-- Module: Mathlib.Geometry.Manifold.VectorField.Pullback
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
H' : Type u_5
inst_6 : TopologicalSpace H'
E' : Type u_6
inst_7 : NormedAddCommGroup E'
inst_8 : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
inst_11 : IsManifold I 2 M
inst_12 : IsManifold I' 2 M'
inst_13 : CompleteSpace E
hV : MDiffAt[t] (T% V) (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : IsInvertible (M := TangentSpace I x₀) (M₂ := TangentSpace I' (f x₀)) (mfderivWithin I I' f s x₀)
hx₀ : x₀ ∈ s
hs : UniqueMDiffOn I s
hmn : 2 ≤ n
b₁ : M → M' := f
b₂ : M → M := id
v : (x : M) → TangentSpace I' (f x) := V ∘ f
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_6, u_3} (RingHom.id 𝕜) (TangentSpace I' (f x)) (TangentSpace I x) :=
  fun (x : M) ↦ inverse (mfderivWithin I I' f s x)
hv : (MDiffAt[s ∩ f ⁻¹' t] fun (x : M) ↦ ⟨f x, v x⟩) x₀
this_1 :
  (MDiffAt[s] fun (x : M) ↦
      inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderivWithin I I' f s x))
    x₀
this :
  MDiffAt[s]
      (inverse ∘ fun (x : M) ↦
        inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderivWithin I I' f s x))
    x₀
A :
  Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
      (trivializationAt E (TangentSpace I) x₀) ∈
    𝓝[s] x₀
B :
  f ⁻¹'
      Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_7, u_6, u_6} (B := M') E' (TangentSpace I'))
        (trivializationAt E' (TangentSpace I') (f x₀)) ∈
    𝓝[s] x₀
x : M
hx :
  x ∈
    Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
      (trivializationAt E (TangentSpace I) x₀)
h'x :
  x ∈
    f ⁻¹'
      Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_7, u_6, u_6} (B := M') E' (TangentSpace I'))
        (trivializationAt E' (TangentSpace I') (f x₀))
h :
  Eq (α := E' →L[𝕜] E)
    ((↑(Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I)
            (trivializationAt E (TangentSpace I) (b₂ x₀)) (b₂ x) hx) :
        TangentSpace I (b₂ x) →L[𝕜] E) ∘SL
      ContinuousLinearMap.comp (M₂ := TangentSpace I' (b₁ x)) (inverse (mfderivWithin I I' f s x))
        (↑(ContinuousLinearEquiv.symm
              (Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I')
                (trivializationAt E' (TangentSpace I') (b₁ x₀)) (b₁ x) h'x)) :
          E' →L[𝕜] TangentSpace I' (b₁ x)))
    (ContinuousLinearMap.comp (M₂ := TangentSpace I' (f x))
      ((↑(Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I) (trivializationAt E (TangentSpace I) x₀)
              x hx) :
          TangentSpace I x →L[𝕜] E) ∘SL
        inverse (mfderivWithin I I' f s x))
      (↑(ContinuousLinearEquiv.symm
            (Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I')
              (trivializationAt E' (TangentSpace I') (f x₀)) (f x) h'x)) :
        E' →L[𝕜] TangentSpace I' (f x)))
⊢ Eq (α := E' →L[𝕜] E)
    ((↑(Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I)
            (trivializationAt E (TangentSpace I) (b₂ x₀)) (b₂ x) hx) :
        TangentSpace I (b₂ x) →L[𝕜] E) ∘SL
      ϕ x ∘SL
        (↑(ContinuousLinearEquiv.symm
              (Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I')
                (trivializationAt E' (TangentSpace I') (b₁ x₀)) (b₁ x) h'x)) :
          E' →L[𝕜] TangentSpace I' (b₁ x)))
    (inverse
      (ContinuousLinearMap.comp (M₂ := TangentSpace I' (f x))
        (↑(Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I')
              (trivializationAt E' (TangentSpace I') (f x₀)) (f x) h'x) :
          TangentSpace I' (f x) →L[𝕜] E')
        (mfderivWithin I I' f s x ∘SL
          (↑(ContinuousLinearEquiv.symm
                (Bundle.Trivialization.continuousLinearEquivAt 𝕜 (E := TangentSpace I)
                  (trivializationAt E (TangentSpace I) x₀) x hx)) :
            E →L[𝕜] TangentSpace I x))))

Tactic:
simp only [inverse_equiv_comp, inverse_comp_equiv, ContinuousLinearEquiv.symm_symm, ϕ]

Premises used: ContinuousLinearMap.inverse_comp_equiv, ContinuousLinearMap.inverse_equiv_comp, ContinuousLinearEquiv.symm_symm