ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Pretrivialization.linearEquivAt_apply
@[simps (config := .asFn)]
def linearEquivAt (e : Pretrivialization 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} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet)
Goal: ⇑(Pretrivialization.linearEquivAt R e b hb) = fun y => (↑e { proj := b, snd := y }).2
-- Pretrivialization.linearEquivAt_symm_apply
@[simps (config := .asFn)]
def linearEquivAt (e : Pretrivialization 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} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet)
Goal: ⇑(Pretrivialization.linearEquivAt R e b hb).symm = e.symm b
-- Pretrivialization.coe_linearMapAt
theorem coe_linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] (b : B) :
    ⇑(e.linearMapAt R b) = fun y => if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] (b : B)
Goal: ⇑(Pretrivialization.linearMapAt R e b) = fun y => if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0
-- Pretrivialization.coe_linearMapAt_of_mem
theorem coe_linearMapAt_of_mem (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) : ⇑(e.linearMapAt R b) = fun y => (e ⟨b, y⟩).2 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: ⇑(Pretrivialization.linearMapAt R e b) = fun y => (↑e { proj := b, snd := y }).2
-- Pretrivialization.linearMapAt_apply
theorem linearMapAt_apply (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B} (y : E b) :
    e.linearMapAt R b y = if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] {b : B} (y : E b)
Goal: (Pretrivialization.linearMapAt R e b) y = if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0
-- Pretrivialization.linearMapAt_def_of_mem
theorem linearMapAt_def_of_mem (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) : e.linearMapAt R b = e.linearEquivAt R b hb 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: Pretrivialization.linearMapAt R e b = ↑(Pretrivialization.linearEquivAt R e b hb)
-- Pretrivialization.linearMapAt_def_of_not_mem
theorem linearMapAt_def_of_not_mem (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] {b : B} (hb : b ∉ e.baseSet)
Goal: Pretrivialization.linearMapAt R e b = 0
-- Pretrivialization.linearMapAt_eq_zero
theorem linearMapAt_eq_zero (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] {b : B} (hb : b ∉ e.baseSet)
Goal: Pretrivialization.linearMapAt R e b = 0
-- Pretrivialization.symmₗ_linearMapAt
theorem symmₗ_linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) (y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet) (y : E b)
Goal: (Pretrivialization.symmₗ R e b) ((Pretrivialization.linearMapAt R e b) y) = y
-- Pretrivialization.linearMapAt_symmₗ
theorem linearMapAt_symmₗ (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) (y : F) : e.linearMapAt 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} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) [Pretrivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: (Pretrivialization.linearMapAt R e b) ((Pretrivialization.symmₗ R e b) y) = y
-- Trivialization.IsLinear.linear
protected class Trivialization.IsLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
  [∀ x, Module R (E x)] (e : Trivialization F (π F E)) : Prop where
  linear : ∀ b ∈ e.baseSet, IsLinearMap R fun x : E b => (e ⟨b, x⟩).2
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] {e : Trivialization F Bundle.TotalSpace.proj} [self : Trivialization.IsLinear R e] (b : B)
Goal: b ∈ e.baseSet → IsLinearMap R fun x => (↑e { proj := b, snd := x }).2
-- Trivialization.linear
protected theorem linear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
    [∀ x, Module R (E x)] [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet) :
    IsLinearMap R fun y : E b => (e ⟨b, y⟩).2 
Arguments: (R : Type u_1) {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] (e : Trivialization F Bundle.TotalSpace.proj) [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: IsLinearMap R fun y => (↑e { proj := b, snd := y }).2
-- Trivialization.linearEquivAt_apply
@[simp]
theorem linearEquivAt_apply (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
    (hb : b ∈ e.baseSet) (v : E b) : e.linearEquivAt R b hb v = (e ⟨b, v⟩).2 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet) (v : E b)
Goal: (Trivialization.linearEquivAt R e b hb) v = (↑e { proj := b, snd := v }).2
-- Trivialization.linearEquivAt_symm_apply
@[simp]
theorem linearEquivAt_symm_apply (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
    (hb : b ∈ e.baseSet) (v : F) : (e.linearEquivAt R b hb).symm v = e.symm b v 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B) (hb : b ∈ e.baseSet) (v : F)
Goal: (Trivialization.linearEquivAt R e b hb).symm v = e.symm b v
-- Trivialization.coe_symmₗ
theorem coe_symmₗ (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) :
    ⇑(e.symmₗ R b) = e.symm b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B)
Goal: ⇑(Trivialization.symmₗ R e b) = e.symm b
-- Trivialization.coe_linearMapAt
theorem coe_linearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) :
    ⇑(e.linearMapAt R b) = fun y => if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] (b : B)
Goal: ⇑(Trivialization.linearMapAt R e b) = fun y => if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0
-- Trivialization.coe_linearMapAt_of_mem
theorem coe_linearMapAt_of_mem (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) : ⇑(e.linearMapAt R b) = fun y => (e ⟨b, y⟩).2 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: ⇑(Trivialization.linearMapAt R e b) = fun y => (↑e { proj := b, snd := y }).2
-- Trivialization.linearMapAt_apply
theorem linearMapAt_apply (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (y : E b) :
    e.linearMapAt R b y = if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (y : E b)
Goal: (Trivialization.linearMapAt R e b) y = if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0
-- Trivialization.linearMapAt_def_of_mem
theorem linearMapAt_def_of_mem (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet) : e.linearMapAt R b = e.linearEquivAt R b hb 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet)
Goal: Trivialization.linearMapAt R e b = ↑(Trivialization.linearEquivAt R e b hb)
-- Trivialization.linearMapAt_def_of_not_mem
theorem linearMapAt_def_of_not_mem (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
    (hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∉ e.baseSet)
Goal: Trivialization.linearMapAt R e b = 0
-- Trivialization.symmₗ_linearMapAt
theorem symmₗ_linearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet)
    (y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (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.linearMapAt R e b) y) = y
-- Trivialization.linearMapAt_symmₗ
theorem linearMapAt_symmₗ (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet)
    (y : F) : e.linearMapAt 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} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: (Trivialization.linearMapAt R e b) ((Trivialization.symmₗ R e b) y) = y
-- Trivialization.coe_coordChangeL
theorem coe_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet ∩ e'.baseSet) :
    ⇑(coordChangeL R e e' b) = (e.linearEquivAt R b hb.1).symm.trans (e'.linearEquivAt R b hb.2) 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (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.coordChangeL R e e' b) =
  ⇑((Trivialization.linearEquivAt R e b ⋯).symm ≪≫ₗ Trivialization.linearEquivAt R e' b ⋯)
-- Trivialization.coe_coordChangeL'
theorem coe_coordChangeL' (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet ∩ e'.baseSet) :
    (coordChangeL R e e' b).toLinearEquiv =
      (e.linearEquivAt R b hb.1).symm.trans (e'.linearEquivAt R b hb.2) 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (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.coordChangeL R e e' b).toLinearEquiv =
  (Trivialization.linearEquivAt R e b ⋯).symm ≪≫ₗ Trivialization.linearEquivAt R e' b ⋯
-- Trivialization.symm_coordChangeL
theorem symm_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
    (hb : b ∈ e'.baseSet ∩ e.baseSet) : (e.coordChangeL R e' b).symm = e'.coordChangeL R e b 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (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.coordChangeL R e e' b).symm = Trivialization.coordChangeL R e' e b
-- Trivialization.coordChangeL_apply
theorem coordChangeL_apply (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
    coordChangeL R e e' b y = (e' ⟨b, e.symm b y⟩).2 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e e' : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] [Trivialization.IsLinear R e'] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F)
Goal: (Trivialization.coordChangeL R e e' b) y = (↑e' { proj := b, snd := e.symm b y }).2
-- Trivialization.mk_coordChangeL
theorem mk_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
    (b, coordChangeL R e e' b y) = e' ⟨b, e.symm b y⟩ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e e' : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] [Trivialization.IsLinear R e'] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F)
Goal: (b, (Trivialization.coordChangeL R e e' b) y) = ↑e' { proj := b, snd := e.symm b y }
-- Trivialization.apply_symm_apply_eq_coordChangeL
theorem apply_symm_apply_eq_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R]
    [e'.IsLinear R] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
    e' (e.toPartialHomeomorph.symm (b, v)) = (b, e.coordChangeL R e' b v) 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e e' : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] [Trivialization.IsLinear R e'] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F)
Goal: ↑e' (↑e.symm (b, v)) = (b, (Trivialization.coordChangeL R e e' b) v)
-- Trivialization.coordChangeL_apply'
theorem coordChangeL_apply' (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
    (hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
    coordChangeL R e e' b y = (e' (e.toPartialHomeomorph.symm (b, y))).2 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (e e' : Trivialization F Bundle.TotalSpace.proj) [Trivialization.IsLinear R e] [Trivialization.IsLinear R e'] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F)
Goal: (Trivialization.coordChangeL R e e' b) y = (↑e' (↑e.symm (b, y))).2
-- Trivialization.coordChangeL_symm_apply
theorem coordChangeL_symm_apply (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R]
    {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) :
    ⇑(coordChangeL R e e' b).symm =
      (e'.linearEquivAt R b hb.2).symm.trans (e.linearEquivAt R b hb.1) 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [Semiring R] [TopologicalSpace F] [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [AddCommMonoid F] [Module R F] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] (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.coordChangeL R e e' b).symm =
  ⇑((Trivialization.linearEquivAt R e' b ⋯).symm ≪≫ₗ Trivialization.linearEquivAt R e b ⋯)
-- Bundle.zeroSection_proj
@[simp, mfld_simps]
theorem zeroSection_proj [∀ x, Zero (E x)] (x : B) : (zeroSection F E x).proj = x 
Arguments: {B : Type u_2} (F : Type u_3) (E : B → Type u_4) [(x : B) → Zero (E x)] (x : B)
Goal: (Bundle.zeroSection F E x).proj = x
-- Bundle.zeroSection_snd
@[simp, mfld_simps]
theorem zeroSection_snd [∀ x, Zero (E x)] (x : B) : (zeroSection F E x).2 = 0 
Arguments: {B : Type u_2} (F : Type u_3) (E : B → Type u_4) [(x : B) → Zero (E x)] (x : B)
Goal: (Bundle.zeroSection F E x).snd = 0
-- VectorBundle.trivialization_linear'
class VectorBundle : Prop where
  trivialization_linear' : ∀ (e : Trivialization F (π F E)) [MemTrivializationAtlas e], e.IsLinear R
  continuousOn_coordChange' :
    ∀ (e e' : Trivialization F (π F E)) [MemTrivializationAtlas e] [MemTrivializationAtlas e'],
      ContinuousOn (fun b => Trivialization.coordChangeL R e e' b : B → F →L[R] F)
        (e.baseSet ∩ e'.baseSet)
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] [self : VectorBundle R F E] (e : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e]
Goal: Trivialization.IsLinear R e
-- VectorBundle.continuousOn_coordChange'
class VectorBundle : Prop where
  trivialization_linear' : ∀ (e : Trivialization F (π F E)) [MemTrivializationAtlas e], e.IsLinear R
  continuousOn_coordChange' :
    ∀ (e e' : Trivialization F (π F E)) [MemTrivializationAtlas e] [MemTrivializationAtlas e'],
      ContinuousOn (fun b => Trivialization.coordChangeL R e e' b : B → F →L[R] F)
        (e.baseSet ∩ e'.baseSet)
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] [self : VectorBundle R F E] (e e' : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e']
Goal: ContinuousOn (fun b => ↑(Trivialization.coordChangeL R e e' b)) (e.baseSet ∩ e'.baseSet)
-- continuousOn_coordChange
theorem continuousOn_coordChange [VectorBundle R F E] (e e' : Trivialization F (π F E))
    [MemTrivializationAtlas e] [MemTrivializationAtlas e'] :
    ContinuousOn (fun b => Trivialization.coordChangeL R e e' b : B → F →L[R] F)
      (e.baseSet ∩ e'.baseSet) 
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] [VectorBundle R F E] (e e' : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e']
Goal: ContinuousOn (fun b => ↑(Trivialization.coordChangeL R e e' b)) (e.baseSet ∩ e'.baseSet)
-- Trivialization.continuousLinearMapAt_apply
@[simps (config := .asFn) apply]
def continuousLinearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : 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)
Goal: ⇑(Trivialization.continuousLinearMapAt R e b) = ⇑(Trivialization.linearMapAt R e b)