i : ι), IsCompact ↑(b i) U : Set α ⊢ IsTopologicalBasis (range fun i => (b i).carrier) ** convert (config := {transparency := .default}) hb ** case h.e'_3 ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_5 b : ι → Opens α hb : IsBasis (range b) hb' : ∀ (i : ι), IsCompact ↑(b i) U : Set α ⊢ (range fun i => (b i).carrier) = SetLike.coe '' range b ** ext ** case h.e'_3.h ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_5 b : ι → Opens α hb : IsBasis (range b) hb' : ∀ (i : ι), IsCompact ↑(b i) U x✝ : Set α ⊢ (x✝ ∈ range fun i => (b i).carrier) ↔ x✝ ∈ SetLike.coe '' range b ** simp ** case hb' ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_5 b : ι → Opens α hb : IsBasis (range b) hb' : ∀ (i : ι), IsCompact ↑(b i) U : Set α ⊢ ∀ (i : ι), IsCompact (b i).carrier ** exact hb' ** Qed
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TopologicalSpace.Opens.isCompactElement_iff ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α ⊢ CompleteLattice.IsCompactElement s ↔ IsCompact ↑s ** rw [isCompact_iff_finite_subcover, CompleteLattice.isCompactElement_iff] ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α ⊢ (∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ Finset.sup t s_1) ↔ ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ** refine' ⟨_, fun H ι U hU => _⟩ ** case refine'_1 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α ⊢ (∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ Finset.sup t s_1) → ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ** introv H hU hU' ** case refine'_1 ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ Finset.sup t s_1 ι : Type u_2 U : ι → Set α hU : ∀ (i : ι), IsOpen (U i) hU' : ↑s ⊆ ⋃ i, U i ⊢ ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ** obtain ⟨t, ht⟩ := H ι (fun i => ⟨U i, hU i⟩) (by simpa) ** case refine'_1.intro ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ Finset.sup t s_1 ι : Type u_2 U : ι → Set α hU : ∀ (i : ι), IsOpen (U i) hU' : ↑s ⊆ ⋃ i, U i t : Finset ι ht : s ≤ Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) } ⊢ ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ** refine' ⟨t, Set.Subset.trans ht _⟩ ** case refine'_1.intro ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ Finset.sup t s_1 ι : Type u_2 U : ι → Set α hU : ∀ (i : ι), IsOpen (U i) hU' : ↑s ⊆ ⋃ i, U i t : Finset ι ht : s ≤ Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) } ⊢ ↑(Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) }) ⊆ ⋃ i ∈ t, U i ** rw [coe_finset_sup, Finset.sup_eq_iSup] ** case refine'_1.intro ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ Finset.sup t s_1 ι : Type u_2 U : ι → Set α hU : ∀ (i : ι), IsOpen (U i) hU' : ↑s ⊆ ⋃ i, U i t : Finset ι ht : s ≤ Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) } ⊢ ⨆ a ∈ t, (SetLike.coe ∘ fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) }) a ⊆ ⋃ i ∈ t, U i ** rfl ** ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ Finset.sup t s_1 ι : Type u_2 U : ι → Set α hU : ∀ (i : ι), IsOpen (U i) hU' : ↑s ⊆ ⋃ i, U i ⊢ s ≤ ⨆ i, { carrier := U i, is_open' := (_ : IsOpen (U i)) } ** simpa ** case refine'_2 ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ι : Type u_2 U : ι → Opens α hU : s ≤ iSup U ⊢ ∃ t, s ≤ Finset.sup t U ** obtain ⟨t, ht⟩ :=
H (fun i => U i) (fun i => (U i).isOpen) (by simpa using show (s : Set α) ⊆ ↑(iSup U) from hU) ** case refine'_2.intro ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ι : Type u_2 U : ι → Opens α hU : s ≤ iSup U t : Finset ι ht : ↑s ⊆ ⋃ i ∈ t, ↑(U i) ⊢ ∃ t, s ≤ Finset.sup t U ** refine' ⟨t, Set.Subset.trans ht _⟩ ** case refine'_2.intro ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ι : Type u_2 U : ι → Opens α hU : s ≤ iSup U t : Finset ι ht : ↑s ⊆ ⋃ i ∈ t, ↑(U i) ⊢ ⋃ i ∈ t, ↑(U i) ⊆ ↑(Finset.sup t U) ** simp only [Set.iUnion_subset_iff] ** case refine'_2.intro ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ι : Type u_2 U : ι → Opens α hU : s ≤ iSup U t : Finset ι ht : ↑s ⊆ ⋃ i ∈ t, ↑(U i) ⊢ ∀ (i : ι), i ∈ t → ↑(U i) ⊆ ↑(Finset.sup t U) ** show ∀ i ∈ t, U i ≤ t.sup U ** case refine'_2.intro ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ι : Type u_2 U : ι → Opens α hU : s ≤ iSup U t : Finset ι ht : ↑s ⊆ ⋃ i ∈ t, ↑(U i) ⊢ ∀ (i : ι), i ∈ t → U i ≤ Finset.sup t U ** exact fun i => Finset.le_sup ** ι✝ : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ s : Opens α H : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i ι : Type u_2 U : ι → Opens α hU : s ≤ iSup U ⊢ ↑s ⊆ ⋃ i, (fun i => ↑(U i)) i ** simpa using show (s : Set α) ⊆ ↑(iSup U) from hU ** Qed
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TopologicalSpace.eq_induced_by_maps_to_sierpinski ** X : Type u_1 t : TopologicalSpace X ⊢ t = ⨅ u, induced (fun x => x ∈ u) sierpinskiSpace ** apply le_antisymm ** case a X : Type u_1 t : TopologicalSpace X ⊢ t ≤ ⨅ u, induced (fun x => x ∈ u) sierpinskiSpace ** rw [le_iInf_iff] ** case a X : Type u_1 t : TopologicalSpace X ⊢ ∀ (i : Opens X), t ≤ induced (fun x => x ∈ i) sierpinskiSpace ** exact fun u => Continuous.le_induced (isOpen_iff_continuous_mem.mp u.2) ** case a X : Type u_1 t : TopologicalSpace X ⊢ ⨅ u, induced (fun x => x ∈ u) sierpinskiSpace ≤ t ** intro u h ** case a X : Type u_1 t : TopologicalSpace X u : Set X h : IsOpen u ⊢ IsOpen u ** apply isOpen_generateFrom_of_mem ** case a.hs X : Type u_1 t : TopologicalSpace X u : Set X h : IsOpen u ⊢ u ∈ ⋃ i, {s | IsOpen s} ** simp only [Set.mem_iUnion, Set.mem_setOf_eq, isOpen_induced_iff] ** case a.hs X : Type u_1 t : TopologicalSpace X u : Set X h : IsOpen u ⊢ ∃ i t_1, IsOpen t_1 ∧ (fun x => x ∈ i) ⁻¹' t_1 = u ** exact ⟨⟨u, h⟩, {True}, isOpen_singleton_true, by simp [Set.preimage]⟩ ** X : Type u_1 t : TopologicalSpace X u : Set X h : IsOpen u ⊢ (fun x => x ∈ { carrier := u, is_open' := h }) ⁻¹' {True} = u ** simp [Set.preimage] ** Qed
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TopologicalSpace.productOfMemOpens_inducing ** X : Type u_1 inst✝ : TopologicalSpace X ⊢ Inducing ↑(productOfMemOpens X) ** convert inducing_iInf_to_pi fun (u : Opens X) (x : X) => x ∈ u ** case h.e'_3 X : Type u_1 inst✝ : TopologicalSpace X ⊢ inst✝ = ⨅ i, induced (fun x => x ∈ i) inferInstance ** apply eq_induced_by_maps_to_sierpinski ** Qed
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TopologicalSpace.productOfMemOpens_injective ** X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : T0Space X ⊢ Function.Injective ↑(productOfMemOpens X) ** intro x1 x2 h ** X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : T0Space X x1 x2 : X h : ↑(productOfMemOpens X) x1 = ↑(productOfMemOpens X) x2 ⊢ x1 = x2 ** apply Inseparable.eq ** case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : T0Space X x1 x2 : X h : ↑(productOfMemOpens X) x1 = ↑(productOfMemOpens X) x2 ⊢ Inseparable x1 x2 ** rw [← Inducing.inseparable_iff (productOfMemOpens_inducing X), h] ** Qed
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ContinuousMap.nullhomotopic_of_constant ** X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : TopologicalSpace Z y : Y ⊢ Homotopic (const X y) (const X y) ** rfl ** Qed
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ContinuousMap.Nullhomotopic.comp_right ** X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : TopologicalSpace Z f : C(X, Y) hf : Nullhomotopic f g : C(Y, Z) ⊢ Nullhomotopic (comp g f) ** cases' hf with y hy ** case intro X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : TopologicalSpace Z f : C(X, Y) g : C(Y, Z) y : Y hy : Homotopic f (const X y) ⊢ Nullhomotopic (comp g f) ** use g y ** case h X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : TopologicalSpace Z f : C(X, Y) g : C(Y, Z) y : Y hy : Homotopic f (const X y) ⊢ Homotopic (comp g f) (const X (↑g y)) ** exact Homotopic.hcomp hy (Homotopic.refl g) ** Qed
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ContinuousMap.Nullhomotopic.comp_left ** X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : TopologicalSpace Z f : C(Y, Z) hf : Nullhomotopic f g : C(X, Y) ⊢ Nullhomotopic (comp f g) ** cases' hf with y hy ** case intro X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : TopologicalSpace Z f : C(Y, Z) g : C(X, Y) y : Z hy : Homotopic f (const Y y) ⊢ Nullhomotopic (comp f g) ** use y ** case h X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝² : TopologicalSpace X inst✝¹ : TopologicalSpace Y inst✝ : TopologicalSpace Z f : C(Y, Z) g : C(X, Y) y : Z hy : Homotopic f (const Y y) ⊢ Homotopic (comp f g) (const X y) ** exact Homotopic.hcomp (Homotopic.refl g) hy ** Qed
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id_nullhomotopic ** X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : ContractibleSpace X ⊢ Nullhomotopic (ContinuousMap.id X) ** obtain ⟨hv⟩ := ContractibleSpace.hequiv_unit X ** case intro X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : ContractibleSpace X hv : X ≃ₕ Unit ⊢ Nullhomotopic (ContinuousMap.id X) ** use hv.invFun () ** case h X : Type u_1 inst✝¹ : TopologicalSpace X inst✝ : ContractibleSpace X hv : X ≃ₕ Unit ⊢ Homotopic (ContinuousMap.id X) (const X (↑hv.invFun ())) ** convert hv.left_inv.symm ** Qed
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contractible_iff_id_nullhomotopic ** Y : Type u_1 inst✝ : TopologicalSpace Y ⊢ ContractibleSpace Y ↔ Nullhomotopic (ContinuousMap.id Y) ** constructor ** case mpr Y : Type u_1 inst✝ : TopologicalSpace Y ⊢ Nullhomotopic (ContinuousMap.id Y) → ContractibleSpace Y ** rintro ⟨p, h⟩ ** case mpr.intro Y : Type u_1 inst✝ : TopologicalSpace Y p : Y h : Homotopic (ContinuousMap.id Y) (const Y p) ⊢ ContractibleSpace Y ** refine
{ hequiv_unit' :=
⟨{ toFun := ContinuousMap.const _ ()
invFun := ContinuousMap.const _ p
left_inv := ?_
right_inv := ?_ }⟩ } ** case mp Y : Type u_1 inst✝ : TopologicalSpace Y ⊢ ContractibleSpace Y → Nullhomotopic (ContinuousMap.id Y) ** intro ** case mp Y : Type u_1 inst✝ : TopologicalSpace Y a✝ : ContractibleSpace Y ⊢ Nullhomotopic (ContinuousMap.id Y) ** apply id_nullhomotopic ** case mpr.intro.refine_1 Y : Type u_1 inst✝ : TopologicalSpace Y p : Y h : Homotopic (ContinuousMap.id Y) (const Y p) ⊢ Homotopic (comp (const Unit p) (const Y ())) (ContinuousMap.id Y) ** exact h.symm ** case mpr.intro.refine_2 Y : Type u_1 inst✝ : TopologicalSpace Y p : Y h : Homotopic (ContinuousMap.id Y) (const Y p) ⊢ Homotopic (comp (const Y ()) (const Unit p)) (ContinuousMap.id Unit) ** convert Homotopic.refl (ContinuousMap.id Unit) ** Qed
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ContractibleSpace.hequiv ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : ContractibleSpace X inst✝ : ContractibleSpace Y ⊢ Nonempty (X ≃ₕ Y) ** rcases ContractibleSpace.hequiv_unit' (X := X) with ⟨h⟩ ** case intro X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : ContractibleSpace X inst✝ : ContractibleSpace Y h : X ≃ₕ Unit ⊢ Nonempty (X ≃ₕ Y) ** rcases ContractibleSpace.hequiv_unit' (X := Y) with ⟨h'⟩ ** case intro.intro X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X inst✝² : TopologicalSpace Y inst✝¹ : ContractibleSpace X inst✝ : ContractibleSpace Y h : X ≃ₕ Unit h' : Y ≃ₕ Unit ⊢ Nonempty (X ≃ₕ Y) ** exact ⟨h.trans h'.symm⟩ ** Qed
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ContinuousMap.prod_apply ** α : Type u_1 β : Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : CommMonoid β inst✝ : ContinuousMul β ι : Type u_3 s : Finset ι f : ι → C(α, β) a : α ⊢ ↑(∏ i in s, f i) a = ∏ i in s, ↑(f i) a ** simp ** Qed
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ContinuousMap.hasSum_apply ** α : Type u_1 β : Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β γ : Type u_3 inst✝¹ : AddCommMonoid β inst✝ : ContinuousAdd β f : γ → C(α, β) g : C(α, β) hf : HasSum f g x : α ⊢ HasSum (fun i => ↑(f i) x) (↑g x) ** let ev : C(α, β) →+ β := (Pi.evalAddMonoidHom _ x).comp coeFnAddMonoidHom ** α : Type u_1 β : Type u_2 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β γ : Type u_3 inst✝¹ : AddCommMonoid β inst✝ : ContinuousAdd β f : γ → C(α, β) g : C(α, β) hf : HasSum f g x : α ev : C(α, β) →+ β := AddMonoidHom.comp (Pi.evalAddMonoidHom (fun a => β) x) coeFnAddMonoidHom ⊢ HasSum (fun i => ↑(f i) x) (↑g x) ** exact hf.map ev (ContinuousMap.continuous_eval_const x) ** Qed
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Subalgebra.separatesPoints_monotone ** α : Type u_1 inst✝⁹ : TopologicalSpace α R : Type u_2 inst✝⁸ : CommSemiring R A : Type u_3 inst✝⁷ : TopologicalSpace A inst✝⁶ : Semiring A inst✝⁵ : Algebra R A inst✝⁴ : TopologicalSemiring A A₂ : Type u_4 inst✝³ : TopologicalSpace A₂ inst✝² : Semiring A₂ inst✝¹ : Algebra R A₂ inst✝ : TopologicalSemiring A₂ s s' : Subalgebra R C(α, A) r : s ≤ s' h : (fun s => SeparatesPoints s) s x y : α n : x ≠ y ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑s' ∧ f x ≠ f y ** obtain ⟨f, m, w⟩ := h n ** case intro.intro α : Type u_1 inst✝⁹ : TopologicalSpace α R : Type u_2 inst✝⁸ : CommSemiring R A : Type u_3 inst✝⁷ : TopologicalSpace A inst✝⁶ : Semiring A inst✝⁵ : Algebra R A inst✝⁴ : TopologicalSemiring A A₂ : Type u_4 inst✝³ : TopologicalSpace A₂ inst✝² : Semiring A₂ inst✝¹ : Algebra R A₂ inst✝ : TopologicalSemiring A₂ s s' : Subalgebra R C(α, A) r : s ≤ s' h : (fun s => SeparatesPoints s) s x y : α n : x ≠ y f : α → A m : f ∈ (fun f => ↑f) '' ↑s w : f x ≠ f y ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑s' ∧ f x ≠ f y ** rcases m with ⟨f, ⟨m, rfl⟩⟩ ** case intro.intro.intro.intro α : Type u_1 inst✝⁹ : TopologicalSpace α R : Type u_2 inst✝⁸ : CommSemiring R A : Type u_3 inst✝⁷ : TopologicalSpace A inst✝⁶ : Semiring A inst✝⁵ : Algebra R A inst✝⁴ : TopologicalSemiring A A₂ : Type u_4 inst✝³ : TopologicalSpace A₂ inst✝² : Semiring A₂ inst✝¹ : Algebra R A₂ inst✝ : TopologicalSemiring A₂ s s' : Subalgebra R C(α, A) r : s ≤ s' h : (fun s => SeparatesPoints s) s x y : α n : x ≠ y f : C(α, A) m : f ∈ ↑s w : (fun f => ↑f) f x ≠ (fun f => ↑f) f y ⊢ ∃ f, f ∈ (fun f => ↑f) '' ↑s' ∧ f x ≠ f y ** exact ⟨_, ⟨f, ⟨r m, rfl⟩⟩, w⟩ ** Qed
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algebraMap_apply ** α : Type u_1 inst✝⁹ : TopologicalSpace α R : Type u_2 inst✝⁸ : CommSemiring R A : Type u_3 inst✝⁷ : TopologicalSpace A inst✝⁶ : Semiring A inst✝⁵ : Algebra R A inst✝⁴ : TopologicalSemiring A A₂ : Type u_4 inst✝³ : TopologicalSpace A₂ inst✝² : Semiring A₂ inst✝¹ : Algebra R A₂ inst✝ : TopologicalSemiring A₂ k : R a : α ⊢ ↑(↑(algebraMap R C(α, A)) k) a = k • 1 ** rw [Algebra.algebraMap_eq_smul_one] ** α : Type u_1 inst✝⁹ : TopologicalSpace α R : Type u_2 inst✝⁸ : CommSemiring R A : Type u_3 inst✝⁷ : TopologicalSpace A inst✝⁶ : Semiring A inst✝⁵ : Algebra R A inst✝⁴ : TopologicalSemiring A A₂ : Type u_4 inst✝³ : TopologicalSpace A₂ inst✝² : Semiring A₂ inst✝¹ : Algebra R A₂ inst✝ : TopologicalSemiring A₂ k : R a : α ⊢ ↑(k • 1) a = k • 1 ** rfl ** Qed
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Subalgebra.SeparatesPoints.strongly ** α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** by_cases n : x = y ** case neg α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** obtain ⟨_, ⟨f, hf, rfl⟩, hxy⟩ := h n ** case neg.intro.intro.intro.intro α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f✝ : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y f : C(α, 𝕜) hf : f ∈ ↑s hxy : (fun f => ↑f) f x ≠ (fun f => ↑f) f y ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** replace hxy : f x - f y ≠ 0 := sub_ne_zero_of_ne hxy ** case neg.intro.intro.intro.intro α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f✝ : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y f : C(α, 𝕜) hf : f ∈ ↑s hxy : ↑f x - ↑f y ≠ 0 ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** let a := v x ** case neg.intro.intro.intro.intro α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f✝ : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y f : C(α, 𝕜) hf : f ∈ ↑s hxy : ↑f x - ↑f y ≠ 0 a : 𝕜 := v x ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** let b := v y ** case neg.intro.intro.intro.intro α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f✝ : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y f : C(α, 𝕜) hf : f ∈ ↑s hxy : ↑f x - ↑f y ≠ 0 a : 𝕜 := v x b : 𝕜 := v y ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** let f' : s :=
((b - a) * (f x - f y)⁻¹) • (algebraMap _ s (f x) - (⟨f, hf⟩ : s)) + algebraMap _ s a ** case neg.intro.intro.intro.intro α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f✝ : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y f : C(α, 𝕜) hf : f ∈ ↑s hxy : ↑f x - ↑f y ≠ 0 a : 𝕜 := v x b : 𝕜 := v y f' : { x // x ∈ s } := ((b - a) * (↑f x - ↑f y)⁻¹) • (↑(algebraMap ((fun x => 𝕜) x) { x // x ∈ s }) (↑f x) - { val := f, property := hf }) + ↑(algebraMap 𝕜 { x // x ∈ s }) a ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** refine' ⟨f', f'.prop, _, _⟩ ** case pos α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : x = y ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f y = v y ** subst n ** case pos α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x : α ⊢ ∃ f, f ∈ ↑s ∧ ↑f x = v x ∧ ↑f x = v x ** refine' ⟨_, (v x • (1 : s) : s).prop, mul_one _, mul_one _⟩ ** case neg.intro.intro.intro.intro.refine'_1 α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f✝ : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y f : C(α, 𝕜) hf : f ∈ ↑s hxy : ↑f x - ↑f y ≠ 0 a : 𝕜 := v x b : 𝕜 := v y f' : { x // x ∈ s } := ((b - a) * (↑f x - ↑f y)⁻¹) • (↑(algebraMap ((fun x => 𝕜) x) { x // x ∈ s }) (↑f x) - { val := f, property := hf }) + ↑(algebraMap 𝕜 { x // x ∈ s }) a ⊢ ↑↑f' x = v x ** simp ** case neg.intro.intro.intro.intro.refine'_2 α : Type u_1 inst✝¹² : TopologicalSpace α R : Type u_2 inst✝¹¹ : CommSemiring R A : Type u_3 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : Algebra R A inst✝⁷ : TopologicalSemiring A A₂ : Type u_4 inst✝⁶ : TopologicalSpace A₂ inst✝⁵ : Semiring A₂ inst✝⁴ : Algebra R A₂ inst✝³ : TopologicalSemiring A₂ 𝕜 : Type u_5 inst✝² : TopologicalSpace 𝕜 s✝ : Set C(α, 𝕜) f✝ : ↑s✝ x✝ : α inst✝¹ : Field 𝕜 inst✝ : TopologicalRing 𝕜 s : Subalgebra 𝕜 C(α, 𝕜) h : SeparatesPoints s v : α → 𝕜 x y : α n : ¬x = y f : C(α, 𝕜) hf : f ∈ ↑s hxy : ↑f x - ↑f y ≠ 0 a : 𝕜 := v x b : 𝕜 := v y f' : { x // x ∈ s } := ((b - a) * (↑f x - ↑f y)⁻¹) • (↑(algebraMap ((fun x => 𝕜) x) { x // x ∈ s }) (↑f x) - { val := f, property := hf }) + ↑(algebraMap 𝕜 { x // x ∈ s }) a ⊢ ↑↑f' y = v y ** simp [inv_mul_cancel_right₀ hxy] ** Qed
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ContinuousMap.periodic_tsum_comp_add_zsmul ** X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p : X ⊢ Function.Periodic (↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p)))) p ** intro x ** X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p x : X ⊢ ↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p))) (x + p) = ↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p))) x ** by_cases h : Summable fun n : ℤ => f.comp (ContinuousMap.addRight (n • p)) ** case pos X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p x : X h : Summable fun n => comp f (ContinuousMap.addRight (n • p)) ⊢ ↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p))) (x + p) = ↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p))) x ** convert congr_arg (fun f : C(X, Y) => f x) ((Equiv.addRight (1 : ℤ)).tsum_eq _) using 1 ** case h.e'_2 X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p x : X h : Summable fun n => comp f (ContinuousMap.addRight (n • p)) ⊢ ↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p))) (x + p) = ↑(∑' (c : ℤ), comp f (ContinuousMap.addRight (↑(Equiv.addRight 1) c • p))) x ** simp_rw [← tsum_apply h] ** case h.e'_2 X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p x : X h : Summable fun n => comp f (ContinuousMap.addRight (n • p)) ⊢ ∑' (i : ℤ), ↑(comp f (ContinuousMap.addRight (i • p))) (x + p) = ↑(∑' (c : ℤ), comp f (ContinuousMap.addRight (↑(Equiv.addRight 1) c • p))) x ** erw [← tsum_apply ((Equiv.addRight (1 : ℤ)).summable_iff.mpr h)] ** case h.e'_2 X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p x : X h : Summable fun n => comp f (ContinuousMap.addRight (n • p)) ⊢ ∑' (i : ℤ), ↑(comp f (ContinuousMap.addRight (i • p))) (x + p) = ∑' (i : ℤ), ↑(((fun n => comp f (ContinuousMap.addRight (n • p))) ∘ ↑(Equiv.addRight 1)) i) x ** simp [coe_addRight, add_one_zsmul, add_comm (_ • p) p, ← add_assoc] ** case neg X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p x : X h : ¬Summable fun n => comp f (ContinuousMap.addRight (n • p)) ⊢ ↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p))) (x + p) = ↑(∑' (n : ℤ), comp f (ContinuousMap.addRight (n • p))) x ** rw [tsum_eq_zero_of_not_summable h] ** case neg X : Type u_1 Y : Type u_2 Z : Type u_3 inst✝¹⁴ : TopologicalSpace X inst✝¹³ : TopologicalSpace Y inst✝¹² : TopologicalSpace Z 𝕜 : Type u_4 inst✝¹¹ : CommSemiring 𝕜 A : Type u_5 inst✝¹⁰ : TopologicalSpace A inst✝⁹ : Semiring A inst✝⁸ : TopologicalSemiring A inst✝⁷ : StarRing A inst✝⁶ : ContinuousStar A inst✝⁵ : Algebra 𝕜 A inst✝⁴ : AddCommGroup X inst✝³ : TopologicalAddGroup X inst✝² : AddCommMonoid Y inst✝¹ : ContinuousAdd Y inst✝ : T2Space Y f : C(X, Y) p x : X h : ¬Summable fun n => comp f (ContinuousMap.addRight (n • p)) ⊢ ↑0 (x + p) = ↑0 x ** simp only [coe_zero, Pi.zero_apply] ** Qed
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Pretrivialization.mem_source ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x : Z ⊢ x ∈ e.source ↔ proj x ∈ e.baseSet ** rw [e.source_eq, mem_preimage] ** Qed
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Pretrivialization.mem_target ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z x : B × F ⊢ x ∈ e.target ↔ x.1 ∈ e.baseSet ** rw [e.target_eq, prod_univ, mem_preimage] ** Qed
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Pretrivialization.proj_symm_apply ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z x : B × F hx : x ∈ e.target ⊢ proj (↑(LocalEquiv.symm e.toLocalEquiv) x) = x.1 ** have := (e.coe_fst (e.map_target hx)).symm ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z x : B × F hx : x ∈ e.target this : proj (↑(LocalEquiv.symm e.toLocalEquiv) x) = (↑e (↑(LocalEquiv.symm e.toLocalEquiv) x)).1 ⊢ proj (↑(LocalEquiv.symm e.toLocalEquiv) x) = x.1 ** rwa [← e.coe_coe, e.right_inv hx] at this ** Qed
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Pretrivialization.symm_apply_mk_proj ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ x : Z ex : x ∈ e.source ⊢ ↑(LocalEquiv.symm e.toLocalEquiv) (proj x, (↑e x).2) = x ** rw [← e.coe_fst ex, ← e.coe_coe, e.left_inv ex] ** Qed
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Pretrivialization.preimage_symm_proj_baseSet ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x : Z ⊢ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' e.baseSet) ∩ e.target = e.target ** refine' inter_eq_right.mpr fun x hx => _ ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z x : B × F hx : x ∈ e.target ⊢ x ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' e.baseSet) ** simp only [mem_preimage, LocalEquiv.invFun_as_coe, e.proj_symm_apply hx] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z x : B × F hx : x ∈ e.target ⊢ x.1 ∈ e.baseSet ** exact e.mem_target.mp hx ** Qed
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Pretrivialization.preimage_symm_proj_inter ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x : Z s : Set B ⊢ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ = (s ∩ e.baseSet) ×ˢ univ ** ext ⟨x, y⟩ ** case h.mk ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z s : Set B x : B y : F ⊢ (x, y) ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ ↔ (x, y) ∈ (s ∩ e.baseSet) ×ˢ univ ** suffices x ∈ e.baseSet → (proj (e.toLocalEquiv.symm (x, y)) ∈ s ↔ x ∈ s) by
simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff] ** case h.mk ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z s : Set B x : B y : F ⊢ x ∈ e.baseSet → (proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s) ** intro h ** case h.mk ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z s : Set B x : B y : F h : x ∈ e.baseSet ⊢ proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s ** rw [e.proj_symm_apply' h] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e : Pretrivialization F proj x✝ : Z s : Set B x : B y : F this : x ∈ e.baseSet → (proj (↑(LocalEquiv.symm e.toLocalEquiv) (x, y)) ∈ s ↔ x ∈ s) ⊢ (x, y) ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ ↔ (x, y) ∈ (s ∩ e.baseSet) ×ˢ univ ** simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff] ** Qed
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Pretrivialization.target_inter_preimage_symm_source_eq ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e f : Pretrivialization F proj ⊢ f.target ∩ ↑(LocalEquiv.symm f.toLocalEquiv) ⁻¹' e.source = (e.baseSet ∩ f.baseSet) ×ˢ univ ** rw [inter_comm, f.target_eq, e.source_eq, f.preimage_symm_proj_inter] ** Qed
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Pretrivialization.trans_source ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e f : Pretrivialization F proj ⊢ (LocalEquiv.trans (LocalEquiv.symm f.toLocalEquiv) e.toLocalEquiv).source = (e.baseSet ∩ f.baseSet) ×ˢ univ ** rw [LocalEquiv.trans_source, LocalEquiv.symm_source, e.target_inter_preimage_symm_source_eq] ** Qed
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Pretrivialization.symm_trans_symm ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e e' : Pretrivialization F proj ⊢ LocalEquiv.symm (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv) = LocalEquiv.trans (LocalEquiv.symm e'.toLocalEquiv) e.toLocalEquiv ** rw [LocalEquiv.trans_symm_eq_symm_trans_symm, LocalEquiv.symm_symm] ** Qed
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Pretrivialization.symm_trans_source_eq ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e e' : Pretrivialization F proj ⊢ (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ univ ** rw [LocalEquiv.trans_source, e'.source_eq, LocalEquiv.symm_source, e.target_eq, inter_comm,
e.preimage_symm_proj_inter, inter_comm] ** Qed
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Pretrivialization.symm_trans_target_eq ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝¹ : TopologicalSpace B inst✝ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e e' : Pretrivialization F proj ⊢ (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ ** rw [← LocalEquiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm] ** Qed
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Pretrivialization.mk_symm ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e' : Pretrivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y✝ : E b✝ inst✝ : (x : B) → Zero (E x) e : Pretrivialization F TotalSpace.proj b : B hb : b ∈ e.baseSet y : F ⊢ { proj := b, snd := Pretrivialization.symm e b y } = ↑(LocalEquiv.symm e.toLocalEquiv) (b, y) ** simp only [e.symm_apply hb, TotalSpace.mk_cast (e.proj_symm_apply' hb), TotalSpace.eta] ** Qed
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Pretrivialization.symm_proj_apply ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e' : Pretrivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b inst✝ : (x : B) → Zero (E x) e : Pretrivialization F TotalSpace.proj z : TotalSpace F E hz : z.proj ∈ e.baseSet ⊢ Pretrivialization.symm e z.proj (↑e z).2 = z.snd ** rw [e.symm_apply hz, cast_eq_iff_heq, e.mk_proj_snd' hz, e.symm_apply_apply (e.mem_source.mpr hz)] ** Qed
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Pretrivialization.apply_mk_symm ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝² : TopologicalSpace B inst✝¹ : TopologicalSpace F proj : Z → B e✝ : Pretrivialization F proj x : Z e' : Pretrivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y✝ : E b✝ inst✝ : (x : B) → Zero (E x) e : Pretrivialization F TotalSpace.proj b : B hb : b ∈ e.baseSet y : F ⊢ ↑e { proj := b, snd := Pretrivialization.symm e b y } = (b, y) ** rw [e.mk_symm hb, e.apply_symm_apply (e.mk_mem_target.mpr hb)] ** Qed
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Trivialization.toPretrivialization_injective ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e e' : Trivialization F proj h : (fun e => toPretrivialization e) e = (fun e => toPretrivialization e) e' ⊢ e = e' ** ext1 ** case h₁ ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e e' : Trivialization F proj h : (fun e => toPretrivialization e) e = (fun e => toPretrivialization e) e' ⊢ e.toLocalHomeomorph = e'.toLocalHomeomorph case h₂ ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e e' : Trivialization F proj h : (fun e => toPretrivialization e) e = (fun e => toPretrivialization e) e' ⊢ e.baseSet = e'.baseSet ** exacts [LocalHomeomorph.toLocalEquiv_injective (congr_arg Pretrivialization.toLocalEquiv h),
congr_arg Pretrivialization.baseSet h] ** Qed
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Trivialization.mem_source ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z ⊢ x ∈ e.source ↔ proj x ∈ e.baseSet ** rw [e.source_eq, mem_preimage] ** Qed
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Trivialization.map_proj_nhds ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z ex : x ∈ e.source ⊢ map proj (𝓝 x) = 𝓝 (proj x) ** rw [← e.coe_fst ex, ← map_congr (e.coe_fst_eventuallyEq_proj ex), ← map_map, ← e.coe_coe,
e.map_nhds_eq ex, map_fst_nhds] ** Qed
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Trivialization.tendsto_nhds_iff ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ e.source ⊢ Tendsto f l (𝓝 z) ↔ Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2) ** rw [e.nhds_eq_comap_inf_principal hz, tendsto_inf, tendsto_comap_iff, Prod.tendsto_iff, coe_coe,
tendsto_principal, coe_fst _ hz] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ e.source ⊢ ((Tendsto (fun n => ((↑e ∘ f) n).1) l (𝓝 (proj z)) ∧ Tendsto (fun n => ((↑e ∘ f) n).2) l (𝓝 (↑e z).2)) ∧ ∀ᶠ (a : α) in l, f a ∈ e.source) ↔ Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2) ** by_cases hl : ∀ᶠ x in l, f x ∈ e.source ** case pos ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ e.source hl : ∀ᶠ (x : α) in l, f x ∈ e.source ⊢ ((Tendsto (fun n => ((↑e ∘ f) n).1) l (𝓝 (proj z)) ∧ Tendsto (fun n => ((↑e ∘ f) n).2) l (𝓝 (↑e z).2)) ∧ ∀ᶠ (a : α) in l, f a ∈ e.source) ↔ Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2) ** simp only [hl, and_true] ** case pos ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ e.source hl : ∀ᶠ (x : α) in l, f x ∈ e.source ⊢ Tendsto (fun n => ((↑e ∘ f) n).1) l (𝓝 (proj z)) ∧ Tendsto (fun n => ((↑e ∘ f) n).2) l (𝓝 (↑e z).2) ↔ Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2) ** refine (tendsto_congr' ?_).and Iff.rfl ** case pos ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ e.source hl : ∀ᶠ (x : α) in l, f x ∈ e.source ⊢ (fun n => ((↑e ∘ f) n).1) =ᶠ[l] proj ∘ f ** exact hl.mono fun x ↦ e.coe_fst ** case neg ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ e.source hl : ¬∀ᶠ (x : α) in l, f x ∈ e.source ⊢ ((Tendsto (fun n => ((↑e ∘ f) n).1) l (𝓝 (proj z)) ∧ Tendsto (fun n => ((↑e ∘ f) n).2) l (𝓝 (↑e z).2)) ∧ ∀ᶠ (a : α) in l, f a ∈ e.source) ↔ Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2) ** simp only [hl, and_false, false_iff, not_and] ** case neg ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ e.source hl : ¬∀ᶠ (x : α) in l, f x ∈ e.source ⊢ Tendsto (proj ∘ f) l (𝓝 (proj z)) → ¬Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2) ** rw [e.source_eq] at hl hz ** case neg ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z α : Type u_6 l : Filter α f : α → Z z : Z hz : z ∈ proj ⁻¹' e.baseSet hl : ¬∀ᶠ (x : α) in l, f x ∈ proj ⁻¹' e.baseSet ⊢ Tendsto (proj ∘ f) l (𝓝 (proj z)) → ¬Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2) ** exact fun h _ ↦ hl <| h <| e.open_baseSet.mem_nhds hz ** Qed
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Trivialization.nhds_eq_inf_comap ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x z : Z hz : z ∈ e.source ⊢ 𝓝 z = comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ ↑e) (𝓝 (↑e z).2) ** refine eq_of_forall_le_iff fun l ↦ ?_ ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x z : Z hz : z ∈ e.source l : Filter Z ⊢ l ≤ 𝓝 z ↔ l ≤ comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ ↑e) (𝓝 (↑e z).2) ** rw [le_inf_iff, ← tendsto_iff_comap, ← tendsto_iff_comap] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x z : Z hz : z ∈ e.source l : Filter Z ⊢ l ≤ 𝓝 z ↔ Tendsto proj l (𝓝 (proj z)) ∧ Tendsto (Prod.snd ∘ ↑e) l (𝓝 (↑e z).2) ** exact e.tendsto_nhds_iff hz ** Qed
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Trivialization.preimageSingletonHomeomorph_symm_apply ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z b : B hb : b ∈ e.baseSet p : F ⊢ ↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, p) ∈ proj ⁻¹' {b} ** rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff] ** Qed
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Trivialization.continuousAt_of_comp_right ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z X : Type u_6 inst✝ : TopologicalSpace X f : Z → X z : Z e : Trivialization F proj he : proj z ∈ e.baseSet hf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z) ⊢ ContinuousAt f z ** have hez : z ∈ e.toLocalEquiv.symm.target := by
rw [LocalEquiv.symm_target, e.mem_source]
exact he ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z X : Type u_6 inst✝ : TopologicalSpace X f : Z → X z : Z e : Trivialization F proj he : proj z ∈ e.baseSet hf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z) hez : z ∈ (LocalEquiv.symm e.toLocalEquiv).target ⊢ ContinuousAt f z ** rwa [e.toLocalHomeomorph.symm.continuousAt_iff_continuousAt_comp_right hez,
LocalHomeomorph.symm_symm] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z X : Type u_6 inst✝ : TopologicalSpace X f : Z → X z : Z e : Trivialization F proj he : proj z ∈ e.baseSet hf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z) ⊢ z ∈ (LocalEquiv.symm e.toLocalEquiv).target ** rw [LocalEquiv.symm_target, e.mem_source] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z X : Type u_6 inst✝ : TopologicalSpace X f : Z → X z : Z e : Trivialization F proj he : proj z ∈ e.baseSet hf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z) ⊢ proj z ∈ e.baseSet ** exact he ** Qed
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Trivialization.continuousAt_of_comp_left ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x✝ : Z X : Type u_6 inst✝ : TopologicalSpace X f : X → Z x : X e : Trivialization F proj hf_proj : ContinuousAt (proj ∘ f) x he : proj (f x) ∈ e.baseSet hf : ContinuousAt (↑e ∘ f) x ⊢ ContinuousAt f x ** rw [e.continuousAt_iff_continuousAt_comp_left] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x✝ : Z X : Type u_6 inst✝ : TopologicalSpace X f : X → Z x : X e : Trivialization F proj hf_proj : ContinuousAt (proj ∘ f) x he : proj (f x) ∈ e.baseSet hf : ContinuousAt (↑e ∘ f) x ⊢ f ⁻¹' e.source ∈ 𝓝 x ** rw [e.source_eq, ← preimage_comp] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x✝ : Z X : Type u_6 inst✝ : TopologicalSpace X f : X → Z x : X e : Trivialization F proj hf_proj : ContinuousAt (proj ∘ f) x he : proj (f x) ∈ e.baseSet hf : ContinuousAt (↑e ∘ f) x ⊢ proj ∘ f ⁻¹' e.baseSet ∈ 𝓝 x ** exact hf_proj.preimage_mem_nhds (e.open_baseSet.mem_nhds he) ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x✝ : Z X : Type u_6 inst✝ : TopologicalSpace X f : X → Z x : X e : Trivialization F proj hf_proj : ContinuousAt (proj ∘ f) x he : proj (f x) ∈ e.baseSet hf : ContinuousAt (↑e ∘ f) x ⊢ ContinuousAt (↑e.toLocalHomeomorph ∘ f) x ** exact hf ** Qed
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Trivialization.continuousOn_symm ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b inst✝ : (x : B) → Zero (E x) e : Trivialization F TotalSpace.proj ⊢ ContinuousOn (fun z => TotalSpace.mk' F z.1 (Trivialization.symm e z.1 z.2)) (e.baseSet ×ˢ univ) ** have : ∀ z ∈ e.baseSet ×ˢ (univ : Set F),
TotalSpace.mk z.1 (e.symm z.1 z.2) = e.toLocalHomeomorph.symm z := by
rintro x ⟨hx : x.1 ∈ e.baseSet, _⟩
rw [e.mk_symm hx] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b inst✝ : (x : B) → Zero (E x) e : Trivialization F TotalSpace.proj this : ∀ (z : B × F), z ∈ e.baseSet ×ˢ univ → { proj := z.1, snd := Trivialization.symm e z.1 z.2 } = ↑(LocalHomeomorph.symm e.toLocalHomeomorph) z ⊢ ContinuousOn (fun z => TotalSpace.mk' F z.1 (Trivialization.symm e z.1 z.2)) (e.baseSet ×ˢ univ) ** refine' ContinuousOn.congr _ this ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b inst✝ : (x : B) → Zero (E x) e : Trivialization F TotalSpace.proj this : ∀ (z : B × F), z ∈ e.baseSet ×ˢ univ → { proj := z.1, snd := Trivialization.symm e z.1 z.2 } = ↑(LocalHomeomorph.symm e.toLocalHomeomorph) z ⊢ ContinuousOn (fun x => ↑(LocalHomeomorph.symm e.toLocalHomeomorph) x) (e.baseSet ×ˢ univ) ** rw [← e.target_eq] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b inst✝ : (x : B) → Zero (E x) e : Trivialization F TotalSpace.proj this : ∀ (z : B × F), z ∈ e.baseSet ×ˢ univ → { proj := z.1, snd := Trivialization.symm e z.1 z.2 } = ↑(LocalHomeomorph.symm e.toLocalHomeomorph) z ⊢ ContinuousOn (fun x => ↑(LocalHomeomorph.symm e.toLocalHomeomorph) x) e.target ** exact e.toLocalHomeomorph.continuousOn_symm ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b inst✝ : (x : B) → Zero (E x) e : Trivialization F TotalSpace.proj ⊢ ∀ (z : B × F), z ∈ e.baseSet ×ˢ univ → { proj := z.1, snd := Trivialization.symm e z.1 z.2 } = ↑(LocalHomeomorph.symm e.toLocalHomeomorph) z ** rintro x ⟨hx : x.1 ∈ e.baseSet, _⟩ ** case intro ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b inst✝ : (x : B) → Zero (E x) e : Trivialization F TotalSpace.proj x : B × F hx : x.1 ∈ e.baseSet right✝ : x.2 ∈ univ ⊢ { proj := x.1, snd := Trivialization.symm e x.1 x.2 } = ↑(LocalHomeomorph.symm e.toLocalHomeomorph) x ** rw [e.mk_symm hx] ** Qed
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Trivialization.mk_coordChange ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet x : F ⊢ (b, coordChange e₁ e₂ b x) = ↑e₂ (↑(LocalHomeomorph.symm e₁.toLocalHomeomorph) (b, x)) ** refine' Prod.ext _ rfl ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet x : F ⊢ (b, coordChange e₁ e₂ b x).1 = (↑e₂ (↑(LocalHomeomorph.symm e₁.toLocalHomeomorph) (b, x))).1 ** rw [e₂.coe_fst', ← e₁.coe_fst', e₁.apply_symm_apply' h₁] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet x : F ⊢ proj (↑(LocalHomeomorph.symm e₁.toLocalHomeomorph) (b, x)) ∈ e₁.baseSet ** rwa [e₁.proj_symm_apply' h₁] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet x : F ⊢ proj (↑(LocalHomeomorph.symm e₁.toLocalHomeomorph) (b, x)) ∈ e₂.baseSet ** rwa [e₁.proj_symm_apply' h₁] ** Qed
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Trivialization.coordChange_apply_snd ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b e₁ e₂ : Trivialization F proj p : Z h : proj p ∈ e₁.baseSet ⊢ coordChange e₁ e₂ (proj p) (↑e₁ p).2 = (↑e₂ p).2 ** rw [coordChange, e₁.symm_apply_mk_proj (e₁.mem_source.2 h)] ** Qed
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Trivialization.coordChange_same_apply ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e : Trivialization F proj b : B h : b ∈ e.baseSet x : F ⊢ coordChange e e b x = x ** rw [coordChange, e.apply_symm_apply' h] ** Qed
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Trivialization.coordChange_coordChange ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ e₃ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet x : F ⊢ coordChange e₂ e₃ b (coordChange e₁ e₂ b x) = coordChange e₁ e₃ b x ** rw [coordChange, e₁.mk_coordChange _ h₁ h₂, ← e₂.coe_coe, e₂.left_inv, coordChange] ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ e₃ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet x : F ⊢ ↑(LocalHomeomorph.symm e₁.toLocalHomeomorph) (b, x) ∈ e₂.source ** rwa [e₂.mem_source, e₁.proj_symm_apply' h₁] ** Qed
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Trivialization.continuous_coordChange ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet ⊢ Continuous (coordChange e₁ e₂ b) ** refine' continuous_snd.comp (e₂.toLocalHomeomorph.continuousOn.comp_continuous
(e₁.toLocalHomeomorph.continuousOn_symm.comp_continuous _ _) _) ** case refine'_1 ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet ⊢ Continuous fun x => (b, x) ** exact continuous_const.prod_mk continuous_id ** case refine'_2 ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet ⊢ ∀ (x : F), (b, x) ∈ e₁.target ** exact fun x => e₁.mem_target.2 h₁ ** case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet ⊢ ∀ (x : F), ↑(LocalHomeomorph.symm e₁.toLocalHomeomorph) (b, x) ∈ e₂.source ** intro x ** case refine'_3 ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace F proj : Z → B inst✝¹ : TopologicalSpace Z inst✝ : TopologicalSpace (TotalSpace F E) e : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b✝ : B y : E b✝ e₁ e₂ : Trivialization F proj b : B h₁ : b ∈ e₁.baseSet h₂ : b ∈ e₂.baseSet x : F ⊢ ↑(LocalHomeomorph.symm e₁.toLocalHomeomorph) (b, x) ∈ e₂.source ** rwa [e₂.mem_source, e₁.proj_symm_apply' h₁] ** Qed
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Trivialization.isImage_preimage_prod ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x✝ : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b B' : Type u_6 inst✝ : TopologicalSpace B' e : Trivialization F proj s : Set B x : Z hx : x ∈ e.source ⊢ ↑e.toLocalHomeomorph x ∈ s ×ˢ univ ↔ x ∈ proj ⁻¹' s ** simp [e.coe_fst', hx] ** Qed
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Trivialization.frontier_preimage ** ι : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 Z : Type u_5 inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace F proj : Z → B inst✝² : TopologicalSpace Z inst✝¹ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F proj x : Z e' : Trivialization F TotalSpace.proj x' : TotalSpace F E b : B y : E b B' : Type u_6 inst✝ : TopologicalSpace B' e : Trivialization F proj s : Set B ⊢ e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s) ** rw [← (e.isImage_preimage_prod s).frontier.preimage_eq, frontier_prod_univ_eq,
(e.isImage_preimage_prod _).preimage_eq, e.source_eq, preimage_inter] ** Qed
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ContinuousMap.uniformInducing_equivBoundedOfCompact ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : MetricSpace β inst✝ : NormedAddCommGroup E ⊢ ∀ (s : Set (C(α, β) × C(α, β))), s ∈ uniformity C(α, β) ↔ ∃ t, t ∈ uniformity (α →ᵇ β) ∧ ∀ (x y : C(α, β)), (↑(equivBoundedOfCompact α β) x, ↑(equivBoundedOfCompact α β) y) ∈ t → (x, y) ∈ s ** simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff] ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : MetricSpace β inst✝ : NormedAddCommGroup E ⊢ ∀ (s : Set (C(α, β) × C(α, β))), (∃ i, (IsCompact i.1 ∧ ∃ i_1, 0 < i_1 ∧ {p | dist p.1 p.2 ≤ i_1} ⊆ i.2) ∧ {fg | ∀ (x : α), x ∈ i.1 → (↑fg.1 x, ↑fg.2 x) ∈ i.2} ⊆ s) ↔ ∃ t, (∃ i, 0 < i ∧ {p | dist p.1 p.2 ≤ i} ⊆ t) ∧ ∀ (x y : C(α, β)), (↑(equivBoundedOfCompact α β) x, ↑(equivBoundedOfCompact α β) y) ∈ t → (x, y) ∈ s ** exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩ ** Qed
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ContinuousMap.dist_apply_le_dist ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : MetricSpace β inst✝ : NormedAddCommGroup E f g : C(α, β) C : ℝ x : α ⊢ dist (↑f x) (↑g x) ≤ dist f g ** simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply] ** Qed
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ContinuousMap.dist_le ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : MetricSpace β inst✝ : NormedAddCommGroup E f g : C(α, β) C : ℝ C0 : 0 ≤ C ⊢ dist f g ≤ C ↔ ∀ (x : α), dist (↑f x) (↑g x) ≤ C ** simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply] ** Qed
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ContinuousMap.dist_le_iff_of_nonempty ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝⁴ : TopologicalSpace α inst✝³ : CompactSpace α inst✝² : MetricSpace β inst✝¹ : NormedAddCommGroup E f g : C(α, β) C : ℝ inst✝ : Nonempty α ⊢ dist f g ≤ C ↔ ∀ (x : α), dist (↑f x) (↑g x) ≤ C ** simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply] ** Qed
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ContinuousMap.dist_lt_iff_of_nonempty ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝⁴ : TopologicalSpace α inst✝³ : CompactSpace α inst✝² : MetricSpace β inst✝¹ : NormedAddCommGroup E f g : C(α, β) C : ℝ inst✝ : Nonempty α ⊢ dist f g < C ↔ ∀ (x : α), dist (↑f x) (↑g x) < C ** simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply] ** Qed
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ContinuousMap.dist_lt_iff ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : MetricSpace β inst✝ : NormedAddCommGroup E f g : C(α, β) C : ℝ C0 : 0 < C ⊢ dist f g < C ↔ ∀ (x : α), dist (↑f x) (↑g x) < C ** rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0] ** α : Type u_1 β : Type u_2 E : Type u_3 inst✝³ : TopologicalSpace α inst✝² : CompactSpace α inst✝¹ : MetricSpace β inst✝ : NormedAddCommGroup E f g : C(α, β) C : ℝ C0 : 0 < C ⊢ (∀ (x : α), dist (↑(mkOfCompact f) x) (↑(mkOfCompact g) x) < C) ↔ ∀ (x : α), dist (↑f x) (↑g x) < C ** simp only [mkOfCompact_apply] ** Qed
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ContinuousLinearMap.toLinear_compLeftContinuousCompact ** X : Type u_1 𝕜 : Type u_2 β : Type u_3 γ : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : CompactSpace X inst✝⁴ : NontriviallyNormedField 𝕜 inst✝³ : NormedAddCommGroup β inst✝² : NormedSpace 𝕜 β inst✝¹ : NormedAddCommGroup γ inst✝ : NormedSpace 𝕜 γ g : β →L[𝕜] γ ⊢ ↑(ContinuousLinearMap.compLeftContinuousCompact X g) = ContinuousLinearMap.compLeftContinuous 𝕜 X g ** ext f ** case h.h X : Type u_1 𝕜 : Type u_2 β : Type u_3 γ : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : CompactSpace X inst✝⁴ : NontriviallyNormedField 𝕜 inst✝³ : NormedAddCommGroup β inst✝² : NormedSpace 𝕜 β inst✝¹ : NormedAddCommGroup γ inst✝ : NormedSpace 𝕜 γ g : β →L[𝕜] γ f : C(X, β) a✝ : X ⊢ ↑(↑↑(ContinuousLinearMap.compLeftContinuousCompact X g) f) a✝ = ↑(↑(ContinuousLinearMap.compLeftContinuous 𝕜 X g) f) a✝ ** rfl ** Qed
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ContinuousMap.summable_of_locally_summable_norm ** X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ ⊢ Summable F ** refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _ ** X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Set X hK : IsCompact K ⊢ ∃ f, Tendsto (fun i => restrict K (∑ b in i, F b)) atTop (𝓝 f) ** lift K to Compacts X using hK ** case intro X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X ⊢ ∃ f, Tendsto (fun i => restrict (↑K) (∑ b in i, F b)) atTop (𝓝 f) ** have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp
congr! ** case intro X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X A : ∀ (s : Finset ι), restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict (↑K) (F i) ⊢ ∃ f, Tendsto (fun i => restrict (↑K) (∑ b in i, F b)) atTop (𝓝 f) ** simpa only [HasSum, A] using summable_of_summable_norm (hF K) ** X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X ⊢ ∀ (s : Finset ι), restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict (↑K) (F i) ** intro s ** X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X s : Finset ι ⊢ restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict (↑K) (F i) ** ext1 x ** case h X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X s : Finset ι x : ↑↑K ⊢ ↑(restrict (↑K) (∑ i in s, F i)) x = ↑(∑ i in s, restrict (↑K) (F i)) x ** simp ** case h X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X s : Finset ι x : ↑↑K ⊢ ↑(restrict (↑K) (∑ i in s, F i)) x = ∑ c in s, ↑(restrict (↑K) (F c)) x ** erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply] ** case h X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X s : Finset ι x : ↑↑K ⊢ ↑(∑ i in s, F i) ↑x = ∑ c in s, ↑(restrict (↑K) (F c)) x ** simp ** case h X : Type u_1 inst✝⁴ : TopologicalSpace X inst✝³ : T2Space X inst✝² : LocallyCompactSpace X E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : CompleteSpace E ι : Type u_3 F : ι → C(X, E) hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖ K : Compacts X s : Finset ι x : ↑↑K ⊢ ∑ c in s, ↑(F c) ↑x = ∑ c in s, ↑(restrict (↑K) (F c)) x ** congr! ** Qed
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unitInterval.continuous_qRight ** ⊢ Continuous fun p => 2 * ↑p.1 ** continuity ** ⊢ Continuous fun p => 1 + ↑p.2 ** continuity ** Qed
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unitInterval.qRight_zero_left ** θ : ↑I ⊢ 2 * ↑(0, θ).1 / (1 + ↑(0, θ).2) ≤ 0 ** simp only [coe_zero, mul_zero, zero_div, le_refl] ** Qed
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unitInterval.qRight_one_left ** θ : ↑I ⊢ 1 * (1 + ↑(1, θ).2) ≤ 2 * ↑(1, θ).1 ** dsimp only ** θ : ↑I ⊢ 1 * (1 + ↑θ) ≤ 2 * ↑1 ** rw [coe_one, one_mul, mul_one, add_comm, ← one_add_one_eq_two] ** θ : ↑I ⊢ ↑θ + 1 ≤ 1 + 1 ** simp only [add_le_add_iff_right] ** θ : ↑I ⊢ ↑θ ≤ 1 ** exact le_one _ ** Qed
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unitInterval.qRight_zero_right ** t : ↑I ⊢ ↑(qRight (t, 0)) = if ↑t ≤ 1 / 2 then 2 * ↑t else 1 ** simp only [qRight, coe_zero, add_zero, div_one] ** t : ↑I ⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = if ↑t ≤ 1 / 2 then 2 * ↑t else 1 ** split_ifs ** case pos t : ↑I h✝ : ↑t ≤ 1 / 2 ⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = 2 * ↑t ** rw [Set.projIcc_of_mem _ ((mul_pos_mem_iff zero_lt_two).2 _)] ** t : ↑I h✝ : ↑t ≤ 1 / 2 ⊢ ↑t ∈ Set.Icc 0 (1 / 2) ** refine' ⟨t.2.1, _⟩ ** t : ↑I h✝ : ↑t ≤ 1 / 2 ⊢ ↑t ≤ 1 / 2 ** tauto ** case neg t : ↑I h✝ : ¬↑t ≤ 1 / 2 ⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = 1 ** rw [(Set.projIcc_eq_right _).2] ** case neg t : ↑I h✝ : ¬↑t ≤ 1 / 2 ⊢ 1 ≤ 2 * ↑t ** linarith ** t : ↑I h✝ : ¬↑t ≤ 1 / 2 ⊢ 0 < 1 ** exact zero_lt_one ** Qed
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unitInterval.qRight_one_right ** t : ↑I ⊢ qRight (t, 1) = Set.projIcc 0 1 (_ : 0 ≤ 1) ↑t ** rw [qRight] ** t : ↑I ⊢ Set.projIcc 0 1 qRight.proof_1 (2 * ↑(t, 1).1 / (1 + ↑(t, 1).2)) = Set.projIcc 0 1 (_ : 0 ≤ 1) ↑t ** congr ** case e_x t : ↑I ⊢ 2 * ↑(t, 1).1 / (1 + ↑(t, 1).2) = ↑t ** norm_num ** case e_x t : ↑I ⊢ 2 * ↑t / 2 = ↑t ** apply mul_div_cancel_left ** case e_x.ha t : ↑I ⊢ 2 ≠ 0 ** exact two_ne_zero ** Qed
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Path.delayReflRight_zero ** X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y ⊢ delayReflRight 0 γ = trans γ (refl y) ** ext t ** case a.h X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I ⊢ ↑(delayReflRight 0 γ) t = ↑(trans γ (refl y)) t ** simp only [delayReflRight, trans_apply, refl_extend, Path.coe_mk_mk, Function.comp_apply,
refl_apply] ** case a.h X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I ⊢ ↑γ (qRight (t, 0)) = if h : ↑t ≤ 1 / 2 then ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } else y ** split_ifs with h ** case pos X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ↑t ≤ 1 / 2 ⊢ ↑γ (qRight (t, 0)) = ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } case neg X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ¬↑t ≤ 1 / 2 ⊢ ↑γ (qRight (t, 0)) = y ** swap ** case neg X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ¬↑t ≤ 1 / 2 ⊢ ↑γ (qRight (t, 0)) = y case pos X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ↑t ≤ 1 / 2 ⊢ ↑γ (qRight (t, 0)) = ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } ** conv_rhs => rw [← γ.target] ** case neg X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ¬↑t ≤ 1 / 2 ⊢ ↑γ (qRight (t, 0)) = ↑γ 1 case pos X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ↑t ≤ 1 / 2 ⊢ ↑γ (qRight (t, 0)) = ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } ** all_goals apply congr_arg γ; ext1; rw [qRight_zero_right] ** case neg.a X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ¬↑t ≤ 1 / 2 ⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑1 case pos.a X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ↑t ≤ 1 / 2 ⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑{ val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } ** exacts [if_neg h, if_pos h] ** case pos X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ↑t ≤ 1 / 2 ⊢ ↑γ (qRight (t, 0)) = ↑γ { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } ** apply congr_arg γ ** case pos X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ↑t ≤ 1 / 2 ⊢ qRight (t, 0) = { val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } ** ext1 ** case pos.a X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I h : ↑t ≤ 1 / 2 ⊢ ↑(qRight (t, 0)) = ↑{ val := 2 * ↑t, property := (_ : 2 * ↑t ∈ I) } ** rw [qRight_zero_right] ** Qed
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Path.delayReflRight_one ** X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y ⊢ delayReflRight 1 γ = γ ** ext t ** case a.h X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y t : ↑I ⊢ ↑(delayReflRight 1 γ) t = ↑γ t ** exact congr_arg γ (qRight_one_right t) ** Qed
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Path.delayReflLeft_zero ** X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y ⊢ delayReflLeft 0 γ = trans (refl x) γ ** simp only [delayReflLeft, delayReflRight_zero, trans_symm, refl_symm, Path.symm_symm] ** Qed
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Path.delayReflLeft_one ** X : Type u inst✝ : TopologicalSpace X x y : X γ : Path x y ⊢ delayReflLeft 1 γ = γ ** simp only [delayReflLeft, delayReflRight_one, Path.symm_symm] ** Qed
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CocompactMap.tendsto_of_forall_preimage ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α → β h : ∀ (s : Set β), IsCompact s → IsCompact (f ⁻¹' s) s : Set β hs : s ∈ cocompact β t : Set β ht : IsCompact t hts : tᶜ ⊆ s ⊢ (f ⁻¹' t)ᶜ ⊆ f ⁻¹' s ** simpa using preimage_mono hts ** Qed
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CocompactMap.isCompact_preimage ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : TopologicalSpace δ inst✝ : T2Space β f : CocompactMap α β s : Set β hs : IsCompact s ⊢ IsCompact (↑f ⁻¹' s) ** obtain ⟨t, ht, hts⟩ :=
mem_cocompact'.mp
(by
simpa only [preimage_image_preimage, preimage_compl] using
mem_map.mp
(cocompact_tendsto f <|
mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr (image_preimage_subset f _)⟩)) ** case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : TopologicalSpace δ inst✝ : T2Space β f : CocompactMap α β s : Set β hs : IsCompact s t : Set α ht : IsCompact t hts : (↑f ⁻¹' s)ᶜᶜ ⊆ t ⊢ IsCompact (↑f ⁻¹' s) ** exact
ht.of_isClosed_subset (hs.isClosed.preimage <| map_continuous f) (by simpa using hts) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : TopologicalSpace δ inst✝ : T2Space β f : CocompactMap α β s : Set β hs : IsCompact s ⊢ ?m.10429 ∈ cocompact ?m.10404 ** simpa only [preimage_image_preimage, preimage_compl] using
mem_map.mp
(cocompact_tendsto f <|
mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr (image_preimage_subset f _)⟩) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁴ : TopologicalSpace α inst✝³ : TopologicalSpace β inst✝² : TopologicalSpace γ inst✝¹ : TopologicalSpace δ inst✝ : T2Space β f : CocompactMap α β s : Set β hs : IsCompact s t : Set α ht : IsCompact t hts : (↑f ⁻¹' s)ᶜᶜ ⊆ t ⊢ ↑f ⁻¹' s ⊆ t ** simpa using hts ** Qed
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SpectralMap.coe_comp_continuousMap ** F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : SpectralMap β γ g : SpectralMap α β ⊢ ↑f ∘ ↑g = ↑↑f ∘ ↑↑g ** rfl ** Qed
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SpectralMap.coe_comp_continuousMap' ** F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : SpectralMap β γ g : SpectralMap α β ⊢ ↑(comp f g) = ContinuousMap.comp ↑f ↑g ** rfl ** Qed
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SpectralMap.cancel_left ** F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 inst✝³ : TopologicalSpace α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : SpectralMap β γ f₁ f₂ : SpectralMap α β hg : Injective ↑g h : comp g f₁ = comp g f₂ a : α ⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a) ** rw [← comp_apply, h, comp_apply] ** Qed
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TopologicalSpace.Compacts.coe_finset_sup ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_4 s : Finset ι f : ι → Compacts α ⊢ ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i) ** refine Finset.cons_induction_on s rfl fun a s _ h => ?_ ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_4 s✝ : Finset ι f : ι → Compacts α a : ι s : Finset ι x✝ : ¬a ∈ s h : ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i) ⊢ ↑(Finset.sup (Finset.cons a s x✝) f) = Finset.sup (Finset.cons a s x✝) fun i => ↑(f i) ** simp_rw [Finset.sup_cons, coe_sup, sup_eq_union] ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : TopologicalSpace α inst✝¹ : TopologicalSpace β inst✝ : TopologicalSpace γ ι : Type u_4 s✝ : Finset ι f : ι → Compacts α a : ι s : Finset ι x✝ : ¬a ∈ s h : ↑(Finset.sup s f) = Finset.sup s fun i => ↑(f i) ⊢ ↑(f a) ∪ ↑(Finset.sup s f) = ↑(f a) ∪ Finset.sup s fun i => ↑(f i) ** congr ** Qed
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ENNReal.embedding_coe ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ OrdConnected (range some) ** rw [range_coe'] ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ OrdConnected (Iio ⊤) ** exact ordConnected_Iio ** Qed
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ENNReal.isOpen_Ico_zero ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ IsOpen (Ico 0 b) ** rw [ENNReal.Ico_eq_Iio] ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ IsOpen (Iio b) ** exact isOpen_Iio ** Qed
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ENNReal.openEmbedding_coe ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ IsOpen (range some) ** rw [range_coe'] ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ IsOpen (Iio ⊤) ** exact isOpen_Iio ** Qed
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ENNReal.tendsto_nhds_coe_iff ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x✝ y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ α : Type u_4 l : Filter α x : ℝ≥0 f : ℝ≥0∞ → α ⊢ Tendsto f (𝓝 ↑x) l ↔ Tendsto (f ∘ some) (𝓝 x) l ** rw [nhds_coe, tendsto_map'_iff] ** Qed
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ENNReal.tendsto_toNNReal ** α : Type u_1 β : Type u_2 γ : Type u_3 a✝ b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ a : ℝ≥0∞ ha : a ≠ ⊤ ⊢ Tendsto ENNReal.toNNReal (𝓝 a) (𝓝 (ENNReal.toNNReal a)) ** lift a to ℝ≥0 using ha ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 a✝ b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ a : ℝ≥0 ⊢ Tendsto ENNReal.toNNReal (𝓝 ↑a) (𝓝 (ENNReal.toNNReal ↑a)) ** rw [nhds_coe, tendsto_map'_iff] ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 a✝ b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ a : ℝ≥0 ⊢ Tendsto (ENNReal.toNNReal ∘ some) (𝓝 a) (𝓝 (ENNReal.toNNReal ↑a)) ** exact tendsto_id ** Qed
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ENNReal.eventuallyEq_of_toReal_eventuallyEq ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ l : Filter α f g : α → ℝ≥0∞ hfi : ∀ᶠ (x : α) in l, f x ≠ ⊤ hgi : ∀ᶠ (x : α) in l, g x ≠ ⊤ hfg : (fun x => ENNReal.toReal (f x)) =ᶠ[l] fun x => ENNReal.toReal (g x) ⊢ f =ᶠ[l] g ** filter_upwards [hfi, hgi, hfg] with _ hfx hgx _ ** case h α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ l : Filter α f g : α → ℝ≥0∞ hfi : ∀ᶠ (x : α) in l, f x ≠ ⊤ hgi : ∀ᶠ (x : α) in l, g x ≠ ⊤ hfg : (fun x => ENNReal.toReal (f x)) =ᶠ[l] fun x => ENNReal.toReal (g x) a✝¹ : α hfx : f a✝¹ ≠ ⊤ hgx : g a✝¹ ≠ ⊤ a✝ : ENNReal.toReal (f a✝¹) = ENNReal.toReal (g a✝¹) ⊢ f a✝¹ = g a✝¹ ** rwa [← ENNReal.toReal_eq_toReal hfx hgx] ** Qed
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ENNReal.nhds_top ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ ⨅ l, ⨅ (_ : l < ⊤), 𝓟 (Ioi l) = ⨅ a, ⨅ (_ : a ≠ ⊤), 𝓟 (Ioi a) ** simp [lt_top_iff_ne_top, Ioi] ** Qed
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ENNReal.tendsto_nhds_top_iff_nnreal ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ m : α → ℝ≥0∞ f : Filter α ⊢ Tendsto m f (𝓝 ⊤) ↔ ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a ** simp only [nhds_top', tendsto_iInf, tendsto_principal, mem_Ioi] ** Qed
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ENNReal.tendsto_nhds_top_iff_nat ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ m : α → ℝ≥0∞ f : Filter α h : ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a n : ℕ ⊢ ∀ᶠ (a : α) in f, ↑n < m a ** simpa only [ENNReal.coe_nat] using h n ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x✝ y✝ z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ m : α → ℝ≥0∞ f : Filter α h : ∀ (n : ℕ), ∀ᶠ (a : α) in f, ↑n < m a x : ℝ≥0 n : ℕ hn : x < ↑n y : α ⊢ ↑x < ↑n ** rwa [← ENNReal.coe_nat, coe_lt_coe] ** Qed
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ENNReal.tendsto_coe_nhds_top ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ f : α → ℝ≥0 l : Filter α ⊢ Tendsto (fun x => ↑(f x)) l (𝓝 ⊤) ↔ Tendsto f l atTop ** rw [tendsto_nhds_top_iff_nnreal, atTop_basis_Ioi.tendsto_right_iff] ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ f : α → ℝ≥0 l : Filter α ⊢ (∀ (x : ℝ≥0), ∀ᶠ (a : α) in l, ↑x < ↑(f a)) ↔ ∀ (i : ℝ≥0), True → ∀ᶠ (x : α) in l, f x ∈ Ioi i ** simp ** Qed
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ENNReal.nhds_zero ** α : Type u_1 β : Type u_2 γ : Type u_3 a b c d : ℝ≥0∞ r p q : ℝ≥0 x y z ε ε₁ ε₂ : ℝ≥0∞ s : Set ℝ≥0∞ ⊢ ⨅ l, ⨅ (_ : ⊥ < l), 𝓟 (Iio l) = ⨅ a, ⨅ (_ : a ≠ 0), 𝓟 (Iio a) ** simp [pos_iff_ne_zero, Iio] ** Qed
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