Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContinuousWithinAt (fun x ↦ { proj := b₁ x, snd := v x }.proj) (insert m₀ s) m₀ ∧
    ContinuousWithinAt
      (fun x ↦ (↑(trivializationAt F₁ E₁ { proj := b₁ m₀, snd := v m₀ }.proj) { proj := b₁ x, snd := v x }).2)
      (insert m₀ s) m₀
hb₂ : ContinuousWithinAt b₂ (insert m₀ s) m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet
A' : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet
m : M
hm : b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet
⊢ b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet →
    (↑(trivializationAt F₂ E₂ (b₂ m₀)) { proj := b₂ m, snd := (ϕ m) (v m) }).2 =
      (inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m))
        (↑(trivializationAt F₁ E₁ (b₁ m₀)) { proj := b₁ m, snd := v m }).2

Tactic:
h'm

Goal state after:
𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContinuousWithinAt (fun x ↦ { proj := b₁ x, snd := v x }.proj) (insert m₀ s) m₀ ∧
    ContinuousWithinAt
      (fun x ↦ (↑(trivializationAt F₁ E₁ { proj := b₁ m₀, snd := v m₀ }.proj) { proj := b₁ x, snd := v x }).2)
      (insert m₀ s) m₀
hb₂ : ContinuousWithinAt b₂ (insert m₀ s) m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet
A' : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet
m : M
hm : b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet
h'm : b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet
⊢ (↑(trivializationAt F₂ E₂ (b₂ m₀)) { proj := b₂ m, snd := (ϕ m) (v m) }).2 =
    (inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m))
      (↑(trivializationAt F₁ E₁ (b₁ m₀)) { proj := b₁ m, snd := v m }).2