Module: Mathlib.Geometry.Manifold.VectorBundle.Riemannian

Goal state before:
pp: EB : Type u_1
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace ℝ EB
HB : Type u_2
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners ℝ EB HB
n : ℕ∞ω
B : Type u_3
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
E : B → Type u_5
inst✝⁹ : TopologicalSpace (TotalSpace F E)
inst✝⁸ : (x : B) → NormedAddCommGroup (E x)
inst✝⁷ : (x : B) → InnerProductSpace ℝ (E x)
inst✝⁶ : FiberBundle F E
inst✝⁵ : VectorBundle ℝ F E
EM : Type u_6
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace ℝ EM
HM : Type u_7
inst✝² : TopologicalSpace HM
IM : ModelWithCorners ℝ EM HM
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
h : IsContMDiffRiemannianBundle IB n F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, v m⟩) s x
hw : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, w m⟩) s x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_smooth : ContMDiff IB (IB.prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) n fun b ↦ ⟨b, g b⟩
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
hb : ContMDiffWithinAt IM IB n b s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, ℝ)) n (fun m ↦ ⟨b m, ((g (b m)) (v m)) (w m)⟩) s x

Tactic:
apply ContMDiffWithinAt.clm_bundle_apply₂ (F₁ := F) (F₂ := F)

Goal state after:
case hψ
EB : Type u_1
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace ℝ EB
HB : Type u_2
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners ℝ EB HB
n : ℕ∞ω
B : Type u_3
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
E : B → Type u_5
inst✝⁹ : TopologicalSpace (TotalSpace F E)
inst✝⁸ : (x : B) → NormedAddCommGroup (E x)
inst✝⁷ : (x : B) → InnerProductSpace ℝ (E x)
inst✝⁶ : FiberBundle F E
inst✝⁵ : VectorBundle ℝ F E
EM : Type u_6
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace ℝ EM
HM : Type u_7
inst✝² : TopologicalSpace HM
IM : ModelWithCorners ℝ EM HM
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
h : IsContMDiffRiemannianBundle IB n F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, v m⟩) s x
hw : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, w m⟩) s x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_smooth : ContMDiff IB (IB.prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) n fun b ↦ ⟨b, g b⟩
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
hb : ContMDiffWithinAt IM IB n b s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) n (fun m ↦ ⟨b m, g (b m)⟩) s x

case hv
EB : Type u_1
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace ℝ EB
HB : Type u_2
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners ℝ EB HB
n : ℕ∞ω
B : Type u_3
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
E : B → Type u_5
inst✝⁹ : TopologicalSpace (TotalSpace F E)
inst✝⁸ : (x : B) → NormedAddCommGroup (E x)
inst✝⁷ : (x : B) → InnerProductSpace ℝ (E x)
inst✝⁶ : FiberBundle F E
inst✝⁵ : VectorBundle ℝ F E
EM : Type u_6
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace ℝ EM
HM : Type u_7
inst✝² : TopologicalSpace HM
IM : ModelWithCorners ℝ EM HM
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
h : IsContMDiffRiemannianBundle IB n F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, v m⟩) s x
hw : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, w m⟩) s x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_smooth : ContMDiff IB (IB.prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) n fun b ↦ ⟨b, g b⟩
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
hb : ContMDiffWithinAt IM IB n b s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, v m⟩) s x

case hw
EB : Type u_1
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace ℝ EB
HB : Type u_2
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners ℝ EB HB
n : ℕ∞ω
B : Type u_3
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
E : B → Type u_5
inst✝⁹ : TopologicalSpace (TotalSpace F E)
inst✝⁸ : (x : B) → NormedAddCommGroup (E x)
inst✝⁷ : (x : B) → InnerProductSpace ℝ (E x)
inst✝⁶ : FiberBundle F E
inst✝⁵ : VectorBundle ℝ F E
EM : Type u_6
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace ℝ EM
HM : Type u_7
inst✝² : TopologicalSpace HM
IM : ModelWithCorners ℝ EM HM
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
h : IsContMDiffRiemannianBundle IB n F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, v m⟩) s x
hw : ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, w m⟩) s x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_smooth : ContMDiff IB (IB.prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) n fun b ↦ ⟨b, g b⟩
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
hb : ContMDiffWithinAt IM IB n b s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(ℝ, F)) n (fun m ↦ ⟨b m, w m⟩) s x