Module: Mathlib.Geometry.Manifold.VectorBundle.Hom

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst✝²⁸ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst✝²⁷ : (x : B) → AddCommGroup (E₁ x)
inst✝²⁶ : (x : B) → Module 𝕜 (E₁ x)
inst✝²⁵ : NormedAddCommGroup F₁
inst✝²⁴ : NormedSpace 𝕜 F₁
inst✝²³ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²² : (x : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst✝²¹ : (x : B) → AddCommGroup (E₂ x)
inst✝²⁰ : (x : B) → Module 𝕜 (E₂ x)
inst✝¹⁹ : NormedAddCommGroup F₂
inst✝¹⁸ : NormedSpace 𝕜 F₂
inst✝¹⁷ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁶ : (x : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_11
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
EM : Type u_12
inst✝¹⁰ : NormedAddCommGroup EM
inst✝⁹ : NormedSpace 𝕜 EM
HM : Type u_13
inst✝⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace HM M
inst✝⁵ : FiberBundle F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst✝¹ : ∀ (x : B), IsTopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
ϕ : (x : M) → E₁ (b x) →L[𝕜] E₂ (b x)
hϕ : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) n (fun m ↦ ⟨b m, ϕ m⟩) s x
hv : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₁)) n (fun m ↦ ⟨b m, v m⟩) s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₂)) n (fun m ↦ ⟨b m, (ϕ m) (v m)⟩) s x

Tactic:
simp only [contMDiffWithinAt_hom_bundle] at hϕ

Goal state after:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst✝²⁸ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst✝²⁷ : (x : B) → AddCommGroup (E₁ x)
inst✝²⁶ : (x : B) → Module 𝕜 (E₁ x)
inst✝²⁵ : NormedAddCommGroup F₁
inst✝²⁴ : NormedSpace 𝕜 F₁
inst✝²³ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²² : (x : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst✝²¹ : (x : B) → AddCommGroup (E₂ x)
inst✝²⁰ : (x : B) → Module 𝕜 (E₂ x)
inst✝¹⁹ : NormedAddCommGroup F₂
inst✝¹⁸ : NormedSpace 𝕜 F₂
inst✝¹⁷ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁶ : (x : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_11
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
EM : Type u_12
inst✝¹⁰ : NormedAddCommGroup EM
inst✝⁹ : NormedSpace 𝕜 EM
HM : Type u_13
inst✝⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace HM M
inst✝⁵ : FiberBundle F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₁ E₁
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst✝¹ : ∀ (x : B), IsTopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)
ϕ : (x : M) → E₁ (b x) →L[𝕜] E₂ (b x)
hv : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₁)) n (fun m ↦ ⟨b m, v m⟩) s x
hϕ :
  ContMDiffWithinAt IM IB n (fun x ↦ b x) s x ∧
    ContMDiffWithinAt IM 𝓘(𝕜, F₁ →L[𝕜] F₂) n (fun x_1 ↦ inCoordinates F₁ E₁ F₂ E₂ (b x) (b x_1) (b x) (b x_1) (ϕ x_1)) s
      x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₂)) n (fun m ↦ ⟨b m, (ϕ m) (v m)⟩) s x