Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
B : Type u_3
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_4
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹³ : (x : B) → AddCommGroup (E₁ x)
inst✝¹² : (x : B) → Module 𝕜 (E₁ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : FiberBundle F₁ E₁
inst✝³ : (x : B) → TopologicalSpace (E₂ x)
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
⊢ cast ⋯ (↑(continuousAlternatingMap 𝕜 ι e₁ e₂).symm (b, L)).snd =
    (Trivialization.symmL 𝕜 e₂ b).compContinuousAlternatingMap
      (L.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 e₁ b))

pp: case hb
𝕜 : Type u_1
ι : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
B : Type u_3
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_4
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹³ : (x : B) → AddCommGroup (E₁ x)
inst✝¹² : (x : B) → Module 𝕜 (E₁ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : FiberBundle F₁ E₁
inst✝³ : (x : B) → TopologicalSpace (E₂ x)
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
⊢ b ∈ (continuousAlternatingMap 𝕜 ι e₁ e₂).baseSet

Tactic:
]

Goal state after:
𝕜 : Type u_1
ι : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
B : Type u_3
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_4
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹³ : (x : B) → AddCommGroup (E₁ x)
inst✝¹² : (x : B) → Module 𝕜 (E₁ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : FiberBundle F₁ E₁
inst✝³ : (x : B) → TopologicalSpace (E₂ x)
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
⊢ cast ⋯ (↑(continuousAlternatingMap 𝕜 ι e₁ e₂).symm (b, L)).snd =
    (Trivialization.symmL 𝕜 e₂ b).compContinuousAlternatingMap
      (L.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 e₁ b))

case hb
𝕜 : Type u_1
ι : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
B : Type u_3
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_4
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹³ : (x : B) → AddCommGroup (E₁ x)
inst✝¹² : (x : B) → Module 𝕜 (E₁ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : FiberBundle F₁ E₁
inst✝³ : (x : B) → TopologicalSpace (E₂ x)
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
⊢ b ∈ (continuousAlternatingMap 𝕜 ι e₁ e₂).baseSet