ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FiberBundleCore.continuousOn_coordChange
-- 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 : ι)
Goal: ContinuousOn (fun p => self.coordChange i j p.1 p.2) ((self.baseSet i ∩ self.baseSet j) ×ˢ Set.univ)
-- FiberBundleCore.isOpen_baseSet
-- 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 : ι)
Goal: IsOpen (self.baseSet i)
-- FiberBundleCore.coordChange_self
-- 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 : ι) (x : B)
Goal: x ∈ self.baseSet i → ∀ (v : F), self.coordChange i i x v = v
-- FiberBundleCore.mem_baseSet_at
-- 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) (x : B)
Goal: x ∈ self.baseSet (self.indexAt x)
-- FiberBundleCore.mem_trivChange_source
@[simp, mfld_simps]
theorem mem_trivChange_source (i j : ι) (p : B × F) :
    p ∈ (Z.trivChange i j).source ↔ p.1 ∈ Z.baseSet i ∩ Z.baseSet j 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i j : ι) (p : B × F)
Goal: p ∈ (Z.trivChange i j).source ↔ p.1 ∈ Z.baseSet i ∩ Z.baseSet j
-- FiberBundleCore.mem_localTrivAsPartialEquiv_source
theorem mem_localTrivAsPartialEquiv_source (p : Z.TotalSpace) :
    p ∈ (Z.localTrivAsPartialEquiv i).source ↔ p.1 ∈ Z.baseSet i 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι) (p : Z.TotalSpace)
Goal: p ∈ (Z.localTrivAsPartialEquiv i).source ↔ p.proj ∈ Z.baseSet i
-- FiberBundleCore.mem_localTrivAsPartialEquiv_target
theorem mem_localTrivAsPartialEquiv_target (p : B × F) :
    p ∈ (Z.localTrivAsPartialEquiv i).target ↔ p.1 ∈ Z.baseSet i 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι) (p : B × F)
Goal: p ∈ (Z.localTrivAsPartialEquiv i).target ↔ p.1 ∈ Z.baseSet i
-- FiberBundleCore.localTrivAsPartialEquiv_apply
theorem localTrivAsPartialEquiv_apply (p : Z.TotalSpace) :
    (Z.localTrivAsPartialEquiv i) p = ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩ 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι) (p : Z.TotalSpace)
Goal: ↑(Z.localTrivAsPartialEquiv i) p = (p.proj, Z.coordChange (Z.indexAt p.proj) i p.proj p.snd)
-- FiberBundleCore.localTrivAsPartialEquiv_trans
theorem localTrivAsPartialEquiv_trans (i j : ι) :
    (Z.localTrivAsPartialEquiv i).symm.trans (Z.localTrivAsPartialEquiv j) ≈
      (Z.trivChange i j).toPartialEquiv 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i j : ι)
Goal: (Z.localTrivAsPartialEquiv i).symm.trans (Z.localTrivAsPartialEquiv j) ≈ (Z.trivChange i j).toPartialEquiv
-- FiberBundleCore.open_source'
theorem open_source' (i : ι) : IsOpen (Z.localTrivAsPartialEquiv i).source 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι)
Goal: IsOpen (Z.localTrivAsPartialEquiv i).source
-- FiberBundleCore.localTrivAt_def
@[simp, mfld_simps]
theorem localTrivAt_def (b : B) : Z.localTriv (Z.indexAt b) = Z.localTrivAt b 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (b : B)
Goal: Z.localTriv (Z.indexAt b) = Z.localTrivAt b
-- FiberBundleCore.localTrivAt_snd
theorem localTrivAt_snd (b : B) (p) :
    (Z.localTrivAt b p).2 = Z.coordChange (Z.indexAt p.1) (Z.indexAt b) p.1 p.2 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (b : B) (p : Bundle.TotalSpace F Z.Fiber)
Goal: (↑(Z.localTrivAt b) p).2 = Z.coordChange (Z.indexAt p.proj) (Z.indexAt b) p.proj p.snd
-- FiberBundleCore.continuous_const_section
theorem continuous_const_section (v : F)
    (h : ∀ i j, ∀ x ∈ Z.baseSet i ∩ Z.baseSet j, Z.coordChange i j x v = v) :
    Continuous (show B → Z.TotalSpace from fun x => ⟨x, v⟩) 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (v : F) (h : ∀ (i j : ι), ∀ x ∈ Z.baseSet i ∩ Z.baseSet j, Z.coordChange i j x v = v)
Goal: Continuous
  (let_fun this := fun x => { proj := x, snd := v };
  this)
-- FiberBundleCore.localTrivAsPartialEquiv_coe
@[simp, mfld_simps]
theorem localTrivAsPartialEquiv_coe : ⇑(Z.localTrivAsPartialEquiv i) = Z.localTriv i 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι)
Goal: ↑(Z.localTrivAsPartialEquiv i) = ↑(Z.localTriv i)
-- FiberBundleCore.localTrivAsPartialEquiv_source
@[simp, mfld_simps]
theorem localTrivAsPartialEquiv_source :
    (Z.localTrivAsPartialEquiv i).source = (Z.localTriv i).source 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι)
Goal: (Z.localTrivAsPartialEquiv i).source = (Z.localTriv i).source
-- FiberBundleCore.localTrivAsPartialEquiv_target
@[simp, mfld_simps]
theorem localTrivAsPartialEquiv_target :
    (Z.localTrivAsPartialEquiv i).target = (Z.localTriv i).target 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι)
Goal: (Z.localTrivAsPartialEquiv i).target = (Z.localTriv i).target
-- FiberBundleCore.localTrivAsPartialEquiv_symm
@[simp, mfld_simps]
theorem localTrivAsPartialEquiv_symm :
    (Z.localTrivAsPartialEquiv i).symm = (Z.localTriv i).toPartialEquiv.symm 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι)
Goal: (Z.localTrivAsPartialEquiv i).symm = (Z.localTriv i).symm
-- FiberBundleCore.baseSet_at
@[simp, mfld_simps]
theorem baseSet_at : Z.baseSet i = (Z.localTriv i).baseSet 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι)
Goal: Z.baseSet i = (Z.localTriv i).baseSet
-- FiberBundleCore.localTriv_apply
@[simp, mfld_simps]
theorem localTriv_apply (p : Z.TotalSpace) :
    (Z.localTriv i) p = ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩ 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι) (p : Z.TotalSpace)
Goal: ↑(Z.localTriv i) p = (p.proj, Z.coordChange (Z.indexAt p.proj) i p.proj p.snd)
-- FiberBundleCore.localTrivAt_apply
@[simp, mfld_simps]
theorem localTrivAt_apply (p : Z.TotalSpace) : (Z.localTrivAt p.1) p = ⟨p.1, p.2⟩ 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (p : Z.TotalSpace)
Goal: ↑(Z.localTrivAt p.proj) p = (p.proj, p.snd)
-- FiberBundleCore.localTrivAt_apply_mk
@[simp, mfld_simps]
theorem localTrivAt_apply_mk (b : B) (a : F) : (Z.localTrivAt b) ⟨b, a⟩ = ⟨b, a⟩ 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (b : B) (a : F)
Goal: ↑(Z.localTrivAt b) { proj := b, snd := a } = (b, a)
-- FiberBundleCore.mem_localTriv_source
@[simp, mfld_simps]
theorem mem_localTriv_source (p : Z.TotalSpace) :
    p ∈ (Z.localTriv i).source ↔ p.1 ∈ (Z.localTriv i).baseSet 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι) (p : Z.TotalSpace)
Goal: p ∈ (Z.localTriv i).source ↔ p.proj ∈ (Z.localTriv i).baseSet
-- FiberBundleCore.mem_localTrivAt_source
@[simp, mfld_simps]
theorem mem_localTrivAt_source (p : Z.TotalSpace) (b : B) :
    p ∈ (Z.localTrivAt b).source ↔ p.1 ∈ (Z.localTrivAt b).baseSet 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (p : Z.TotalSpace) (b : B)
Goal: p ∈ (Z.localTrivAt b).source ↔ p.proj ∈ (Z.localTrivAt b).baseSet
-- FiberBundleCore.mem_localTriv_target
@[simp, mfld_simps]
theorem mem_localTriv_target (p : B × F) :
    p ∈ (Z.localTriv i).target ↔ p.1 ∈ (Z.localTriv i).baseSet 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι) (p : B × F)
Goal: p ∈ (Z.localTriv i).target ↔ p.1 ∈ (Z.localTriv i).baseSet
-- FiberBundleCore.mem_localTrivAt_target
@[simp, mfld_simps]
theorem mem_localTrivAt_target (p : B × F) (b : B) :
    p ∈ (Z.localTrivAt b).target ↔ p.1 ∈ (Z.localTrivAt b).baseSet 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (p : B × F) (b : B)
Goal: p ∈ (Z.localTrivAt b).target ↔ p.1 ∈ (Z.localTrivAt b).baseSet
-- FiberBundleCore.localTriv_symm_apply
@[simp, mfld_simps]
theorem localTriv_symm_apply (p : B × F) :
    (Z.localTriv i).toPartialHomeomorph.symm p = ⟨p.1, Z.coordChange i (Z.indexAt p.1) p.1 p.2⟩ 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (i : ι) (p : B × F)
Goal: ↑(Z.localTriv i).symm p = { proj := p.1, snd := Z.coordChange i (Z.indexAt p.1) p.1 p.2 }
-- FiberBundleCore.mem_localTrivAt_baseSet
@[simp, mfld_simps]
theorem mem_localTrivAt_baseSet (b : B) : b ∈ (Z.localTrivAt b).baseSet 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (b : B)
Goal: b ∈ (Z.localTrivAt b).baseSet
-- FiberBundleCore.mk_mem_localTrivAt_source
theorem mk_mem_localTrivAt_source : (⟨b, a⟩ : Z.TotalSpace) ∈ (Z.localTrivAt b).source 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (b : B) (a : F)
Goal: { proj := b, snd := a } ∈ (Z.localTrivAt b).source
-- FiberBundleCore.continuous_totalSpaceMk
@[continuity]
theorem continuous_totalSpaceMk (b : B) :
    Continuous (TotalSpace.mk b : Z.Fiber b → Bundle.TotalSpace F Z.Fiber) 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F) (b : B)
Goal: Continuous (Bundle.TotalSpace.mk b)
-- FiberBundleCore.continuous_proj
nonrec theorem continuous_proj : Continuous Z.proj 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F)
Goal: Continuous Z.proj
-- FiberBundleCore.isOpenMap_proj
nonrec theorem isOpenMap_proj : IsOpenMap Z.proj 
Arguments: {ι : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [TopologicalSpace F] (Z : FiberBundleCore ι B F)
Goal: IsOpenMap Z.proj
-- FiberPrebundle.totalSpaceMk_inducing
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure FiberPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  continuous_trivChange : ∀ e, e ∈ pretrivializationAtlas → ∀ e', e' ∈ pretrivializationAtlas →
    ContinuousOn (e ∘ e'.toPartialEquiv.symm) (e'.target ∩ e'.toPartialEquiv.symm ⁻¹' e.source)
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ TotalSpace.mk b)
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (self : FiberPrebundle F E) (b : B)
Goal: Inducing (↑(self.pretrivializationAt b) ∘ Bundle.TotalSpace.mk b)
-- FiberPrebundle.continuous_trivChange
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure FiberPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  continuous_trivChange : ∀ e, e ∈ pretrivializationAtlas → ∀ e', e' ∈ pretrivializationAtlas →
    ContinuousOn (e ∘ e'.toPartialEquiv.symm) (e'.target ∩ e'.toPartialEquiv.symm ⁻¹' e.source)
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ TotalSpace.mk b)
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (self : FiberPrebundle F E) (e : Pretrivialization F Bundle.TotalSpace.proj)
Goal: e ∈ self.pretrivializationAtlas →
  ∀ e' ∈ self.pretrivializationAtlas, ContinuousOn (↑e ∘ ↑e'.symm) (e'.target ∩ ↑e'.symm ⁻¹' e.source)
-- FiberPrebundle.pretrivialization_mem_atlas
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure FiberPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  continuous_trivChange : ∀ e, e ∈ pretrivializationAtlas → ∀ e', e' ∈ pretrivializationAtlas →
    ContinuousOn (e ∘ e'.toPartialEquiv.symm) (e'.target ∩ e'.toPartialEquiv.symm ⁻¹' e.source)
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ TotalSpace.mk b)
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (self : FiberPrebundle F E) (x : B)
Goal: self.pretrivializationAt x ∈ self.pretrivializationAtlas
-- FiberPrebundle.mem_base_pretrivializationAt
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure FiberPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  continuous_trivChange : ∀ e, e ∈ pretrivializationAtlas → ∀ e', e' ∈ pretrivializationAtlas →
    ContinuousOn (e ∘ e'.toPartialEquiv.symm) (e'.target ∩ e'.toPartialEquiv.symm ⁻¹' e.source)
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ TotalSpace.mk b)
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (self : FiberPrebundle F E) (x : B)
Goal: x ∈ (self.pretrivializationAt x).baseSet
-- FiberPrebundle.continuous_symm_of_mem_pretrivializationAtlas
theorem continuous_symm_of_mem_pretrivializationAtlas (he : e ∈ a.pretrivializationAtlas) :
    @ContinuousOn _ _ _ a.totalSpaceTopology e.toPartialEquiv.symm e.target 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) {e : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas)
Goal: ContinuousOn (↑e.symm) e.target
-- FiberPrebundle.isOpen_source
theorem isOpen_source (e : Pretrivialization F (π F E)) :
    IsOpen[a.totalSpaceTopology] e.source 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) (e : Pretrivialization F Bundle.TotalSpace.proj)
Goal: IsOpen e.source
-- FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter
theorem isOpen_target_of_mem_pretrivializationAtlas_inter (e e' : Pretrivialization F (π F E))
    (he' : e' ∈ a.pretrivializationAtlas) :
    IsOpen (e'.toPartialEquiv.target ∩ e'.toPartialEquiv.symm ⁻¹' e.source) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) (e e' : Pretrivialization F Bundle.TotalSpace.proj) (he' : e' ∈ a.pretrivializationAtlas)
Goal: IsOpen (e'.target ∩ ↑e'.symm ⁻¹' e.source)
-- FiberPrebundle.mem_pretrivializationAt_source
theorem mem_pretrivializationAt_source (b : B) (x : E b) :
    ⟨b, x⟩ ∈ (a.pretrivializationAt b).source 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) (b : B) (x : E b)
Goal: { proj := b, snd := x } ∈ (a.pretrivializationAt b).source
-- FiberPrebundle.totalSpaceMk_preimage_source
@[simp]
theorem totalSpaceMk_preimage_source (b : B) :
    TotalSpace.mk b ⁻¹' (a.pretrivializationAt b).source = univ 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) (b : B)
Goal: Bundle.TotalSpace.mk b ⁻¹' (a.pretrivializationAt b).source = Set.univ
-- FiberPrebundle.continuous_totalSpaceMk
@[continuity]
theorem continuous_totalSpaceMk (b : B) :
    Continuous[_, a.totalSpaceTopology] (TotalSpace.mk b) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) (b : B)
Goal: Continuous (Bundle.TotalSpace.mk b)
-- FiberPrebundle.inducing_totalSpaceMk_of_inducing_comp
theorem inducing_totalSpaceMk_of_inducing_comp (b : B)
    (h : Inducing (a.pretrivializationAt b ∘ TotalSpace.mk b)) :
    @Inducing _ _ _ a.totalSpaceTopology (TotalSpace.mk b) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) (b : B) (h : Inducing (↑(a.pretrivializationAt b) ∘ Bundle.TotalSpace.mk b))
Goal: Inducing (Bundle.TotalSpace.mk b)
-- FiberPrebundle.continuous_proj
theorem continuous_proj : @Continuous _ _ a.totalSpaceTopology _ (π F E) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E)
Goal: Continuous Bundle.TotalSpace.proj
-- FiberPrebundle.continuousOn_of_comp_right
theorem continuousOn_of_comp_right {X : Type*} [TopologicalSpace X] {f : TotalSpace F E → X}
    {s : Set B} (hs : IsOpen s) (hf : ∀ b ∈ s,
      ContinuousOn (f ∘ (a.pretrivializationAt b).toPartialEquiv.symm)
        ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ (Set.univ : Set F))) :
    @ContinuousOn _ _ a.totalSpaceTopology _ f (π F E ⁻¹' s) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → TopologicalSpace (E x)] (a : FiberPrebundle F E) {X : Type u_6} [TopologicalSpace X] {f : Bundle.TotalSpace F E → X} {s : Set B} (hs : IsOpen s) (hf : ∀ b ∈ s, ContinuousOn (f ∘ ↑(a.pretrivializationAt b).symm) ((s ∩ (a.pretrivializationAt b).baseSet) ×ˢ Set.univ))
Goal: ContinuousOn f (Bundle.TotalSpace.proj ⁻¹' s)
-- Filter.HasBasis.specializes_iff_uniformity
theorem Filter.HasBasis.specializes_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
    (h : (𝓤 α).HasBasis p s) {x y : α} : x ⤳ y ↔ ∀ i, p i → (x, y) ∈ s i 
Arguments: {α : Type u} [UniformSpace α] {ι : Sort u_1} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) {x y : α}
Goal: x ⤳ y ↔ ∀ (i : ι), p i → (x, y) ∈ s i
-- Filter.HasBasis.inseparable_iff_uniformity
theorem Filter.HasBasis.inseparable_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
    (h : (𝓤 α).HasBasis p s) {x y : α} : Inseparable x y ↔ ∀ i, p i → (x, y) ∈ s i 
Arguments: {α : Type u} [UniformSpace α] {ι : Sort u_1} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) {x y : α}
Goal: Inseparable x y ↔ ∀ (i : ι), p i → (x, y) ∈ s i
-- inseparable_iff_ker_uniformity
theorem inseparable_iff_ker_uniformity {x y : α} : Inseparable x y ↔ (x, y) ∈ (𝓤 α).ker 
Arguments: {α : Type u} [UniformSpace α] {x y : α}
Goal: Inseparable x y ↔ (x, y) ∈ (𝓤 α).ker
-- Inseparable.nhds_le_uniformity
protected theorem Inseparable.nhds_le_uniformity {x y : α} (h : Inseparable x y) :
    𝓝 (x, y) ≤ 𝓤 α 
Arguments: {α : Type u} [UniformSpace α] {x y : α} (h : Inseparable x y)
Goal: 𝓝 (x, y) ≤ 𝓤 α
-- inseparable_iff_clusterPt_uniformity
theorem inseparable_iff_clusterPt_uniformity {x y : α} :
    Inseparable x y ↔ ClusterPt (x, y) (𝓤 α) 
Arguments: {α : Type u} [UniformSpace α] {x y : α}
Goal: Inseparable x y ↔ ClusterPt (x, y) (𝓤 α)
-- t0Space_iff_uniformity
theorem t0Space_iff_uniformity :
    T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y 
Arguments: {α : Type u} [UniformSpace α]
Goal: T0Space α ↔ ∀ (x y : α), (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y
-- t0Space_iff_uniformity'
theorem t0Space_iff_uniformity' :
    T0Space α ↔ Pairwise fun x y ↦ ∃ r ∈ 𝓤 α, (x, y) ∉ r 
Arguments: {α : Type u} [UniformSpace α]
Goal: T0Space α ↔ Pairwise fun x y => ∃ r ∈ 𝓤 α, (x, y) ∉ r
-- t0Space_iff_ker_uniformity
theorem t0Space_iff_ker_uniformity : T0Space α ↔ (𝓤 α).ker = diagonal α 
Arguments: {α : Type u} [UniformSpace α]
Goal: T0Space α ↔ (𝓤 α).ker = Set.diagonal α
-- eq_of_uniformity
theorem eq_of_uniformity {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
    (h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y 
Arguments: {α : Type u_1} [UniformSpace α] [T0Space α] {x y : α} (h : ∀ {V : Set (α × α)}, V ∈ 𝓤 α → (x, y) ∈ V)
Goal: x = y