ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMap.abs_mem_subalgebra_closure
theorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
    |(f : C(X, ℝ))| ∈ A.topologicalClosure 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f : ↥A)
Goal: |↑f| ∈ A.topologicalClosure
-- ContinuousMap.inf_mem_subalgebra_closure
theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
    (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f g : ↥A)
Goal: ↑f ⊓ ↑g ∈ A.topologicalClosure
-- ContinuousMap.inf_mem_closed_subalgebra
theorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
    (f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed ↑A) (f g : ↥A)
Goal: ↑f ⊓ ↑g ∈ A
-- ContinuousMap.sup_mem_subalgebra_closure
theorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
    (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f g : ↥A)
Goal: ↑f ⊔ ↑g ∈ A.topologicalClosure
-- ContinuousMap.sup_mem_closed_subalgebra
theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
    (f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed ↑A) (f g : ↥A)
Goal: ↑f ⊔ ↑g ∈ A
-- ContinuousMap.sublattice_closure_eq_top
theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
    (inf_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊓ g ∈ L)
    (sup_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊔ g ∈ L) (sep : L.SeparatesPointsStrongly) :
    closure L = ⊤ 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (L : Set C(X, ℝ)) (nA : L.Nonempty) (inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L) (sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L) (sep : L.SeparatesPointsStrongly)
Goal: closure L = ⊤
-- ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints
theorem subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
    (w : A.SeparatesPoints) : A.topologicalClosure = ⊤ 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (w : A.SeparatesPoints)
Goal: A.topologicalClosure = ⊤
-- ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints
theorem continuousMap_mem_subalgebra_closure_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
    (w : A.SeparatesPoints) (f : C(X, ℝ)) : f ∈ A.topologicalClosure 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (w : A.SeparatesPoints) (f : C(X, ℝ))
Goal: f ∈ A.topologicalClosure
-- ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints
theorem exists_mem_subalgebra_near_continuousMap_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
    (w : A.SeparatesPoints) (f : C(X, ℝ)) (ε : ℝ) (pos : 0 < ε) :
    ∃ g : A, ‖(g : C(X, ℝ)) - f‖ < ε 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (w : A.SeparatesPoints) (f : C(X, ℝ)) (ε : ℝ) (pos : 0 < ε)
Goal: ∃ g, ‖↑g - f‖ < ε
-- ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints
theorem exists_mem_subalgebra_near_continuous_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
    (w : A.SeparatesPoints) (f : X → ℝ) (c : Continuous f) (ε : ℝ) (pos : 0 < ε) :
    ∃ g : A, ∀ x, ‖(g : X → ℝ) x - f x‖ < ε 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (w : A.SeparatesPoints) (f : X → ℝ) (c : Continuous f) (ε : ℝ) (pos : 0 < ε)
Goal: ∃ g, ∀ (x : X), ‖↑g x - f x‖ < ε
-- Subalgebra.SeparatesPoints.rclike_to_real
theorem Subalgebra.SeparatesPoints.rclike_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
    (hA : A.SeparatesPoints) :
      ((A.restrictScalars ℝ).comap
        (ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints 
Arguments: {𝕜 : Type u_1} {X : Type u_2} [RCLike 𝕜] [TopologicalSpace X] {A : StarSubalgebra 𝕜 C(X, 𝕜)} (hA : A.SeparatesPoints)
Goal: (Subalgebra.comap (AlgHom.compLeftContinuous ℝ RCLike.ofRealAm ⋯)
    (Subalgebra.restrictScalars ℝ A.toSubalgebra)).SeparatesPoints
-- ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
    (A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints) : A.topologicalClosure = ⊤ 
Arguments: {𝕜 : Type u_1} {X : Type u_2} [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] (A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints)
Goal: A.topologicalClosure = ⊤
-- polynomialFunctions.topologicalClosure
theorem polynomialFunctions.topologicalClosure (s : Set ℝ)
    [CompactSpace s] : (polynomialFunctions s).topologicalClosure = ⊤ 
Arguments: (s : Set ℝ) [CompactSpace ↑s]
Goal: (polynomialFunctions s).topologicalClosure = ⊤
-- polynomialFunctions.starClosure_topologicalClosure
theorem polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type*} [RCLike 𝕜] (s : Set 𝕜)
    [CompactSpace s] : (polynomialFunctions s).starClosure.topologicalClosure = ⊤ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] (s : Set 𝕜) [CompactSpace ↑s]
Goal: (polynomialFunctions s).starClosure.topologicalClosure = ⊤
-- ContinuousMap.algHom_ext_map_X
@[ext]
theorem ContinuousMap.algHom_ext_map_X {A : Type*} [Ring A]
    [Algebra ℝ A] [TopologicalSpace A] [T2Space A] {s : Set ℝ} [CompactSpace s]
    {φ ψ : C(s, ℝ) →ₐ[ℝ] A} (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ 
Arguments: {A : Type u_1} [Ring A] [Algebra ℝ A] [TopologicalSpace A] [T2Space A] {s : Set ℝ} [CompactSpace ↑s] {φ ψ : C(↑s, ℝ) →ₐ[ℝ] A} (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h : φ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X))
Goal: φ = ψ
-- ContinuousMap.starAlgHom_ext_map_X
@[ext]
theorem ContinuousMap.starAlgHom_ext_map_X {𝕜 A : Type*} [RCLike 𝕜] [Ring A] [StarRing A]
    [Algebra 𝕜 A] [TopologicalSpace A] [T2Space A] {s : Set 𝕜} [CompactSpace s]
    {φ ψ : C(s, 𝕜) →⋆ₐ[𝕜] A} (hφ : Continuous φ) (hψ : Continuous ψ)
    (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [Ring A] [StarRing A] [Algebra 𝕜 A] [TopologicalSpace A] [T2Space A] {s : Set 𝕜} [CompactSpace ↑s] {φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A} (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ) (h : φ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X))
Goal: φ = ψ
-- ContinuousMap.adjoin_id_eq_span_one_union
lemma adjoin_id_eq_span_one_union (s : Set 𝕜) :
    ((StarAlgebra.adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))}) : Set C(s, 𝕜)) =
      span 𝕜 ({(1 : C(s, 𝕜))} ∪ (adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))})) 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (s : Set 𝕜)
Goal: ↑(StarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)}) =
  ↑(Submodule.span 𝕜 ({1} ∪ ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)})))
-- ContinuousMap.adjoin_id_eq_span_one_add
lemma adjoin_id_eq_span_one_add (s : Set 𝕜) :
    ((StarAlgebra.adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))}) : Set C(s, 𝕜)) =
      (span 𝕜 {(1 : C(s, 𝕜))} : Set C(s, 𝕜)) + (adjoin 𝕜 {(restrict s (.id 𝕜) : C(s, 𝕜))}) 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (s : Set 𝕜)
Goal: ↑(StarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)}) =
  ↑(Submodule.span 𝕜 {1}) + ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)})
-- ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom
lemma nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom {s : Set 𝕜} (h0 : 0 ∈ s) :
    (adjoin 𝕜 {restrict s (.id 𝕜)} : Set C(s, 𝕜)) ⊆
      RingHom.ker (evalStarAlgHom 𝕜 𝕜 (⟨0, h0⟩ : s)) 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] {s : Set 𝕜} (h0 : 0 ∈ s)
Goal: ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)}) ⊆
  ↑(RingHom.ker (ContinuousMap.evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))
-- ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id
lemma ker_evalStarAlgHom_inter_adjoin_id (s : Set 𝕜) (h0 : 0 ∈ s) :
    (StarAlgebra.adjoin 𝕜 {restrict s (.id 𝕜)} : Set C(s, 𝕜)) ∩
      RingHom.ker (evalStarAlgHom 𝕜 𝕜 (⟨0, h0⟩ : s)) = adjoin 𝕜 {restrict s (.id 𝕜)} 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (s : Set 𝕜) (h0 : 0 ∈ s)
Goal: ↑(StarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)}) ∩
    ↑(RingHom.ker (ContinuousMap.evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) =
  ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)})
-- ContinuousMap.AlgHom.closure_ker_inter
theorem AlgHom.closure_ker_inter {F S K A : Type*} [CommRing K] [Ring A] [Algebra K A]
    [TopologicalSpace K] [T1Space K] [TopologicalSpace A] [ContinuousSub A] [ContinuousSMul K A]
    [FunLike F A K] [AlgHomClass F K A K] [SetLike S A] [OneMemClass S A] [AddSubgroupClass S A]
    [SMulMemClass S K A] (φ : F) (hφ : Continuous φ) (s : S) :
    closure (s ∩ RingHom.ker φ) = closure s ∩ (ker φ : Set A) 
Arguments: {F : Type u_3} {S : Type u_4} {K : Type u_5} {A : Type u_6} [CommRing K] [Ring A] [Algebra K A] [TopologicalSpace K] [T1Space K] [TopologicalSpace A] [ContinuousSub A] [ContinuousSMul K A] [FunLike F A K] [AlgHomClass F K A K] [SetLike S A] [OneMemClass S A] [AddSubgroupClass S A] [SMulMemClass S K A] (φ : F) (hφ : Continuous ⇑φ) (s : S)
Goal: closure (↑s ∩ ↑(RingHom.ker φ)) = closure ↑s ∩ ↑(RingHom.ker φ)
-- ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id
lemma ker_evalStarAlgHom_eq_closure_adjoin_id (s : Set 𝕜) (h0 : 0 ∈ s) [CompactSpace s] :
    (RingHom.ker (evalStarAlgHom 𝕜 𝕜 (⟨0, h0⟩ : s)) : Set C(s, 𝕜)) =
      closure (adjoin 𝕜 {(restrict s (.id 𝕜))}) 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (s : Set 𝕜) (h0 : 0 ∈ s) [CompactSpace ↑s]
Goal: ↑(RingHom.ker (ContinuousMap.evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) =
  closure ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)})
-- ContinuousMapZero.adjoin_id_dense
lemma ContinuousMapZero.adjoin_id_dense {s : Set 𝕜} [Zero s] (h0 : ((0 : s) : 𝕜) = 0)
    [CompactSpace s] : Dense (adjoin 𝕜 {(.id h0 : C(s, 𝕜)₀)} : Set C(s, 𝕜)₀) 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] {s : Set 𝕜} [Zero ↑s] (h0 : ↑0 = 0) [CompactSpace ↑s]
Goal: Dense ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id h0})
-- continuous_ofMul
theorem continuous_ofMul : Continuous (ofMul : X → Additive X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ⇑Additive.ofMul
-- continuous_toMul
theorem continuous_toMul : Continuous (toMul : Additive X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ⇑Additive.toMul
-- continuous_ofAdd
theorem continuous_ofAdd : Continuous (ofAdd : X → Multiplicative X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ⇑Multiplicative.ofAdd
-- continuous_toAdd
theorem continuous_toAdd : Continuous (toAdd : Multiplicative X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ⇑Multiplicative.toAdd
-- isOpenMap_ofMul
theorem isOpenMap_ofMul : IsOpenMap (ofMul : X → Additive X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsOpenMap ⇑Additive.ofMul
-- isOpenMap_toMul
theorem isOpenMap_toMul : IsOpenMap (toMul : Additive X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsOpenMap ⇑Additive.toMul
-- isOpenMap_ofAdd
theorem isOpenMap_ofAdd : IsOpenMap (ofAdd : X → Multiplicative X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsOpenMap ⇑Multiplicative.ofAdd
-- isOpenMap_toAdd
theorem isOpenMap_toAdd : IsOpenMap (toAdd : Multiplicative X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsOpenMap ⇑Multiplicative.toAdd
-- isClosedMap_ofMul
theorem isClosedMap_ofMul : IsClosedMap (ofMul : X → Additive X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosedMap ⇑Additive.ofMul
-- isClosedMap_toMul
theorem isClosedMap_toMul : IsClosedMap (toMul : Additive X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosedMap ⇑Additive.toMul
-- isClosedMap_ofAdd
theorem isClosedMap_ofAdd : IsClosedMap (ofAdd : X → Multiplicative X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosedMap ⇑Multiplicative.ofAdd
-- isClosedMap_toAdd
theorem isClosedMap_toAdd : IsClosedMap (toAdd : Multiplicative X → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosedMap ⇑Multiplicative.toAdd
-- nhds_ofMul
theorem nhds_ofMul (x : X) : 𝓝 (ofMul x) = map ofMul (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: 𝓝 (Additive.ofMul x) = Filter.map (⇑Additive.ofMul) (𝓝 x)
-- nhds_ofAdd
theorem nhds_ofAdd (x : X) : 𝓝 (ofAdd x) = map ofAdd (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: 𝓝 (Multiplicative.ofAdd x) = Filter.map (⇑Multiplicative.ofAdd) (𝓝 x)
-- nhds_toMul
theorem nhds_toMul (x : Additive X) : 𝓝 (toMul x) = map toMul (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] (x : Additive X)
Goal: 𝓝 (Additive.toMul x) = Filter.map (⇑Additive.toMul) (𝓝 x)
-- nhds_toAdd
theorem nhds_toAdd (x : Multiplicative X) : 𝓝 (toAdd x) = map toAdd (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] (x : Multiplicative X)
Goal: 𝓝 (Multiplicative.toAdd x) = Filter.map (⇑Multiplicative.toAdd) (𝓝 x)
-- continuous_toDual
theorem continuous_toDual : Continuous (toDual : X → Xᵒᵈ) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ⇑OrderDual.toDual
-- continuous_ofDual
theorem continuous_ofDual : Continuous (ofDual : Xᵒᵈ → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: Continuous ⇑OrderDual.ofDual
-- isOpenMap_toDual
theorem isOpenMap_toDual : IsOpenMap (toDual : X → Xᵒᵈ) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsOpenMap ⇑OrderDual.toDual
-- isOpenMap_ofDual
theorem isOpenMap_ofDual : IsOpenMap (ofDual : Xᵒᵈ → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsOpenMap ⇑OrderDual.ofDual
-- isClosedMap_toDual
theorem isClosedMap_toDual : IsClosedMap (toDual : X → Xᵒᵈ) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosedMap ⇑OrderDual.toDual
-- isClosedMap_ofDual
theorem isClosedMap_ofDual : IsClosedMap (ofDual : Xᵒᵈ → X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClosedMap ⇑OrderDual.ofDual
-- nhds_toDual
theorem nhds_toDual (x : X) : 𝓝 (toDual x) = map toDual (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: 𝓝 (OrderDual.toDual x) = Filter.map (⇑OrderDual.toDual) (𝓝 x)
-- nhds_ofDual
theorem nhds_ofDual (x : X) : 𝓝 (ofDual x) = map ofDual (𝓝 x) 
Arguments: {X : Type u} [TopologicalSpace X] (x : X)
Goal: 𝓝 (OrderDual.ofDual x) = Filter.map (⇑OrderDual.ofDual) (𝓝 x)
-- Quotient.preimage_mem_nhds
theorem Quotient.preimage_mem_nhds [TopologicalSpace X] [s : Setoid X] {V : Set <| Quotient s}
    {x : X} (hs : V ∈ 𝓝 (Quotient.mk' x)) : Quotient.mk' ⁻¹' V ∈ 𝓝 x 
Arguments: {X : Type u} [TopologicalSpace X] [s : Setoid X] {V : Set (Quotient s)} {x : X} (hs : V ∈ 𝓝 (Quotient.mk' x))
Goal: Quotient.mk' ⁻¹' V ∈ 𝓝 x
-- Dense.quotient
theorem Dense.quotient [Setoid X] [TopologicalSpace X] {s : Set X} (H : Dense s) :
    Dense (Quotient.mk' '' s) 
Arguments: {X : Type u} [Setoid X] [TopologicalSpace X] {s : Set X} (H : Dense s)
Goal: Dense (Quotient.mk' '' s)
-- DenseRange.quotient
theorem DenseRange.quotient [Setoid X] [TopologicalSpace X] {f : Y → X} (hf : DenseRange f) :
    DenseRange (Quotient.mk' ∘ f) 
Arguments: {X : Type u} {Y : Type v} [Setoid X] [TopologicalSpace X] {f : Y → X} (hf : DenseRange f)
Goal: DenseRange (Quotient.mk' ∘ f)
-- continuous_map_of_le
theorem continuous_map_of_le {α : Type*} [TopologicalSpace α]
    {s t : Setoid α} (h : s ≤ t) : Continuous (Setoid.map_of_le h) 
Arguments: {α : Type u_5} [TopologicalSpace α] {s t : Setoid α} (h : s ≤ t)
Goal: Continuous (Setoid.map_of_le h)
-- continuous_map_sInf
theorem continuous_map_sInf {α : Type*} [TopologicalSpace α]
    {S : Set (Setoid α)} {s : Setoid α} (h : s ∈ S) : Continuous (Setoid.map_sInf h) 
Arguments: {α : Type u_5} [TopologicalSpace α] {S : Set (Setoid α)} {s : Setoid α} (h : s ∈ S)
Goal: Continuous (Setoid.map_sInf h)
-- comap_nhdsWithin_range
@[simp] lemma comap_nhdsWithin_range {α β} [TopologicalSpace β] (f : α → β) (y : β) :
    comap f (𝓝[range f] y) = comap f (𝓝 y) 
Arguments: {α : Type u_5} {β : Type u_6} [TopologicalSpace β] (f : α → β) (y : β)
Goal: Filter.comap f (𝓝[Set.range f] y) = Filter.comap f (𝓝 y)
-- mem_nhds_subtype
theorem mem_nhds_subtype (s : Set X) (x : { x // x ∈ s }) (t : Set { x // x ∈ s }) :
    t ∈ 𝓝 x ↔ ∃ u ∈ 𝓝 (x : X), Subtype.val ⁻¹' u ⊆ t 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X) (x : { x // x ∈ s }) (t : Set { x // x ∈ s })
Goal: t ∈ 𝓝 x ↔ ∃ u ∈ 𝓝 ↑x, Subtype.val ⁻¹' u ⊆ t
-- nhds_subtype
theorem nhds_subtype (s : Set X) (x : { x // x ∈ s }) : 𝓝 x = comap (↑) (𝓝 (x : X)) 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X) (x : { x // x ∈ s })
Goal: 𝓝 x = Filter.comap Subtype.val (𝓝 ↑x)
-- nhds_subtype_eq_comap_nhdsWithin
lemma nhds_subtype_eq_comap_nhdsWithin (s : Set X) (x : { x // x ∈ s }) :
    𝓝 x = comap (↑) (𝓝[s] (x : X)) 
Arguments: {X : Type u} [TopologicalSpace X] (s : Set X) (x : { x // x ∈ s })
Goal: 𝓝 x = Filter.comap Subtype.val (𝓝[s] ↑x)
-- nhdsWithin_subtype_eq_bot_iff
theorem nhdsWithin_subtype_eq_bot_iff {s t : Set X} {x : s} :
    𝓝[((↑) : s → X) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : X) ⊓ 𝓟 s = ⊥ 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} {x : ↑s}
Goal: 𝓝[Subtype.val ⁻¹' t] x = ⊥ ↔ 𝓝[t] ↑x ⊓ 𝓟 s = ⊥
-- nhds_ne_subtype_eq_bot_iff
theorem nhds_ne_subtype_eq_bot_iff {S : Set X} {x : S} :
    𝓝[≠] x = ⊥ ↔ 𝓝[≠] (x : X) ⊓ 𝓟 S = ⊥ 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set X} {x : ↑S}
Goal: 𝓝[≠] x = ⊥ ↔ 𝓝[≠] ↑x ⊓ 𝓟 S = ⊥
-- nhds_ne_subtype_neBot_iff
theorem nhds_ne_subtype_neBot_iff {S : Set X} {x : S} :
    (𝓝[≠] x).NeBot ↔ (𝓝[≠] (x : X) ⊓ 𝓟 S).NeBot 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set X} {x : ↑S}
Goal: (𝓝[≠] x).NeBot ↔ (𝓝[≠] ↑x ⊓ 𝓟 S).NeBot
-- discreteTopology_subtype_iff
theorem discreteTopology_subtype_iff {S : Set X} :
    DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ 
Arguments: {X : Type u} [TopologicalSpace X] {S : Set X}
Goal: DiscreteTopology ↑S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥
-- CofiniteTopology.isOpen_iff
theorem isOpen_iff {s : Set (CofiniteTopology X)} : IsOpen s ↔ s.Nonempty → sᶜ.Finite 
Arguments: {X : Type u} {s : Set (CofiniteTopology X)}
Goal: IsOpen s ↔ s.Nonempty → sᶜ.Finite
-- CofiniteTopology.isOpen_iff'
theorem isOpen_iff' {s : Set (CofiniteTopology X)} : IsOpen s ↔ s = ∅ ∨ sᶜ.Finite 
Arguments: {X : Type u} {s : Set (CofiniteTopology X)}
Goal: IsOpen s ↔ s = ∅ ∨ sᶜ.Finite
-- CofiniteTopology.isClosed_iff
theorem isClosed_iff {s : Set (CofiniteTopology X)} : IsClosed s ↔ s = univ ∨ s.Finite 
Arguments: {X : Type u} {s : Set (CofiniteTopology X)}
Goal: IsClosed s ↔ s = Set.univ ∨ s.Finite
-- CofiniteTopology.nhds_eq
theorem nhds_eq (x : CofiniteTopology X) : 𝓝 x = pure x ⊔ cofinite 
Arguments: {X : Type u} (x : CofiniteTopology X)
Goal: 𝓝 x = pure x ⊔ Filter.cofinite
-- CofiniteTopology.mem_nhds_iff
theorem mem_nhds_iff {x : CofiniteTopology X} {s : Set (CofiniteTopology X)} :
    s ∈ 𝓝 x ↔ x ∈ s ∧ sᶜ.Finite 
Arguments: {X : Type u} {x : CofiniteTopology X} {s : Set (CofiniteTopology X)}
Goal: s ∈ 𝓝 x ↔ x ∈ s ∧ sᶜ.Finite
-- continuous_prod_mk
@[simp] theorem continuous_prod_mk {f : X → Y} {g : X → Z} :
    (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : X → Z}
Goal: (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g
-- continuous_fst
@[continuity]
theorem continuous_fst : Continuous (@Prod.fst X Y) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Continuous Prod.fst
-- Continuous.fst
@[fun_prop]
theorem Continuous.fst {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).1 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y × Z} (hf : Continuous f)
Goal: Continuous fun x => (f x).1
-- Continuous.fst'
theorem Continuous.fst' {f : X → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.fst 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} (hf : Continuous f)
Goal: Continuous fun x => f x.1
-- continuousAt_fst
theorem continuousAt_fst {p : X × Y} : ContinuousAt Prod.fst p 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {p : X × Y}
Goal: ContinuousAt Prod.fst p
-- ContinuousAt.fst
@[fun_prop]
theorem ContinuousAt.fst {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
    ContinuousAt (fun x : X => (f x).1) x 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y × Z} {x : X} (hf : ContinuousAt f x)
Goal: ContinuousAt (fun x => (f x).1) x
-- ContinuousAt.fst'
theorem ContinuousAt.fst' {f : X → Z} {x : X} {y : Y} (hf : ContinuousAt f x) :
    ContinuousAt (fun x : X × Y => f x.fst) (x, y) 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {x : X} {y : Y} (hf : ContinuousAt f x)
Goal: ContinuousAt (fun x => f x.1) (x, y)
-- ContinuousAt.fst''
theorem ContinuousAt.fst'' {f : X → Z} {x : X × Y} (hf : ContinuousAt f x.fst) :
    ContinuousAt (fun x : X × Y => f x.fst) x 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {x : X × Y} (hf : ContinuousAt f x.1)
Goal: ContinuousAt (fun x => f x.1) x
-- Filter.Tendsto.fst_nhds
theorem Filter.Tendsto.fst_nhds {l : Filter X} {f : X → Y × Z} {p : Y × Z}
    (h : Tendsto f l (𝓝 p)) : Tendsto (fun a ↦ (f a).1) l (𝓝 <| p.1) 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace Y] [TopologicalSpace Z] {l : Filter X} {f : X → Y × Z} {p : Y × Z} (h : Filter.Tendsto f l (𝓝 p))
Goal: Filter.Tendsto (fun a => (f a).1) l (𝓝 p.1)
-- continuous_snd
@[continuity]
theorem continuous_snd : Continuous (@Prod.snd X Y) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Continuous Prod.snd
-- Continuous.snd
@[fun_prop]
theorem Continuous.snd {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).2 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y × Z} (hf : Continuous f)
Goal: Continuous fun x => (f x).2
-- Continuous.snd'
theorem Continuous.snd' {f : Y → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.snd 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : Y → Z} (hf : Continuous f)
Goal: Continuous fun x => f x.2
-- continuousAt_snd
theorem continuousAt_snd {p : X × Y} : ContinuousAt Prod.snd p 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {p : X × Y}
Goal: ContinuousAt Prod.snd p
-- ContinuousAt.snd
@[fun_prop]
theorem ContinuousAt.snd {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
    ContinuousAt (fun x : X => (f x).2) x 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y × Z} {x : X} (hf : ContinuousAt f x)
Goal: ContinuousAt (fun x => (f x).2) x
-- ContinuousAt.snd'
theorem ContinuousAt.snd' {f : Y → Z} {x : X} {y : Y} (hf : ContinuousAt f y) :
    ContinuousAt (fun x : X × Y => f x.snd) (x, y) 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : Y → Z} {x : X} {y : Y} (hf : ContinuousAt f y)
Goal: ContinuousAt (fun x => f x.2) (x, y)
-- ContinuousAt.snd''
theorem ContinuousAt.snd'' {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.snd) :
    ContinuousAt (fun x : X × Y => f x.snd) x 
Arguments: {X : Type u} {Y : Type v} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.2)
Goal: ContinuousAt (fun x => f x.2) x