Theorem: VectorPrebundle.smoothVectorBundle
File: Mathlib/Geometry/Manifold/VectorBundle/Basic.lean

Proof steps:
state_before: 𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e e' : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
⊢ ∀ (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],
    SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b)) (e.baseSet ∩ e'.baseSet)
state_after: case mk.intro.intro.mk.intro.intro
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ SmoothOn IB 𝓘(𝕜, F →L[𝕜] F)
    (fun b =>
      ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b))
    ((a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet)
tactic: rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
state_before: case mk.intro.intro.mk.intro.intro
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ SmoothOn IB 𝓘(𝕜, F →L[𝕜] F)
    (fun b =>
      ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b))
    ((a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet)
state_after: case mk.intro.intro.mk.intro.intro
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ ∀ y ∈ e.baseSet ∩ e'.baseSet,
    ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') y) =
      smoothCoordChange IB a he he' y
tactic: refine (a.smoothOn_smoothCoordChange he he').congr ?_
state_before: case mk.intro.intro.mk.intro.intro
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ ∀ y ∈ e.baseSet ∩ e'.baseSet,
    ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') y) =
      smoothCoordChange IB a he he' y
state_after: case mk.intro.intro.mk.intro.intro
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
⊢ ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
        (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b) =
    smoothCoordChange IB a he he' b
tactic: intro b hb
state_before: case mk.intro.intro.mk.intro.intro
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
⊢ ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
        (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b) =
    smoothCoordChange IB a he he' b
state_after: case mk.intro.intro.mk.intro.intro.h
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (smoothCoordChange IB a he he' b) v
tactic: ext v
state_before: case mk.intro.intro.mk.intro.intro.h
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(Trivialization.coordChangeL 𝕜 (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (smoothCoordChange IB a he he' b) v
state_after: case mk.intro.intro.mk.intro.intro.h
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (↑(a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
        { proj := b, snd := (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).symm b v }).2 =
    (↑e' { proj := b, snd := e.symm b v }).2

case mk.intro.intro.mk.intro.intro.h.hb
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ b ∈
    (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet
tactic: rw [a.smoothCoordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe,
        Trivialization.coordChangeL_apply]
state_before: case mk.intro.intro.mk.intro.intro.h
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (↑(a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
        { proj := b, snd := (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).symm b v }).2 =
    (↑e' { proj := b, snd := e.symm b v }).2

case mk.intro.intro.mk.intro.intro.h.hb
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ b ∈
    (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet
state_after: no goals
tactic: exacts [rfl, hb]