[GOAL]
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝³⁵ : NontriviallyNormedField 𝕜
inst✝³⁴ : (x : B) → AddCommGroup (E₁ x)
inst✝³³ : (x : B) → Module 𝕜 (E₁ x)
inst✝³² : NormedAddCommGroup F₁
inst✝³¹ : NormedSpace 𝕜 F₁
inst✝³⁰ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝²⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝²⁶ : NormedAddCommGroup F₂
inst✝²⁵ : NormedSpace 𝕜 F₂
inst✝²⁴ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝²³ : (x : B) → TopologicalSpace (E₂ x)
inst✝²² : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝²¹ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_10
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : VectorBundle 𝕜 F₁ E₁
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁵ : SmoothVectorBundle F₁ E₁ IB
inst✝⁴ : SmoothVectorBundle F₂ E₂ IB
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
⊢ SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂) (continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))
[PROOFSTEP]
have h₁ := smoothOn_coordChangeL IB e₁' e₁
[GOAL]
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝³⁵ : NontriviallyNormedField 𝕜
inst✝³⁴ : (x : B) → AddCommGroup (E₁ x)
inst✝³³ : (x : B) → Module 𝕜 (E₁ x)
inst✝³² : NormedAddCommGroup F₁
inst✝³¹ : NormedSpace 𝕜 F₁
inst✝³⁰ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝²⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝²⁶ : NormedAddCommGroup F₂
inst✝²⁵ : NormedSpace 𝕜 F₂
inst✝²⁴ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝²³ : (x : B) → TopologicalSpace (E₂ x)
inst✝²² : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝²¹ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_10
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : VectorBundle 𝕜 F₁ E₁
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁵ : SmoothVectorBundle F₁ E₁ IB
inst✝⁴ : SmoothVectorBundle F₂ E₂ IB
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : SmoothOn IB 𝓘(𝕜, F₁ →L[𝕜] F₁) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
⊢ SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂) (continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))
[PROOFSTEP]
have h₂ := smoothOn_coordChangeL IB e₂ e₂'
[GOAL]
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝³⁵ : NontriviallyNormedField 𝕜
inst✝³⁴ : (x : B) → AddCommGroup (E₁ x)
inst✝³³ : (x : B) → Module 𝕜 (E₁ x)
inst✝³² : NormedAddCommGroup F₁
inst✝³¹ : NormedSpace 𝕜 F₁
inst✝³⁰ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝²⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝²⁶ : NormedAddCommGroup F₂
inst✝²⁵ : NormedSpace 𝕜 F₂
inst✝²⁴ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝²³ : (x : B) → TopologicalSpace (E₂ x)
inst✝²² : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝²¹ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_10
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : VectorBundle 𝕜 F₁ E₁
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁵ : SmoothVectorBundle F₁ E₁ IB
inst✝⁴ : SmoothVectorBundle F₂ E₂ IB
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : SmoothOn IB 𝓘(𝕜, F₁ →L[𝕜] F₁) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
h₂ : SmoothOn IB 𝓘(𝕜, F₂ →L[𝕜] F₂) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet)
⊢ SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂) (continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))
[PROOFSTEP]
refine (h₁.mono ?_).cle_arrowCongr (h₂.mono ?_)
[GOAL]
case refine_1
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝³⁵ : NontriviallyNormedField 𝕜
inst✝³⁴ : (x : B) → AddCommGroup (E₁ x)
inst✝³³ : (x : B) → Module 𝕜 (E₁ x)
inst✝³² : NormedAddCommGroup F₁
inst✝³¹ : NormedSpace 𝕜 F₁
inst✝³⁰ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝²⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝²⁶ : NormedAddCommGroup F₂
inst✝²⁵ : NormedSpace 𝕜 F₂
inst✝²⁴ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝²³ : (x : B) → TopologicalSpace (E₂ x)
inst✝²² : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝²¹ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_10
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : VectorBundle 𝕜 F₁ E₁
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁵ : SmoothVectorBundle F₁ E₁ IB
inst✝⁴ : SmoothVectorBundle F₂ E₂ IB
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : SmoothOn IB 𝓘(𝕜, F₁ →L[𝕜] F₁) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
h₂ : SmoothOn IB 𝓘(𝕜, F₂ →L[𝕜] F₂) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet)
⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₁'.baseSet ∩ e₁.baseSet
[PROOFSTEP]
mfld_set_tac
[GOAL]
case refine_2
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝³⁵ : NontriviallyNormedField 𝕜
inst✝³⁴ : (x : B) → AddCommGroup (E₁ x)
inst✝³³ : (x : B) → Module 𝕜 (E₁ x)
inst✝³² : NormedAddCommGroup F₁
inst✝³¹ : NormedSpace 𝕜 F₁
inst✝³⁰ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝²⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝²⁶ : NormedAddCommGroup F₂
inst✝²⁵ : NormedSpace 𝕜 F₂
inst✝²⁴ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝²³ : (x : B) → TopologicalSpace (E₂ x)
inst✝²² : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝²¹ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_10
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : VectorBundle 𝕜 F₁ E₁
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁵ : SmoothVectorBundle F₁ E₁ IB
inst✝⁴ : SmoothVectorBundle F₂ E₂ IB
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : SmoothOn IB 𝓘(𝕜, F₁ →L[𝕜] F₁) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
h₂ : SmoothOn IB 𝓘(𝕜, F₂ →L[𝕜] F₂) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet)
⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₂.baseSet ∩ e₂'.baseSet
[PROOFSTEP]
mfld_set_tac
[GOAL]
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝²⁹ : NontriviallyNormedField 𝕜
inst✝²⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝²⁷ : (x : B) → Module 𝕜 (E₁ x)
inst✝²⁶ : NormedAddCommGroup F₁
inst✝²⁵ : NormedSpace 𝕜 F₁
inst✝²⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²³ : (x : B) → TopologicalSpace (E₁ x)
inst✝²² : (x : B) → AddCommGroup (E₂ x)
inst✝²¹ : (x : B) → Module 𝕜 (E₂ x)
inst✝²⁰ : NormedAddCommGroup F₂
inst✝¹⁹ : NormedSpace 𝕜 F₂
inst✝¹⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁷ : (x : B) → TopologicalSpace (E₂ x)
inst✝¹⁶ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝¹⁵ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
EM : Type u_10
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
inst✝⁴ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
y₀ y : LE₁E₂
⊢ ↑(chartAt (ModelProd HB (F₁ →L[𝕜] F₂)) y₀) y =
    (↑(chartAt HB y₀.proj) y.proj, inCoordinates F₁ E₁ F₂ E₂ y₀.proj y.proj y₀.proj y.proj y.snd)
[PROOFSTEP]
rw [FiberBundle.chartedSpace_chartAt, trans_apply, LocalHomeomorph.prod_apply, Trivialization.coe_coe,
  LocalHomeomorph.refl_apply, Function.id_def, hom_trivializationAt_apply]
[GOAL]
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝³¹ : NontriviallyNormedField 𝕜
inst✝³⁰ : (x : B) → AddCommGroup (E₁ x)
inst✝²⁹ : (x : B) → Module 𝕜 (E₁ x)
inst✝²⁸ : NormedAddCommGroup F₁
inst✝²⁷ : NormedSpace 𝕜 F₁
inst✝²⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝²³ : (x : B) → Module 𝕜 (E₂ x)
inst✝²² : NormedAddCommGroup F₂
inst✝²¹ : NormedSpace 𝕜 F₂
inst✝²⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁹ : (x : B) → TopologicalSpace (E₂ x)
inst✝¹⁸ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝¹⁷ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
EM : Type u_10
inst✝¹¹ : NormedAddCommGroup EM
inst✝¹⁰ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁹ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace HM M
inst✝⁶ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁵ : FiberBundle F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝¹ : SmoothVectorBundle F₁ E₁ IB
inst✝ : SmoothVectorBundle F₂ E₂ IB
⊢ ∀ (e : Pretrivialization (F₁ →L[𝕜] F₂) TotalSpace.proj),
    e ∈ (vectorPrebundle (RingHom.id 𝕜) F₁ E₁ F₂ E₂).pretrivializationAtlas →
      ∀ (e' : Pretrivialization (F₁ →L[𝕜] F₂) TotalSpace.proj),
        e' ∈ (vectorPrebundle (RingHom.id 𝕜) F₁ E₁ F₂ E₂).pretrivializationAtlas →
          ∃ f,
            SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂) f (e.baseSet ∩ e'.baseSet) ∧
              ∀ (b : B),
                b ∈ e.baseSet ∩ e'.baseSet →
                  ∀ (v : F₁ →L[𝕜] F₂), ↑(f b) v = (↑e' { proj := b, snd := Pretrivialization.symm e b v }).snd
[PROOFSTEP]
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩ _ ⟨e₁', e₂', he₁', he₂', rfl⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_5
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝³¹ : NontriviallyNormedField 𝕜
inst✝³⁰ : (x : B) → AddCommGroup (E₁ x)
inst✝²⁹ : (x : B) → Module 𝕜 (E₁ x)
inst✝²⁸ : NormedAddCommGroup F₁
inst✝²⁷ : NormedSpace 𝕜 F₁
inst✝²⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝²³ : (x : B) → Module 𝕜 (E₂ x)
inst✝²² : NormedAddCommGroup F₂
inst✝²¹ : NormedSpace 𝕜 F₂
inst✝²⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁹ : (x : B) → TopologicalSpace (E₂ x)
inst✝¹⁸ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝¹⁷ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
EB : Type u_8
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
EM : Type u_10
inst✝¹¹ : NormedAddCommGroup EM
inst✝¹⁰ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁹ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace HM M
inst✝⁶ : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁵ : FiberBundle F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜 F₂ E₂
e₁✝ e₁'✝ : Trivialization F₁ TotalSpace.proj
e₂✝ e₂'✝ : Trivialization F₂ TotalSpace.proj
inst✝¹ : SmoothVectorBundle F₁ E₁ IB
inst✝ : SmoothVectorBundle F₂ E₂ IB
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
he₁ : MemTrivializationAtlas e₁
he₂ : MemTrivializationAtlas e₂
e₁' : Trivialization F₁ TotalSpace.proj
e₂' : Trivialization F₂ TotalSpace.proj
he₁' : MemTrivializationAtlas e₁'
he₂' : MemTrivializationAtlas e₂'
⊢ ∃ f,
    SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂) f
        ((continuousLinearMap (RingHom.id 𝕜) e₁ e₂).baseSet ∩ (continuousLinearMap (RingHom.id 𝕜) e₁' e₂').baseSet) ∧
      ∀ (b : B),
        b ∈ (continuousLinearMap (RingHom.id 𝕜) e₁ e₂).baseSet ∩ (continuousLinearMap (RingHom.id 𝕜) e₁' e₂').baseSet →
          ∀ (v : F₁ →L[𝕜] F₂),
            ↑(f b) v =
              (↑(continuousLinearMap (RingHom.id 𝕜) e₁' e₂')
                  { proj := b, snd := Pretrivialization.symm (continuousLinearMap (RingHom.id 𝕜) e₁ e₂) b v }).snd
[PROOFSTEP]
exact
  ⟨continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂', smoothOn_continuousLinearMapCoordChange IB,
    continuousLinearMapCoordChange_apply (RingHom.id 𝕜) e₁ e₁' e₂ e₂'⟩
