-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
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'
inst_11 : IsManifold I 2 M
inst_12 : IsManifold I' 2 M'
inst_13 : CompleteSpace E
inst_14 : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiffOn I s
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 ((↑(extChartAt I' (f x₀)) : M' → E') ∘ f ∘ (↑(extChartAt I x₀).symm : E → M))
    ((↑(extChartAt I x₀).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(extChartAt I x₀) : M → E) x₀)
A : (↑(extChartAt I x₀).symm : E → M) ((↑(extChartAt I x₀) : M → E) x₀) = x₀
A' : x₀ = (↑(extChartAt I x₀).symm : E → M) ((↑(extChartAt I x₀) : M → E) x₀)
h'f : MDiffAt[s] f x₀
this :
  Eq (α :=
    TangentSpace I ((↑(extChartAt I x₀).symm : E → M) ((↑(extChartAt I x₀) : M → E) x₀)) →L[𝕜]
      TangentSpace 𝓘(𝕜, E) ((↑(extChartAt I x₀) : M → E) x₀))
    (ContinuousLinearMap.inverse
      (mfderivWithin 𝓘(𝕜, E) I (↑(extChartAt I x₀).symm : E → M) (range (↑I : H → E))
        ((↑(extChartAt I x₀) : M → E) x₀)))
    (mfderiv% (↑(extChartAt I x₀) : M → E) x₀)
V' : (x : E') → TangentSpace 𝓘(𝕜, E') x :=
  mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm : E' → M') V (range (↑I' : H' → E'))
hV' : V' = mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm : E' → M') V (range (↑I' : H' → E'))
W' : (x : E') → TangentSpace 𝓘(𝕜, E') x :=
  mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm : E' → M') W (range (↑I' : H' → E'))
hW' : W' = mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm : E' → M') W (range (↑I' : H' → E'))
F : E → E' := ((↑(extChartAt I' (f x₀)) : M' → E') ∘ f) ∘ (↑(extChartAt I x₀).symm : E → M)
hF : F = ((↑(extChartAt I' (f x₀)) : M' → E') ∘ f) ∘ (↑(extChartAt I x₀).symm : E → M)
hFx₀ : (↑(extChartAt I' (f x₀)) : M' → E') (f x₀) = F ((↑(extChartAt I x₀) : M → E) x₀)
⊢ ∀ (Y : (x : M') → TangentSpace I' x),
    mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x₀).symm : E → M) (mpullbackWithin I I' f Y s)
        (range
          (↑I :
            H → E)) =ᶠ[𝓝[(↑(extChartAt I x₀).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)] (↑(extChartAt I x₀) : M → E) x₀]
      mpullbackWithin 𝓘(𝕜, E) 𝓘(𝕜, E') F
        (mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm : E' → M') Y (range (↑I' : H' → E')))
        ((↑(extChartAt I x₀).symm : E → M) ⁻¹' s ∩ range (↑I : H → E))

Tactic:
have P (Y : (x : M') → TangentSpace I' x) :
      (mpullbackWithin 𝓘(𝕜, E) I (extChartAt I x₀).symm (mpullbackWithin I I' f Y s)
      (range I)) =ᶠ[𝓝[(extChartAt I x₀).symm ⁻¹' s ∩ range I] (extChartAt I x₀ x₀)]
        mpullbackWithin 𝓘(𝕜, E) 𝓘(𝕜, E') F
          (mpullbackWithin 𝓘(𝕜, E') I' ((extChartAt I' (f x₀)).symm) Y (range I'))
          ((extChartAt I x₀).symm ⁻¹' s ∩ range I) := by
    have : (extChartAt I x₀).target
        ∈ 𝓝[(extChartAt I x₀).symm ⁻¹' s ∩ range I] (extChartAt I x₀ x₀) :=
      nhdsWithin_mono _ inter_subset_right (extChartAt_target_mem_nhdsWithin x₀)
    filter_upwards [self_mem_nhdsWithin, this] with y hy h'''y
    have h'y : f ((extChartAt I x₀).symm y) ∈ (extChartAt I' (f x₀)).source := ht (hst hy.1)
    have h''y : f ((extChartAt I x₀).symm y) ∈ (chartAt H' (f x₀)).source := by simpa using h'y
    have huy : UniqueMDiffWithinAt 𝓘(𝕜, E) ((extChartAt I x₀).symm ⁻¹' s ∩ range I) y := by
      apply UniqueDiffWithinAt.uniqueMDiffWithinAt
      rw [inter_comm]
      apply hu.uniqueDiffWithinAt_range_inter
      exact ⟨h'''y, hy.1⟩
    simp only [mpullbackWithin_apply, hF, comp_apply]
    rw [mfderivWithin_comp (I' := I) (u := s)]; rotate_left
    · apply (mdifferentiableAt_extChartAt h''y).comp_mdifferentiableWithinAt (I' := I')
      exact (hf _ hy.1).mdifferentiableWithinAt two_ne_zero
    · exact (mdifferentiableWithinAt_extChartAt_symm h'''y).mono inter_subset_right
    · exact inter_subset_left
    · exact huy
    rw [mfderiv_comp_mfderivWithin (I' := I')]; rotate_left
    · exact mdifferentiableAt_extChartAt h''y
    · exact (hf _ hy.1).mdifferentiableWithinAt two_ne_zero
    · exact hu _ hy.1
    rw [← ContinuousLinearMap.IsInvertible.inverse_comp_apply_of_right]; swap
    · exact isInvertible_mfderivWithin_extChartAt_symm h'''y
    rw [← ContinuousLinearMap.IsInvertible.inverse_comp_apply_of_left]; swap
    · exact isInvertible_mfderivWithin_extChartAt_symm (PartialEquiv.map_source _ h'y)
    have : f ((extChartAt I x₀).symm y)
        = (extChartAt I' (f x₀)).symm ((extChartAt I' (f x₀)) (f ((extChartAt I x₀).symm y))) :=
      (PartialEquiv.left_inv (extChartAt I' (f x₀)) h'y).symm
    congr 2
    have : (mfderiv[range I'] ((extChartAt I' (f x₀)).symm)
        (extChartAt I' (f x₀) (f ((extChartAt I x₀).symm y)))) ∘L
        (mfderiv% (extChartAt I' (f x₀)) (f ((extChartAt I x₀).symm y))) =
        ContinuousLinearMap.id _ _ := by
      convert!
        mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt ((PartialEquiv.map_source _ h'y))
    simp only [← ContinuousLinearMap.comp_assoc, this, ContinuousLinearMap.id_comp]
    congr 1
    exact ((mdifferentiableWithinAt_extChartAt_symm h'''y).mfderivWithin_mono huy
      inter_subset_right).symm