-- Module: Mathlib.Topology.VectorBundle.Riemannian
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < HPow.hPow (β := ℕ) r' 2
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' y - g' x‖ ≤ δ
v : E x
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
      (a : E x) → F)
    v
hgx :
  Eq (α := ℝ) ((((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) v : ℝ)
    ((((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
hgy :
  Eq (α := ℝ)
    ((((g y : (a : E y) → E y →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) y w) : E y →L[ℝ] ℝ) :
          (a : E y) → ℝ)
        (Trivialization.symm (E := E) (trivializationAt F E x) y w) :
      ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
e_a :
  Eq (α := ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ)
              ((G.symm : (a : F) → E x)
                  (((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
                        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
                      (a : E x) → F)
                    v) :
                E x) :
            E x →L[ℝ] ℝ) :
          (a : E x) → ℝ)
        ((G.symm : (a : F) → E x)
            (((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
                  ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
                (a : E x) → F)
              v) :
          E x) :
      ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ)
              (Trivialization.symm (E := E) (trivializationAt F E x) x
                (((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
                      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
                    (a : E x) → F)
                  v)) :
            E x →L[ℝ] ℝ) :
          (a : E x) → ℝ)
        (Trivialization.symm (E := E) (trivializationAt F E x) x
          (((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
                ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
              (a : E x) → F)
            v)) :
      ℝ)
⊢ Eq (α := ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ) ((G.symm : (a : F) → E x) w : E x) : E x →L[ℝ] ℝ) : (a : E x) → ℝ)
        ((G.symm : (a : F) → E x) w : E x) :
      ℝ)
    (((Trivialization.linearMapAt ℝ (E := fun (x : B) ↦ ℝ) (trivializationAt ℝ (fun (x : B) ↦ ℝ) x) x : ℝ →ₗ[ℝ] ℝ) :
          (a : ℝ) → ℝ)
        ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
              (a : E x) → ℝ)
            (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
          ℝ) :
      ℝ)

Tactic:
simp only [Trivial.fiberBundle_trivializationAt', Trivial.linearMapAt_trivialization,
        LinearMap.id_coe, id_eq, w]

Premises used: LinearMap.id_coe, Bundle.Trivial.linearMapAt_trivialization, Bundle.Trivial.fiberBundle_trivializationAt', id_eq