ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Trivialization.symmL_apply
@[simps (config := .asFn) apply]
def symmL (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : F →L[R] E b 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B)
Goal: ⇑(Trivialization.symmL R e b) = e.symm b
-- Trivialization.symmL_continuousLinearMapAt
theorem symmL_continuousLinearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) (y : E b) : e.symmL R b (e.continuousLinearMapAt R b y) = y 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet) (y : E b)
Goal: (Trivialization.symmL R e b) ((Trivialization.continuousLinearMapAt R e b) y) = y
-- Trivialization.continuousLinearMapAt_symmL
theorem continuousLinearMapAt_symmL (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) (y : F) : e.continuousLinearMapAt R b (e.symmL R b y) = y 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: (Trivialization.continuousLinearMapAt R e b) ((Trivialization.symmL R e b) y) = y
-- Trivialization.continuousLinearEquivAt_apply
@[simps (config := .asFn) apply symm_apply]
def continuousLinearEquivAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
    (hb : b ∈ e.baseSet) : E b ≃L[R] F 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet)
Goal: ⇑(Trivialization.continuousLinearEquivAt R e b hb) = fun y => (↑e { proj := b, snd := y }).2
-- Trivialization.continuousLinearEquivAt_symm_apply
@[simps (config := .asFn) apply symm_apply]
def continuousLinearEquivAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
    (hb : b ∈ e.baseSet) : E b ≃L[R] F 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet)
Goal: ⇑(Trivialization.continuousLinearEquivAt R e b hb).symm = e.symm b
-- Trivialization.coe_continuousLinearEquivAt_eq
theorem coe_continuousLinearEquivAt_eq (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) :
    (e.continuousLinearEquivAt R b hb : E b → F) = e.continuousLinearMapAt R b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: ⇑(Trivialization.continuousLinearEquivAt R e b hb) = ⇑(Trivialization.continuousLinearMapAt R e b)
-- Trivialization.symm_continuousLinearEquivAt_eq
theorem symm_continuousLinearEquivAt_eq (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) : ((e.continuousLinearEquivAt R b hb).symm : F → E b) = e.symmL R b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: ⇑(Trivialization.continuousLinearEquivAt R e b hb).symm = ⇑(Trivialization.symmL R e b)
-- Trivialization.continuousLinearEquivAt_apply'
@[simp]
theorem continuousLinearEquivAt_apply' (e : Trivialization F (π F E)) [e.IsLinear R]
    (x : TotalSpace F E) (hx : x ∈ e.source) :
    e.continuousLinearEquivAt R x.proj (e.mem_source.1 hx) x.2 = (e x).2 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (x : Bundle.TotalSpace F E) (hx : x ∈ e.source)
Goal: (Trivialization.continuousLinearEquivAt R e x.proj ⋯) x.snd = (↑e x).2
-- Trivialization.apply_eq_prod_continuousLinearEquivAt
theorem apply_eq_prod_continuousLinearEquivAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
    (hb : b ∈ e.baseSet) (z : E b) : e ⟨b, z⟩ = (b, e.continuousLinearEquivAt R b hb z) 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet) (z : E b)
Goal: ↑e { proj := b, snd := z } = (b, (Trivialization.continuousLinearEquivAt R e b hb) z)
-- Trivialization.zeroSection
protected theorem zeroSection (e : Trivialization F (π F E)) [e.IsLinear R] {x : B}
    (hx : x ∈ e.baseSet) : e (zeroSection F E x) = (x, 0) 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {x : B} (hx : x ∈ e.baseSet)
Goal: ↑e (Bundle.zeroSection F E x) = (x, 0)
-- Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm
theorem symm_apply_eq_mk_continuousLinearEquivAt_symm (e : Trivialization F (π F E)) [e.IsLinear R]
    (b : B) (hb : b ∈ e.baseSet) (z : F) :
    e.toPartialHomeomorph.symm ⟨b, z⟩ = ⟨b, (e.continuousLinearEquivAt R b hb).symm z⟩ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet) (z : F)
Goal: ↑e.symm (b, z) = { proj := b, snd := (Trivialization.continuousLinearEquivAt R e b hb).symm z }
-- Trivialization.comp_continuousLinearEquivAt_eq_coord_change
theorem comp_continuousLinearEquivAt_eq_coord_change (e e' : Trivialization F (π F E))
    [e.IsLinear R] [e'.IsLinear R] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) :
    (e.continuousLinearEquivAt R b hb.1).symm.trans (e'.continuousLinearEquivAt R b hb.2) =
      coordChangeL R e e' b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (e e' : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] [Trivialization.IsLinear R e'] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet)
Goal: (Trivialization.continuousLinearEquivAt R e b ⋯).symm.trans (Trivialization.continuousLinearEquivAt R e' b ⋯) =
  Trivialization.coordChangeL R e e' b
-- VectorBundleCore.coordChange_self
structure VectorBundleCore (ι : Type*) 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 →L[R] F
  coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coordChange i i x v = v
  continuousOn_coordChange : ∀ i j, ContinuousOn (coordChange i j) (baseSet i ∩ baseSet j)
  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: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (self : VectorBundleCore R B F ι) (i : ι) (x : B)
Goal: x ∈ self.baseSet i → ∀ (v : F), (self.coordChange i i x) v = v
-- VectorBundleCore.mem_baseSet_at
structure VectorBundleCore (ι : Type*) 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 →L[R] F
  coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coordChange i i x v = v
  continuousOn_coordChange : ∀ i j, ContinuousOn (coordChange i j) (baseSet i ∩ baseSet j)
  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: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (self : VectorBundleCore R B F ι) (x : B)
Goal: x ∈ self.baseSet (self.indexAt x)
-- VectorBundleCore.isOpen_baseSet
structure VectorBundleCore (ι : Type*) 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 →L[R] F
  coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coordChange i i x v = v
  continuousOn_coordChange : ∀ i j, ContinuousOn (coordChange i j) (baseSet i ∩ baseSet j)
  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: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (self : VectorBundleCore R B F ι) (i : ι)
Goal: IsOpen (self.baseSet i)
-- VectorBundleCore.continuousOn_coordChange
structure VectorBundleCore (ι : Type*) 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 →L[R] F
  coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coordChange i i x v = v
  continuousOn_coordChange : ∀ i j, ContinuousOn (coordChange i j) (baseSet i ∩ baseSet j)
  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: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (self : VectorBundleCore R B F ι) (i j : ι)
Goal: ContinuousOn (self.coordChange i j) (self.baseSet i ∩ self.baseSet j)
-- VectorBundleCore.coordChange_comp
structure VectorBundleCore (ι : Type*) 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 →L[R] F
  coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coordChange i i x v = v
  continuousOn_coordChange : ∀ i j, ContinuousOn (coordChange i j) (baseSet i ∩ baseSet j)
  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: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (self : VectorBundleCore R B F ι) (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
-- VectorBundleCore.toFiberBundleCore_indexAt
@[simps (config := mfld_cfg)]
def toFiberBundleCore : FiberBundleCore ι B F 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι)
Goal: Z.toFiberBundleCore.indexAt = Z.indexAt
-- VectorBundleCore.toFiberBundleCore_coordChange
@[simps (config := mfld_cfg)]
def toFiberBundleCore : FiberBundleCore ι B F 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι)
Goal: Z.toFiberBundleCore.coordChange = fun i j b => ⇑(Z.coordChange i j b)
-- VectorBundleCore.toFiberBundleCore_baseSet
@[simps (config := mfld_cfg)]
def toFiberBundleCore : FiberBundleCore ι B F 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι)
Goal: Z.toFiberBundleCore.baseSet = Z.baseSet
-- VectorBundleCore.coordChange_linear_comp
theorem coordChange_linear_comp (i j k : ι) :
    ∀ x ∈ Z.baseSet i ∩ Z.baseSet j ∩ Z.baseSet k,
      (Z.coordChange j k x).comp (Z.coordChange i j x) = Z.coordChange i k x 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (i j k : ι) (x : B)
Goal: x ∈ Z.baseSet i ∩ Z.baseSet j ∩ Z.baseSet k → (Z.coordChange j k x).comp (Z.coordChange i j x) = Z.coordChange i k x
-- VectorBundleCore.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: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (i j : ι) (p : B × F)
Goal: p ∈ (Z.trivChange i j).source ↔ p.1 ∈ Z.baseSet i ∩ Z.baseSet j
-- VectorBundleCore.coe_coordChange
@[simp, mfld_simps]
theorem coe_coordChange (i j : ι) : Z.toFiberBundleCore.coordChange i j b = Z.coordChange i j b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (b : B) (i j : ι)
Goal: Z.toFiberBundleCore.coordChange i j b = ⇑(Z.coordChange i j b)
-- VectorBundleCore.localTriv_apply
@[simp, mfld_simps]
theorem localTriv_apply {i : ι} (p : Z.TotalSpace) :
    (Z.localTriv i) p = ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) {i : ι} (p : Z.TotalSpace)
Goal: ↑(Z.localTriv i) p = (p.proj, (Z.coordChange (Z.indexAt p.proj) i p.proj) p.snd)
-- VectorBundleCore.mem_localTriv_source
@[simp, mfld_simps]
theorem mem_localTriv_source (p : Z.TotalSpace) : p ∈ (Z.localTriv i).source ↔ p.1 ∈ Z.baseSet i 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (i : ι) (p : Z.TotalSpace)
Goal: p ∈ (Z.localTriv i).source ↔ p.proj ∈ Z.baseSet i
-- VectorBundleCore.baseSet_at
@[simp, mfld_simps]
theorem baseSet_at : Z.baseSet i = (Z.localTriv i).baseSet 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (i : ι)
Goal: Z.baseSet i = (Z.localTriv i).baseSet
-- VectorBundleCore.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: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (i : ι) (p : B × F)
Goal: p ∈ (Z.localTriv i).target ↔ p.1 ∈ (Z.localTriv i).baseSet
-- VectorBundleCore.localTriv_symm_fst
@[simp, mfld_simps]
theorem localTriv_symm_fst (p : B × F) :
    (Z.localTriv i).toPartialHomeomorph.symm p = ⟨p.1, Z.coordChange i (Z.indexAt p.1) p.1 p.2⟩ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (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 }
-- VectorBundleCore.localTriv_symm_apply
@[simp, mfld_simps]
theorem localTriv_symm_apply {b : B} (hb : b ∈ Z.baseSet i) (v : F) :
    (Z.localTriv i).symm b v = Z.coordChange i (Z.indexAt b) b v 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (i : ι) {b : B} (hb : b ∈ Z.baseSet i) (v : F)
Goal: (Z.localTriv i).symm b v = (Z.coordChange i (Z.indexAt b) b) v
-- VectorBundleCore.localTriv_coordChange_eq
@[simp, mfld_simps]
theorem localTriv_coordChange_eq {b : B} (hb : b ∈ Z.baseSet i ∩ Z.baseSet j) (v : F) :
    (Z.localTriv i).coordChangeL R (Z.localTriv j) b v = Z.coordChange i j b v 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (i j : ι) {b : B} (hb : b ∈ Z.baseSet i ∩ Z.baseSet j) (v : F)
Goal: (Trivialization.coordChangeL R (Z.localTriv i) (Z.localTriv j) b) v = (Z.coordChange i j b) v
-- VectorBundleCore.localTrivAt_def
@[simp, mfld_simps]
theorem localTrivAt_def : Z.localTriv (Z.indexAt b) = Z.localTrivAt b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (b : B)
Goal: Z.localTriv (Z.indexAt b) = Z.localTrivAt b
-- VectorBundleCore.mem_source_at
@[simp, mfld_simps]
theorem mem_source_at : (⟨b, a⟩ : Z.TotalSpace) ∈ (Z.localTrivAt b).source 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (b : B) (a : F)
Goal: { proj := b, snd := a } ∈ (Z.localTrivAt b).source
-- VectorBundleCore.localTrivAt_apply
@[simp, mfld_simps]
theorem localTrivAt_apply (p : Z.TotalSpace) : Z.localTrivAt p.1 p = ⟨p.1, p.2⟩ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (p : Z.TotalSpace)
Goal: ↑(Z.localTrivAt p.proj) p = (p.proj, p.snd)
-- VectorBundleCore.localTrivAt_apply_mk
@[simp, mfld_simps]
theorem localTrivAt_apply_mk (b : B) (a : F) : Z.localTrivAt b ⟨b, a⟩ = ⟨b, a⟩ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (b : B) (a : F)
Goal: ↑(Z.localTrivAt b) { proj := b, snd := a } = (b, a)
-- VectorBundleCore.mem_localTrivAt_baseSet
@[simp, mfld_simps]
theorem mem_localTrivAt_baseSet : b ∈ (Z.localTrivAt b).baseSet 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) (b : B)
Goal: b ∈ (Z.localTrivAt b).baseSet
-- VectorBundleCore.continuous_proj
@[continuity]
theorem continuous_proj : Continuous Z.proj 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι)
Goal: Continuous Z.proj
-- VectorBundleCore.isOpenMap_proj
theorem isOpenMap_proj : IsOpenMap Z.proj 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι)
Goal: IsOpenMap Z.proj
-- VectorBundleCore.localTriv_continuousLinearMapAt
@[simp, mfld_simps]
theorem localTriv_continuousLinearMapAt {b : B} (hb : b ∈ Z.baseSet i) :
    (Z.localTriv i).continuousLinearMapAt R b = Z.coordChange (Z.indexAt b) i b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) {i : ι} {b : B} (hb : b ∈ Z.baseSet i)
Goal: Trivialization.continuousLinearMapAt R (Z.localTriv i) b = Z.coordChange (Z.indexAt b) i b
-- VectorBundleCore.trivializationAt_continuousLinearMapAt
@[simp, mfld_simps]
theorem trivializationAt_continuousLinearMapAt {b₀ b : B}
    (hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet) :
    (trivializationAt F Z.Fiber b₀).continuousLinearMapAt R b =
      Z.coordChange (Z.indexAt b) (Z.indexAt b₀) b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) {b₀ b : B} (hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet)
Goal: Trivialization.continuousLinearMapAt R (trivializationAt F Z.Fiber b₀) b = Z.coordChange (Z.indexAt b) (Z.indexAt b₀) b
-- VectorBundleCore.localTriv_symmL
@[simp, mfld_simps]
theorem localTriv_symmL {b : B} (hb : b ∈ Z.baseSet i) :
    (Z.localTriv i).symmL R b = Z.coordChange i (Z.indexAt b) b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) {i : ι} {b : B} (hb : b ∈ Z.baseSet i)
Goal: Trivialization.symmL R (Z.localTriv i) b = Z.coordChange i (Z.indexAt b) b
-- VectorBundleCore.trivializationAt_symmL
@[simp, mfld_simps]
theorem trivializationAt_symmL {b₀ b : B} (hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet) :
    (trivializationAt F Z.Fiber b₀).symmL R b = Z.coordChange (Z.indexAt b₀) (Z.indexAt b) b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) {b₀ b : B} (hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet)
Goal: Trivialization.symmL R (trivializationAt F Z.Fiber b₀) b = Z.coordChange (Z.indexAt b₀) (Z.indexAt b) b