Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: n : ℕ∞ω
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✝³⁸ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝³⁷ : NormedAddCommGroup EB
inst✝³⁶ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³⁵ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝³⁴ : TopologicalSpace B
inst✝³³ : ChartedSpace HB B
EM : Type u_9
inst✝³² : NormedAddCommGroup EM
inst✝³¹ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝³⁰ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝²⁹ : TopologicalSpace M
inst✝²⁸ : ChartedSpace HM M
inst✝²⁷ : (x : B) → AddCommMonoid (E x)
inst✝²⁶ : (x : B) → Module 𝕜 (E x)
inst✝²⁵ : NormedAddCommGroup F
inst✝²⁴ : NormedSpace 𝕜 F
inst✝²³ : TopologicalSpace (TotalSpace F E)
inst✝²² : (x : B) → TopologicalSpace (E x)
inst✝²¹ : FiberBundle F E
inst✝²⁰ : VectorBundle 𝕜 F E
inst✝¹⁹ : ContMDiffVectorBundle n F E IB
F₁ : Type u_11
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜 F₁
E₁ : B → Type u_12
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹⁵ : (x : B) → AddCommMonoid (E₁ x)
inst✝¹⁴ : (x : B) → Module 𝕜 (E₁ x)
F₂ : Type u_13
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜 F₂
E₂ : B → Type u_14
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁹ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁸ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁷ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : FiberBundle F₂ E₂
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : VectorBundle 𝕜 F₂ E₂
inst✝² : ContMDiffVectorBundle n F₁ E₁ IB
inst✝¹ : ContMDiffVectorBundle n F₂ E₂ IB
inst✝ : IsManifold IB n B
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
i₁ : MemTrivializationAtlas e₁
i₂ : MemTrivializationAtlas e₂
e₁' : Trivialization F₁ TotalSpace.proj
e₂' : Trivialization F₂ TotalSpace.proj
i₁' : MemTrivializationAtlas e₁'
i₂' : MemTrivializationAtlas e₂'
⊢ ContMDiffOn IB 𝓘(𝕜, F₁ × F₂ →L[𝕜] F₁ × F₂) n
    (fun y ↦ (↑(Trivialization.coordChangeL 𝕜 e₁ e₁' y)).prodMap ↑(Trivialization.coordChangeL 𝕜 e₂ e₂' y))
    ((e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet)

Tactic:
refine ContMDiffOn.clm_prodMap ?_ ?_

Goal state after:
case refine_1
n : ℕ∞ω
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✝³⁸ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝³⁷ : NormedAddCommGroup EB
inst✝³⁶ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³⁵ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝³⁴ : TopologicalSpace B
inst✝³³ : ChartedSpace HB B
EM : Type u_9
inst✝³² : NormedAddCommGroup EM
inst✝³¹ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝³⁰ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝²⁹ : TopologicalSpace M
inst✝²⁸ : ChartedSpace HM M
inst✝²⁷ : (x : B) → AddCommMonoid (E x)
inst✝²⁶ : (x : B) → Module 𝕜 (E x)
inst✝²⁵ : NormedAddCommGroup F
inst✝²⁴ : NormedSpace 𝕜 F
inst✝²³ : TopologicalSpace (TotalSpace F E)
inst✝²² : (x : B) → TopologicalSpace (E x)
inst✝²¹ : FiberBundle F E
inst✝²⁰ : VectorBundle 𝕜 F E
inst✝¹⁹ : ContMDiffVectorBundle n F E IB
F₁ : Type u_11
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜 F₁
E₁ : B → Type u_12
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹⁵ : (x : B) → AddCommMonoid (E₁ x)
inst✝¹⁴ : (x : B) → Module 𝕜 (E₁ x)
F₂ : Type u_13
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜 F₂
E₂ : B → Type u_14
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁹ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁸ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁷ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : FiberBundle F₂ E₂
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : VectorBundle 𝕜 F₂ E₂
inst✝² : ContMDiffVectorBundle n F₁ E₁ IB
inst✝¹ : ContMDiffVectorBundle n F₂ E₂ IB
inst✝ : IsManifold IB n B
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
i₁ : MemTrivializationAtlas e₁
i₂ : MemTrivializationAtlas e₂
e₁' : Trivialization F₁ TotalSpace.proj
e₂' : Trivialization F₂ TotalSpace.proj
i₁' : MemTrivializationAtlas e₁'
i₂' : MemTrivializationAtlas e₂'
⊢ ContMDiffOn IB 𝓘(𝕜, F₁ →L[𝕜] F₁) n (fun y ↦ ↑(Trivialization.coordChangeL 𝕜 e₁ e₁' y))
    ((e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet)

case refine_2
n : ℕ∞ω
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✝³⁸ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝³⁷ : NormedAddCommGroup EB
inst✝³⁶ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³⁵ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝³⁴ : TopologicalSpace B
inst✝³³ : ChartedSpace HB B
EM : Type u_9
inst✝³² : NormedAddCommGroup EM
inst✝³¹ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝³⁰ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝²⁹ : TopologicalSpace M
inst✝²⁸ : ChartedSpace HM M
inst✝²⁷ : (x : B) → AddCommMonoid (E x)
inst✝²⁶ : (x : B) → Module 𝕜 (E x)
inst✝²⁵ : NormedAddCommGroup F
inst✝²⁴ : NormedSpace 𝕜 F
inst✝²³ : TopologicalSpace (TotalSpace F E)
inst✝²² : (x : B) → TopologicalSpace (E x)
inst✝²¹ : FiberBundle F E
inst✝²⁰ : VectorBundle 𝕜 F E
inst✝¹⁹ : ContMDiffVectorBundle n F E IB
F₁ : Type u_11
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜 F₁
E₁ : B → Type u_12
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹⁵ : (x : B) → AddCommMonoid (E₁ x)
inst✝¹⁴ : (x : B) → Module 𝕜 (E₁ x)
F₂ : Type u_13
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜 F₂
E₂ : B → Type u_14
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁹ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁸ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁷ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : FiberBundle F₂ E₂
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : VectorBundle 𝕜 F₂ E₂
inst✝² : ContMDiffVectorBundle n F₁ E₁ IB
inst✝¹ : ContMDiffVectorBundle n F₂ E₂ IB
inst✝ : IsManifold IB n B
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
i₁ : MemTrivializationAtlas e₁
i₂ : MemTrivializationAtlas e₂
e₁' : Trivialization F₁ TotalSpace.proj
e₂' : Trivialization F₂ TotalSpace.proj
i₁' : MemTrivializationAtlas e₁'
i₂' : MemTrivializationAtlas e₂'
⊢ ContMDiffOn IB 𝓘(𝕜, F₂ →L[𝕜] F₂) n (fun y ↦ ↑(Trivialization.coordChangeL 𝕜 e₂ e₂' y))
    ((e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet)