ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMap.hasBasis_compactConvergenceUniformity
theorem hasBasis_compactConvergenceUniformity :
    HasBasis (𝓤 C(α, β)) (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β]
Goal: (𝓤 C(α, β)).HasBasis (fun p => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p => {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2}
-- ContinuousMap.mem_compactConvergence_entourage_iff
theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
    X ∈ 𝓤 C(α, β) ↔
      ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧
        { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] (X : Set (C(α, β) × C(α, β)))
Goal: X ∈ 𝓤 C(α, β) ↔ ∃ K V, IsCompact K ∧ V ∈ 𝓤 β ∧ {fg | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V} ⊆ X
-- CompactExhaustion.hasBasis_compactConvergenceUniformity
theorem _root_.CompactExhaustion.hasBasis_compactConvergenceUniformity {ι : Type*}
    {p : ι → Prop} {V : ι → Set (β × β)} (K : CompactExhaustion α) (hb : (𝓤 β).HasBasis p V) :
    HasBasis (𝓤 C(α, β)) (fun i : ℕ × ι ↦ p i.2) fun i ↦
      {fg | ∀ x ∈ K i.1, (fg.1 x, fg.2 x) ∈ V i.2} 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {ι : Type u_1} {p : ι → Prop} {V : ι → Set (β × β)} (K : CompactExhaustion α) (hb : (𝓤 β).HasBasis p V)
Goal: (𝓤 C(α, β)).HasBasis (fun i => p i.2) fun i => {fg | ∀ x ∈ K i.1, (fg.1 x, fg.2 x) ∈ V i.2}
-- CompactExhaustion.hasAntitoneBasis_compactConvergenceUniformity
theorem _root_.CompactExhaustion.hasAntitoneBasis_compactConvergenceUniformity
    {V : ℕ → Set (β × β)} (K : CompactExhaustion α) (hb : (𝓤 β).HasAntitoneBasis V) :
    HasAntitoneBasis (𝓤 C(α, β)) fun n ↦ {fg | ∀ x ∈ K n, (fg.1 x, fg.2 x) ∈ V n} 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {V : ℕ → Set (β × β)} (K : CompactExhaustion α) (hb : (𝓤 β).HasAntitoneBasis V)
Goal: (𝓤 C(α, β)).HasAntitoneBasis fun n => {fg | ∀ x ∈ K n, (fg.1 x, fg.2 x) ∈ V n}
-- ContinuousMap.tendsto_of_tendstoLocallyUniformly
theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a => F i a) f p) :
    Tendsto F p (𝓝 f) 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {f : C(α, β)} {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} (h : TendstoLocallyUniformly (fun i a => (F i) a) (⇑f) p)
Goal: Filter.Tendsto F p (𝓝 f)
-- ContinuousMap.tendsto_iff_tendstoLocallyUniformly
theorem tendsto_iff_tendstoLocallyUniformly [WeaklyLocallyCompactSpace α] :
    Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a => F i a) f p 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {f : C(α, β)} {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} [WeaklyLocallyCompactSpace α]
Goal: Filter.Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a => (F i) a) (⇑f) p
-- ContinuousMap.tendstoLocallyUniformly_of_tendsto
@[deprecated tendsto_iff_tendstoLocallyUniformly (since := "2023-09-03")]
theorem tendstoLocallyUniformly_of_tendsto [WeaklyLocallyCompactSpace α] (h : Tendsto F p (𝓝 f)) :
    TendstoLocallyUniformly (fun i a => F i a) f p 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {f : C(α, β)} {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} [WeaklyLocallyCompactSpace α] (h : Filter.Tendsto F p (𝓝 f))
Goal: TendstoLocallyUniformly (fun i a => (F i) a) (⇑f) p
-- ContinuousMap.uniformContinuous_comp
theorem uniformContinuous_comp (g : C(β, δ)) (hg : UniformContinuous g) :
    UniformContinuous (ContinuousMap.comp g : C(α, β) → C(α, δ)) 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ : Type u_2} [UniformSpace δ] (g : C(β, δ)) (hg : UniformContinuous ⇑g)
Goal: UniformContinuous g.comp
-- ContinuousMap.uniformInducing_comp
theorem uniformInducing_comp (g : C(β, δ)) (hg : UniformInducing g) :
    UniformInducing (ContinuousMap.comp g : C(α, β) → C(α, δ)) 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ : Type u_2} [UniformSpace δ] (g : C(β, δ)) (hg : UniformInducing ⇑g)
Goal: UniformInducing g.comp
-- ContinuousMap.uniformEmbedding_comp
theorem uniformEmbedding_comp (g : C(β, δ)) (hg : UniformEmbedding g) :
    UniformEmbedding (ContinuousMap.comp g : C(α, β) → C(α, δ)) 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ : Type u_2} [UniformSpace δ] (g : C(β, δ)) (hg : UniformEmbedding ⇑g)
Goal: UniformEmbedding g.comp
-- ContinuousMap.uniformContinuous_comp_left
theorem uniformContinuous_comp_left (g : C(α, γ)) :
    UniformContinuous (fun f ↦ f.comp g : C(γ, β) → C(α, β)) 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {γ : Type u_1} [TopologicalSpace γ] (g : C(α, γ))
Goal: UniformContinuous fun f => f.comp g
-- ContinuousMap.hasBasis_compactConvergenceUniformity_of_compact
theorem hasBasis_compactConvergenceUniformity_of_compact :
    HasBasis (𝓤 C(α, β)) (fun V : Set (β × β) => V ∈ 𝓤 β) fun V =>
      { fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V } 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] [CompactSpace α]
Goal: (𝓤 C(α, β)).HasBasis (fun V => V ∈ 𝓤 β) fun V => {fg | ∀ (x : α), (fg.1 x, fg.2 x) ∈ V}
-- ContinuousMap.tendsto_iff_tendstoUniformly
theorem tendsto_iff_tendstoUniformly :
    Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a => F i a) f p 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {f : C(α, β)} {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} [CompactSpace α]
Goal: Filter.Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a => (F i) a) (⇑f) p
-- ContinuousMap.uniformSpace_eq_inf_precomp_of_cover
theorem uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} [TopologicalSpace δ₁]
    [TopologicalSpace δ₂] (φ₁ : C(δ₁, α)) (φ₂ : C(δ₂, α)) (h_proper₁ : IsProperMap φ₁)
    (h_proper₂ : IsProperMap φ₂) (h_cover : range φ₁ ∪ range φ₂ = univ) :
    (inferInstanceAs <| UniformSpace C(α, β)) =
      .comap (comp · φ₁) inferInstance ⊓
      .comap (comp · φ₂) inferInstance 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ₁ : Type u_1} {δ₂ : Type u_2} [TopologicalSpace δ₁] [TopologicalSpace δ₂] (φ₁ : C(δ₁, α)) (φ₂ : C(δ₂, α)) (h_proper₁ : IsProperMap ⇑φ₁) (h_proper₂ : IsProperMap ⇑φ₂) (h_cover : Set.range ⇑φ₁ ∪ Set.range ⇑φ₂ = Set.univ)
Goal: inferInstanceAs (UniformSpace C(α, β)) =
  UniformSpace.comap (fun x => x.comp φ₁) inferInstance ⊓ UniformSpace.comap (fun x => x.comp φ₂) inferInstance
-- ContinuousMap.uniformSpace_eq_iInf_precomp_of_cover
theorem uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} [∀ i, TopologicalSpace (δ i)]
    (φ : Π i, C(δ i, α)) (h_proper : ∀ i, IsProperMap (φ i))
    (h_lf : LocallyFinite fun i ↦ range (φ i)) (h_cover : ⋃ i, range (φ i) = univ) :
    (inferInstanceAs <| UniformSpace C(α, β)) = ⨅ i, .comap (comp · (φ i)) inferInstance 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {ι : Type u₃} {δ : ι → Type u_1} [(i : ι) → TopologicalSpace (δ i)] (φ : (i : ι) → C(δ i, α)) (h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)) (h_lf : LocallyFinite fun i => Set.range ⇑(φ i)) (h_cover : ⋃ i, Set.range ⇑(φ i) = Set.univ)
Goal: inferInstanceAs (UniformSpace C(α, β)) = ⨅ i, UniformSpace.comap (fun x => x.comp (φ i)) inferInstance
-- nhdsSet_diagonal_eq_uniformity
theorem nhdsSet_diagonal_eq_uniformity [CompactSpace α] : 𝓝ˢ (diagonal α) = 𝓤 α 
Arguments: {α : Type u_1} [UniformSpace α] [CompactSpace α]
Goal: 𝓝ˢ (Set.diagonal α) = 𝓤 α
-- compactSpace_uniformity
theorem compactSpace_uniformity [CompactSpace α] : 𝓤 α = ⨆ x, 𝓝 (x, x) 
Arguments: {α : Type u_1} [UniformSpace α] [CompactSpace α]
Goal: 𝓤 α = ⨆ x, 𝓝 (x, x)
-- unique_uniformity_of_compact
theorem unique_uniformity_of_compact [t : TopologicalSpace γ] [CompactSpace γ]
    {u u' : UniformSpace γ} (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) :
    u = u' 
Arguments: {γ : Type u_3} [t : TopologicalSpace γ] [CompactSpace γ] {u u' : UniformSpace γ} (h : UniformSpace.toTopologicalSpace = t) (h' : UniformSpace.toTopologicalSpace = t)
Goal: u = u'
-- CompactSpace.uniformContinuous_of_continuous
theorem CompactSpace.uniformContinuous_of_continuous [CompactSpace α] {f : α → β}
    (h : Continuous f) : UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] [CompactSpace α] {f : α → β} (h : Continuous f)
Goal: UniformContinuous f
-- IsCompact.uniformContinuousOn_of_continuous
theorem IsCompact.uniformContinuousOn_of_continuous {s : Set α} {f : α → β} (hs : IsCompact s)
    (hf : ContinuousOn f s) : UniformContinuousOn f s 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {s : Set α} {f : α → β} (hs : IsCompact s) (hf : ContinuousOn f s)
Goal: UniformContinuousOn f s
-- IsCompact.uniformContinuousAt_of_continuousAt
theorem IsCompact.uniformContinuousAt_of_continuousAt {r : Set (β × β)} {s : Set α}
    (hs : IsCompact s) (f : α → β) (hf : ∀ a ∈ s, ContinuousAt f a) (hr : r ∈ 𝓤 β) :
    { x : α × α | x.1 ∈ s → (f x.1, f x.2) ∈ r } ∈ 𝓤 α 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {r : Set (β × β)} {s : Set α} (hs : IsCompact s) (f : α → β) (hf : ∀ a ∈ s, ContinuousAt f a) (hr : r ∈ 𝓤 β)
Goal: {x | x.1 ∈ s → (f x.1, f x.2) ∈ r} ∈ 𝓤 α
-- Continuous.uniformContinuous_of_tendsto_cocompact
theorem Continuous.uniformContinuous_of_tendsto_cocompact {f : α → β} {x : β}
    (h_cont : Continuous f) (hx : Tendsto f (cocompact α) (𝓝 x)) : UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} {x : β} (h_cont : Continuous f) (hx : Filter.Tendsto f (Filter.cocompact α) (𝓝 x))
Goal: UniformContinuous f
-- HasCompactSupport.uniformContinuous_of_continuous
@[to_additive]
theorem HasCompactMulSupport.uniformContinuous_of_continuous {f : α → β} [One β]
    (h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} [Zero β] (h1 : HasCompactSupport f) (h2 : Continuous f)
Goal: UniformContinuous f
-- HasCompactMulSupport.uniformContinuous_of_continuous
@[to_additive]
theorem HasCompactMulSupport.uniformContinuous_of_continuous {f : α → β} [One β]
    (h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {f : α → β} [One β] (h1 : HasCompactMulSupport f) (h2 : Continuous f)
Goal: UniformContinuous f
-- ContinuousOn.tendstoUniformly
theorem ContinuousOn.tendstoUniformly [LocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
    {f : α → β → γ} {x : α} {U : Set α} (hxU : U ∈ 𝓝 x) (h : ContinuousOn (↿f) (U ×ˢ univ)) :
    TendstoUniformly f (f x) (𝓝 x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [LocallyCompactSpace α] [CompactSpace β] [UniformSpace γ] {f : α → β → γ} {x : α} {U : Set α} (hxU : U ∈ 𝓝 x) (h : ContinuousOn (↿f) (U ×ˢ Set.univ))
Goal: TendstoUniformly f (f x) (𝓝 x)
-- Continuous.tendstoUniformly
theorem Continuous.tendstoUniformly [WeaklyLocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
    (f : α → β → γ) (h : Continuous ↿f) (x : α) : TendstoUniformly f (f x) (𝓝 x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [WeaklyLocallyCompactSpace α] [CompactSpace β] [UniformSpace γ] (f : α → β → γ) (h : Continuous ↿f) (x : α)
Goal: TendstoUniformly f (f x) (𝓝 x)
-- IsCompact.mem_uniformity_of_prod
lemma IsCompact.mem_uniformity_of_prod
    {α β E : Type*} [TopologicalSpace α] [TopologicalSpace β] [UniformSpace E]
    {f : α → β → E} {s : Set α} {k : Set β} {q : α} {u : Set (E × E)}
    (hk : IsCompact k) (hf : ContinuousOn f.uncurry (s ×ˢ k)) (hq : q ∈ s) (hu : u ∈ 𝓤 E) :
    ∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ k, (f p x, f q x) ∈ u 
Arguments: {α : Type u_4} {β : Type u_5} {E : Type u_6} [TopologicalSpace α] [TopologicalSpace β] [UniformSpace E] {f : α → β → E} {s : Set α} {k : Set β} {q : α} {u : Set (E × E)} (hk : IsCompact k) (hf : ContinuousOn (Function.uncurry f) (s ×ˢ k)) (hq : q ∈ s) (hu : u ∈ 𝓤 E)
Goal: ∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ k, (f p x, f q x) ∈ u
-- CompactSpace.uniformEquicontinuous_of_equicontinuous
theorem CompactSpace.uniformEquicontinuous_of_equicontinuous {ι : Type*} {F : ι → β → α}
    [CompactSpace β] (h : Equicontinuous F) : UniformEquicontinuous F 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {ι : Type u_4} {F : ι → β → α} [CompactSpace β] (h : Equicontinuous F)
Goal: UniformEquicontinuous F
-- FiberBundle.trivialization_mem_atlas'
class FiberBundle where
  totalSpaceMk_inducing' : ∀ b : B, Inducing (@TotalSpace.mk B F E b)
  trivializationAtlas' : Set (Trivialization F (π F E))
  trivializationAt' : B → Trivialization F (π F E)
  mem_baseSet_trivializationAt' : ∀ b : B, b ∈ (trivializationAt' b).baseSet
  trivialization_mem_atlas' : ∀ b : B, trivializationAt' b ∈ trivializationAtlas'
Arguments: {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [self : FiberBundle F E] (b : B)
Goal: FiberBundle.trivializationAt' b ∈ FiberBundle.trivializationAtlas'
-- FiberBundle.totalSpaceMk_inducing'
class FiberBundle where
  totalSpaceMk_inducing' : ∀ b : B, Inducing (@TotalSpace.mk B F E b)
  trivializationAtlas' : Set (Trivialization F (π F E))
  trivializationAt' : B → Trivialization F (π F E)
  mem_baseSet_trivializationAt' : ∀ b : B, b ∈ (trivializationAt' b).baseSet
  trivialization_mem_atlas' : ∀ b : B, trivializationAt' b ∈ trivializationAtlas'
Arguments: {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [self : FiberBundle F E] (b : B)
Goal: Inducing (Bundle.TotalSpace.mk b)
-- FiberBundle.mem_baseSet_trivializationAt'
class FiberBundle where
  totalSpaceMk_inducing' : ∀ b : B, Inducing (@TotalSpace.mk B F E b)
  trivializationAtlas' : Set (Trivialization F (π F E))
  trivializationAt' : B → Trivialization F (π F E)
  mem_baseSet_trivializationAt' : ∀ b : B, b ∈ (trivializationAt' b).baseSet
  trivialization_mem_atlas' : ∀ b : B, trivializationAt' b ∈ trivializationAtlas'
Arguments: {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [self : FiberBundle F E] (b : B)
Goal: b ∈ (FiberBundle.trivializationAt' b).baseSet
-- FiberBundle.totalSpaceMk_inducing
theorem totalSpaceMk_inducing : Inducing (@TotalSpace.mk B F E b) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (b : B)
Goal: Inducing (Bundle.TotalSpace.mk b)
-- FiberBundle.mem_baseSet_trivializationAt
theorem mem_baseSet_trivializationAt : b ∈ (trivializationAt F E b).baseSet 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (b : B)
Goal: b ∈ (trivializationAt F E b).baseSet
-- FiberBundle.trivialization_mem_atlas
theorem trivialization_mem_atlas : trivializationAt F E b ∈ trivializationAtlas F E 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (b : B)
Goal: trivializationAt F E b ∈ trivializationAtlas F E
-- MemTrivializationAtlas.out
@[mk_iff]
class MemTrivializationAtlas [FiberBundle F E] (e : Trivialization F (π F E)) : Prop where
  out : e ∈ trivializationAtlas F E
Arguments: {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] {e : Trivialization F Bundle.TotalSpace.proj} [self : MemTrivializationAtlas e]
Goal: e ∈ trivializationAtlas F E
-- memTrivializationAtlas_iff
@[mk_iff]
class MemTrivializationAtlas [FiberBundle F E] (e : Trivialization F (π F E)) : Prop where
  out : e ∈ trivializationAtlas F E
Arguments: {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj)
Goal: MemTrivializationAtlas e ↔ e ∈ trivializationAtlas F E
-- FiberBundle.map_proj_nhds
theorem map_proj_nhds (x : TotalSpace F E) : map (π F E) (𝓝 x) = 𝓝 x.proj 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (x : Bundle.TotalSpace F E)
Goal: Filter.map Bundle.TotalSpace.proj (𝓝 x) = 𝓝 x.proj
-- FiberBundle.continuous_proj
@[continuity]
theorem continuous_proj : Continuous (π F E) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E]
Goal: Continuous Bundle.TotalSpace.proj
-- FiberBundle.isOpenMap_proj
theorem isOpenMap_proj : IsOpenMap (π F E) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E]
Goal: IsOpenMap Bundle.TotalSpace.proj
-- FiberBundle.surjective_proj
theorem surjective_proj [Nonempty F] : Function.Surjective (π F E) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] [Nonempty F]
Goal: Function.Surjective Bundle.TotalSpace.proj
-- FiberBundle.quotientMap_proj
theorem quotientMap_proj [Nonempty F] : QuotientMap (π F E) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] [Nonempty F]
Goal: QuotientMap Bundle.TotalSpace.proj
-- FiberBundle.continuous_totalSpaceMk
theorem continuous_totalSpaceMk (x : B) : Continuous (@TotalSpace.mk B F E x) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (x : B)
Goal: Continuous (Bundle.TotalSpace.mk x)
-- FiberBundle.totalSpaceMk_embedding
theorem totalSpaceMk_embedding (x : B) : Embedding (@TotalSpace.mk B F E x) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (x : B)
Goal: Embedding (Bundle.TotalSpace.mk x)
-- FiberBundle.totalSpaceMk_closedEmbedding
theorem totalSpaceMk_closedEmbedding [T1Space B] (x : B) :
    ClosedEmbedding (@TotalSpace.mk B F E x) 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] [T1Space B] (x : B)
Goal: ClosedEmbedding (Bundle.TotalSpace.mk x)
-- FiberBundle.mem_trivializationAt_proj_source
@[simp, mfld_simps]
theorem mem_trivializationAt_proj_source {x : TotalSpace F E} :
    x ∈ (trivializationAt F E x.proj).source 
Arguments: {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] {x : Bundle.TotalSpace F E}
Goal: x ∈ (trivializationAt F E x.proj).source
-- FiberBundle.trivializationAt_proj_fst
theorem trivializationAt_proj_fst {x : TotalSpace F E} :
    ((trivializationAt F E x.proj) x).1 = x.proj 
Arguments: {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] {x : Bundle.TotalSpace F E}
Goal: (↑(trivializationAt F E x.proj) x).1 = x.proj
-- FiberBundle.continuousWithinAt_totalSpace
theorem continuousWithinAt_totalSpace (f : X → TotalSpace F E) {s : Set X} {x₀ : X} :
    ContinuousWithinAt f s x₀ ↔
      ContinuousWithinAt (fun x => (f x).proj) s x₀ ∧
        ContinuousWithinAt (fun x => ((trivializationAt F E (f x₀).proj) (f x)).2) s x₀ 
Arguments: {B : Type u_2} (F : Type u_3) {X : Type u_4} [TopologicalSpace X] [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (f : X → Bundle.TotalSpace F E) {s : Set X} {x₀ : X}
Goal: ContinuousWithinAt f s x₀ ↔
  ContinuousWithinAt (fun x => (f x).proj) s x₀ ∧
    ContinuousWithinAt (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) s x₀
-- FiberBundle.continuousAt_totalSpace
theorem continuousAt_totalSpace (f : X → TotalSpace F E) {x₀ : X} :
    ContinuousAt f x₀ ↔
      ContinuousAt (fun x => (f x).proj) x₀ ∧
        ContinuousAt (fun x => ((trivializationAt F E (f x₀).proj) (f x)).2) x₀ 
Arguments: {B : Type u_2} (F : Type u_3) {X : Type u_4} [TopologicalSpace X] [TopologicalSpace B] [TopologicalSpace F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [FiberBundle F E] (f : X → Bundle.TotalSpace F E) {x₀ : X}
Goal: ContinuousAt f x₀ ↔
  ContinuousAt (fun x => (f x).proj) x₀ ∧ ContinuousAt (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) x₀
-- FiberBundle.exists_trivialization_Icc_subset
theorem FiberBundle.exists_trivialization_Icc_subset [ConditionallyCompleteLinearOrder B]
    [OrderTopology B] [FiberBundle F E] (a b : B) :
    ∃ e : Trivialization F (π F E), Icc a b ⊆ e.baseSet 
Arguments: {B : Type u_2} (F : Type u_3) [TopologicalSpace B] [TopologicalSpace F] (E : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F E)] [(b : B) → TopologicalSpace (E b)] [ConditionallyCompleteLinearOrder B] [OrderTopology B] [FiberBundle F E] (a b : B)
Goal: ∃ e, Set.Icc a b ⊆ e.baseSet
-- FiberBundleCore.coordChange_comp
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure FiberBundleCore (ι : Type*) (B : Type*) [TopologicalSpace B] (F : Type*)
    [TopologicalSpace F] where
  baseSet : ι → Set B
  isOpen_baseSet : ∀ i, IsOpen (baseSet i)
  indexAt : B → ι
  mem_baseSet_at : ∀ x, x ∈ baseSet (indexAt x)
  coordChange : ι → ι → B → F → F
  coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coordChange i i x v = v
  continuousOn_coordChange : ∀ i j,
    ContinuousOn (fun p : B × F => coordChange i j p.1 p.2) ((baseSet i ∩ baseSet j) ×ˢ univ)
  coordChange_comp : ∀ i j k, ∀ x ∈ baseSet i ∩ baseSet j ∩ baseSet k, ∀ v,
    (coordChange j k x) (coordChange i j x v) = coordChange i k x v
Arguments: {ι : Type u_5} {B : Type u_6} [TopologicalSpace B] {F : Type u_7} [TopologicalSpace F] (self : FiberBundleCore ι B F) (i j k : ι) (x : B)
Goal: x ∈ self.baseSet i ∩ self.baseSet j ∩ self.baseSet k →
  ∀ (v : F), self.coordChange j k x (self.coordChange i j x v) = self.coordChange i k x v