ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Pretrivialization.apply_mk_symm
theorem apply_mk_symm (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
    e ⟨b, e.symm b y⟩ = (b, y) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → Zero (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: ↑e { proj := b, snd := e.symm b y } = (b, y)
-- Trivialization.source_eq
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure Trivialization (proj : Z → B) extends PartialHomeomorph Z (B × F) where
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toPartialHomeomorph p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace Z] {proj : Z → B} (self : Trivialization F proj)
Goal: self.source = proj ⁻¹' self.baseSet
-- Trivialization.proj_toFun
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure Trivialization (proj : Z → B) extends PartialHomeomorph Z (B × F) where
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toPartialHomeomorph p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace Z] {proj : Z → B} (self : Trivialization F proj) (p : Z)
Goal: p ∈ self.source → (↑self.toPartialHomeomorph p).1 = proj p
-- Trivialization.target_eq
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure Trivialization (proj : Z → B) extends PartialHomeomorph Z (B × F) where
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toPartialHomeomorph p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace Z] {proj : Z → B} (self : Trivialization F proj)
Goal: self.target = self.baseSet ×ˢ Set.univ
-- Trivialization.open_baseSet
-- Porting note (#5171): was @[nolint has_nonempty_instance]
structure Trivialization (proj : Z → B) extends PartialHomeomorph Z (B × F) where
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toPartialHomeomorph p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace Z] {proj : Z → B} (self : Trivialization F proj)
Goal: IsOpen self.baseSet
-- Trivialization.ext'
@[ext]
lemma ext' (e e' : Trivialization F proj) (h₁ : e.toPartialHomeomorph = e'.toPartialHomeomorph)
    (h₂ : e.baseSet = e'.baseSet) : e = e' 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e e' : Trivialization F proj) (h₁ : e.toPartialHomeomorph = e'.toPartialHomeomorph) (h₂ : e.baseSet = e'.baseSet)
Goal: e = e'
-- Trivialization.toPretrivialization_injective
theorem toPretrivialization_injective :
    Function.Injective fun e : Trivialization F proj => e.toPretrivialization 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z]
Goal: Function.Injective fun e => e.toPretrivialization
-- Trivialization.coe_coe
@[simp, mfld_simps]
theorem coe_coe : ⇑e.toPartialHomeomorph = e 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj)
Goal: ↑e.toPartialHomeomorph = ↑e
-- Trivialization.coe_fst
@[simp, mfld_simps]
theorem coe_fst (ex : x ∈ e.source) : (e x).1 = proj x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: (↑e x).1 = proj x
-- Trivialization.eqOn
protected theorem eqOn : EqOn (Prod.fst ∘ e) proj e.source 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj)
Goal: Set.EqOn (Prod.fst ∘ ↑e) proj e.source
-- Trivialization.mem_source
theorem mem_source : x ∈ e.source ↔ proj x ∈ e.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z}
Goal: x ∈ e.source ↔ proj x ∈ e.baseSet
-- Trivialization.coe_fst'
theorem coe_fst' (ex : proj x ∈ e.baseSet) : (e x).1 = proj x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : proj x ∈ e.baseSet)
Goal: (↑e x).1 = proj x
-- Trivialization.mk_proj_snd
theorem mk_proj_snd (ex : x ∈ e.source) : (proj x, (e x).2) = e x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: (proj x, (↑e x).2) = ↑e x
-- Trivialization.mk_proj_snd'
theorem mk_proj_snd' (ex : proj x ∈ e.baseSet) : (proj x, (e x).2) = e x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : proj x ∈ e.baseSet)
Goal: (proj x, (↑e x).2) = ↑e x
-- Trivialization.source_inter_preimage_target_inter
theorem source_inter_preimage_target_inter (s : Set (B × F)) :
    e.source ∩ e ⁻¹' (e.target ∩ s) = e.source ∩ e ⁻¹' s 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) (s : Set (B × F))
Goal: e.source ∩ ↑e ⁻¹' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' s
-- Trivialization.coe_mk
@[simp, mfld_simps]
theorem coe_mk (e : PartialHomeomorph Z (B × F)) (i j k l m) (x : Z) :
    (Trivialization.mk e i j k l m : Trivialization F proj) x = e x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : PartialHomeomorph Z (B × F)) (i : Set B) (j : IsOpen i) (k : e.source = proj ⁻¹' i) (l : e.target = i ×ˢ Set.univ) (m : ∀ p ∈ e.source, (↑e p).1 = proj p) (x : Z)
Goal: ↑{ toPartialHomeomorph := e, baseSet := i, open_baseSet := j, source_eq := k, target_eq := l, proj_toFun := m } x = ↑e x
-- Trivialization.mem_target
theorem mem_target {x : B × F} : x ∈ e.target ↔ x.1 ∈ e.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : B × F}
Goal: x ∈ e.target ↔ x.1 ∈ e.baseSet
-- Trivialization.map_target
theorem map_target {x : B × F} (hx : x ∈ e.target) : e.toPartialHomeomorph.symm x ∈ e.source 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : B × F} (hx : x ∈ e.target)
Goal: ↑e.symm x ∈ e.source
-- Trivialization.proj_symm_apply
theorem proj_symm_apply {x : B × F} (hx : x ∈ e.target) :
    proj (e.toPartialHomeomorph.symm x) = x.1 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : B × F} (hx : x ∈ e.target)
Goal: proj (↑e.symm x) = x.1
-- Trivialization.proj_symm_apply'
theorem proj_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
    proj (e.toPartialHomeomorph.symm (b, x)) = b 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {b : B} {x : F} (hx : b ∈ e.baseSet)
Goal: proj (↑e.symm (b, x)) = b
-- Trivialization.proj_surjOn_baseSet
theorem proj_surjOn_baseSet [Nonempty F] : Set.SurjOn proj e.source e.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) [Nonempty F]
Goal: Set.SurjOn proj e.source e.baseSet
-- Trivialization.apply_symm_apply
theorem apply_symm_apply {x : B × F} (hx : x ∈ e.target) : e (e.toPartialHomeomorph.symm x) = x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : B × F} (hx : x ∈ e.target)
Goal: ↑e (↑e.symm x) = x
-- Trivialization.apply_symm_apply'
theorem apply_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
    e (e.toPartialHomeomorph.symm (b, x)) = (b, x) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {b : B} {x : F} (hx : b ∈ e.baseSet)
Goal: ↑e (↑e.symm (b, x)) = (b, x)
-- Trivialization.symm_apply_mk_proj
@[simp, mfld_simps]
theorem symm_apply_mk_proj (ex : x ∈ e.source) : e.toPartialHomeomorph.symm (proj x, (e x).2) = x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: ↑e.symm (proj x, (↑e x).2) = x
-- Trivialization.symm_trans_source_eq
theorem symm_trans_source_eq (e e' : Trivialization F proj) :
    (e.toPartialEquiv.symm.trans e'.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e e' : Trivialization F proj)
Goal: (e.symm.trans e'.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ Set.univ
-- Trivialization.symm_trans_target_eq
theorem symm_trans_target_eq (e e' : Trivialization F proj) :
    (e.toPartialEquiv.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e e' : Trivialization F proj)
Goal: (e.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ Set.univ
-- Trivialization.coe_fst_eventuallyEq_proj
theorem coe_fst_eventuallyEq_proj (ex : x ∈ e.source) : Prod.fst ∘ e =ᶠ[𝓝 x] proj 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: Prod.fst ∘ ↑e =ᶠ[𝓝 x] proj
-- Trivialization.coe_fst_eventuallyEq_proj'
theorem coe_fst_eventuallyEq_proj' (ex : proj x ∈ e.baseSet) : Prod.fst ∘ e =ᶠ[𝓝 x] proj 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : proj x ∈ e.baseSet)
Goal: Prod.fst ∘ ↑e =ᶠ[𝓝 x] proj
-- Trivialization.map_proj_nhds
theorem map_proj_nhds (ex : x ∈ e.source) : map proj (𝓝 x) = 𝓝 (proj x) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: Filter.map proj (𝓝 x) = 𝓝 (proj x)
-- Trivialization.preimage_subset_source
theorem preimage_subset_source {s : Set B} (hb : s ⊆ e.baseSet) : proj ⁻¹' s ⊆ e.source 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {s : Set B} (hb : s ⊆ e.baseSet)
Goal: proj ⁻¹' s ⊆ e.source
-- Trivialization.image_preimage_eq_prod_univ
theorem image_preimage_eq_prod_univ {s : Set B} (hb : s ⊆ e.baseSet) :
    e '' (proj ⁻¹' s) = s ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {s : Set B} (hb : s ⊆ e.baseSet)
Goal: ↑e '' (proj ⁻¹' s) = s ×ˢ Set.univ
-- Trivialization.tendsto_nhds_iff
theorem tendsto_nhds_iff {α : Type*} {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) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {α : Type u_6} {l : Filter α} {f : α → Z} {z : Z} (hz : z ∈ e.source)
Goal: Filter.Tendsto f l (𝓝 z) ↔
  Filter.Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Filter.Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2)
-- Trivialization.nhds_eq_inf_comap
theorem nhds_eq_inf_comap {z : Z} (hz : z ∈ e.source) :
    𝓝 z = comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ e) (𝓝 (e z).2) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {z : Z} (hz : z ∈ e.source)
Goal: 𝓝 z = Filter.comap proj (𝓝 (proj z)) ⊓ Filter.comap (Prod.snd ∘ ↑e) (𝓝 (↑e z).2)
-- Trivialization.preimageHomeomorph_apply
@[simp]
theorem preimageHomeomorph_apply {s : Set B} (hb : s ⊆ e.baseSet) (p : proj ⁻¹' s) :
    e.preimageHomeomorph hb p = (⟨proj p, p.2⟩, (e p).2) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {s : Set B} (hb : s ⊆ e.baseSet) (p : ↑(proj ⁻¹' s))
Goal: (e.preimageHomeomorph hb) p = (⟨proj ↑p, ⋯⟩, (↑e ↑p).2)
-- Trivialization.preimageHomeomorph_symm_apply
@[simp]
theorem preimageHomeomorph_symm_apply {s : Set B} (hb : s ⊆ e.baseSet) (p : s × F) :
    (e.preimageHomeomorph hb).symm p =
      ⟨e.symm (p.1, p.2), ((preimageHomeomorph_symm_apply.aux e hb) p).2⟩ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {s : Set B} (hb : s ⊆ e.baseSet) (p : ↑s × F)
Goal: (e.preimageHomeomorph hb).symm p = ⟨↑e.symm (↑p.1, p.2), ⋯⟩
-- Trivialization.sourceHomeomorphBaseSetProd_apply
@[simp]
theorem sourceHomeomorphBaseSetProd_apply (p : e.source) :
    e.sourceHomeomorphBaseSetProd p = (⟨proj p, e.mem_source.mp p.2⟩, (e p).2) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) (p : ↑e.source)
Goal: e.sourceHomeomorphBaseSetProd p = (⟨proj ↑p, ⋯⟩, (↑e ↑p).2)
-- Trivialization.sourceHomeomorphBaseSetProd_symm_apply
@[simp]
theorem sourceHomeomorphBaseSetProd_symm_apply (p : e.baseSet × F) :
    e.sourceHomeomorphBaseSetProd.symm p =
      ⟨e.symm (p.1, p.2), (sourceHomeomorphBaseSetProd_symm_apply.aux e p).2⟩ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) (p : ↑e.baseSet × F)
Goal: e.sourceHomeomorphBaseSetProd.symm p = ⟨↑e.symm (↑p.1, p.2), ⋯⟩
-- Trivialization.preimageSingletonHomeomorph_apply
@[simp]
theorem preimageSingletonHomeomorph_apply {b : B} (hb : b ∈ e.baseSet) (p : proj ⁻¹' {b}) :
    e.preimageSingletonHomeomorph hb p = (e p).2 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {b : B} (hb : b ∈ e.baseSet) (p : ↑(proj ⁻¹' {b}))
Goal: (e.preimageSingletonHomeomorph hb) p = (↑e ↑p).2
-- Trivialization.preimageSingletonHomeomorph_symm_apply
@[simp]
theorem preimageSingletonHomeomorph_symm_apply {b : B} (hb : b ∈ e.baseSet) (p : F) :
    (e.preimageSingletonHomeomorph hb).symm p =
      ⟨e.symm (b, p), by rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff]⟩ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {b : B} (hb : b ∈ e.baseSet) (p : F)
Goal: (e.preimageSingletonHomeomorph hb).symm p = ⟨↑e.symm (b, p), ⋯⟩
-- Trivialization.continuousAt_proj
theorem continuousAt_proj (ex : x ∈ e.source) : ContinuousAt proj x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Trivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: ContinuousAt proj x
-- Trivialization.continuousAt_of_comp_right
theorem continuousAt_of_comp_right {X : Type*} [TopologicalSpace X] {f : Z → X} {z : Z}
    (e : Trivialization F proj) (he : proj z ∈ e.baseSet)
    (hf : ContinuousAt (f ∘ e.toPartialEquiv.symm) (e z)) : ContinuousAt f z 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] {X : Type u_6} [TopologicalSpace X] {f : Z → X} {z : Z} (e : Trivialization F proj) (he : proj z ∈ e.baseSet) (hf : ContinuousAt (f ∘ ↑e.symm) (↑e z))
Goal: ContinuousAt f z
-- Trivialization.continuousAt_of_comp_left
theorem continuousAt_of_comp_left {X : Type*} [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 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] {X : Type u_6} [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)
Goal: ContinuousAt f x
-- Trivialization.continuousOn
protected theorem continuousOn : ContinuousOn e' e'.source 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] (e' : Trivialization F Bundle.TotalSpace.proj)
Goal: ContinuousOn (↑e') e'.source
-- Trivialization.coe_mem_source
theorem coe_mem_source : ↑y ∈ e'.source ↔ b ∈ e'.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] (e' : Trivialization F Bundle.TotalSpace.proj) {b : B} {y : E b}
Goal: { proj := b, snd := y } ∈ e'.source ↔ b ∈ e'.baseSet
-- Trivialization.coe_coe_fst
@[simp, mfld_simps]
theorem coe_coe_fst (hb : b ∈ e'.baseSet) : (e' y).1 = b 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] (e' : Trivialization F Bundle.TotalSpace.proj) {b : B} {y : E b} (hb : b ∈ e'.baseSet)
Goal: (↑e' { proj := b, snd := y }).1 = b
-- Trivialization.mk_mem_target
theorem mk_mem_target {y : F} : (b, y) ∈ e'.target ↔ b ∈ e'.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] (e' : Trivialization F Bundle.TotalSpace.proj) {b : B} {y : F}
Goal: (b, y) ∈ e'.target ↔ b ∈ e'.baseSet
-- Trivialization.symm_apply_apply
theorem symm_apply_apply {x : TotalSpace F E} (hx : x ∈ e'.source) :
    e'.toPartialHomeomorph.symm (e' x) = x 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] (e' : Trivialization F Bundle.TotalSpace.proj) {x : Bundle.TotalSpace F E} (hx : x ∈ e'.source)
Goal: ↑e'.symm (↑e' x) = x
-- Trivialization.symm_coe_proj
@[simp, mfld_simps]
theorem symm_coe_proj {x : B} {y : F} (e : Trivialization F (π F E)) (h : x ∈ e.baseSet) :
    (e.toPartialHomeomorph.symm (x, y)).1 = x 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] {x : B} {y : F} (e : Trivialization F Bundle.TotalSpace.proj) (h : x ∈ e.baseSet)
Goal: (↑e.symm (x, y)).proj = x
-- Trivialization.symm_apply
theorem symm_apply (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
    e.symm b y = cast (congr_arg E (e.symm_coe_proj hb)) (e.toPartialHomeomorph.symm (b, y)).2 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → Zero (E x)] (e : Trivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: e.symm b y = cast ⋯ (↑e.symm (b, y)).snd
-- Trivialization.symm_apply_of_not_mem
theorem symm_apply_of_not_mem (e : Trivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet) (y : F) :
    e.symm b y = 0 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → Zero (E x)] (e : Trivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∉ e.baseSet) (y : F)
Goal: e.symm b y = 0
-- Trivialization.mk_symm
theorem mk_symm (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
    TotalSpace.mk b (e.symm b y) = e.toPartialHomeomorph.symm (b, y) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → Zero (E x)] (e : Trivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: { proj := b, snd := e.symm b y } = ↑e.symm (b, y)
-- Trivialization.symm_proj_apply
theorem symm_proj_apply (e : Trivialization F (π F E)) (z : TotalSpace F E)
    (hz : z.proj ∈ e.baseSet) : e.symm z.proj (e z).2 = z.2 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → Zero (E x)] (e : Trivialization F Bundle.TotalSpace.proj) (z : Bundle.TotalSpace F E) (hz : z.proj ∈ e.baseSet)
Goal: e.symm z.proj (↑e z).2 = z.snd
-- Trivialization.symm_apply_apply_mk
theorem symm_apply_apply_mk (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : E b) :
    e.symm b (e ⟨b, y⟩).2 = y 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → Zero (E x)] (e : Trivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : E b)
Goal: e.symm b (↑e { proj := b, snd := y }).2 = y
-- Trivialization.apply_mk_symm
theorem apply_mk_symm (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
    e ⟨b, e.symm b y⟩ = (b, y) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → Zero (E x)] (e : Trivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: ↑e { proj := b, snd := e.symm b y } = (b, y)
-- Trivialization.continuousOn_symm
theorem continuousOn_symm (e : Trivialization F (π F E)) :
    ContinuousOn (fun z : B × F => TotalSpace.mk' F z.1 (e.symm z.1 z.2)) (e.baseSet ×ˢ univ) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → Zero (E x)] (e : Trivialization F Bundle.TotalSpace.proj)
Goal: ContinuousOn (fun z => Bundle.TotalSpace.mk' F z.1 (e.symm z.1 z.2)) (e.baseSet ×ˢ Set.univ)
-- Trivialization.transFiberHomeomorph_apply
@[simp]
theorem transFiberHomeomorph_apply {F' : Type*} [TopologicalSpace F'] (e : Trivialization F proj)
    (h : F ≃ₜ F') (x : Z) : e.transFiberHomeomorph h x = ((e x).1, h (e x).2) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] {F' : Type u_6} [TopologicalSpace F'] (e : Trivialization F proj) (h : F ≃ₜ F') (x : Z)
Goal: ↑(e.transFiberHomeomorph h) x = ((↑e x).1, h (↑e x).2)
-- Trivialization.mk_coordChange
theorem mk_coordChange (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
    (h₂ : b ∈ e₂.baseSet) (x : F) :
    (b, e₁.coordChange e₂ b x) = e₂ (e₁.toPartialHomeomorph.symm (b, x)) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet) (h₂ : b ∈ e₂.baseSet) (x : F)
Goal: (b, e₁.coordChange e₂ b x) = ↑e₂ (↑e₁.symm (b, x))