Module: Mathlib.Geometry.Manifold.GroupLieAlgebra

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
A : minSmoothness 𝕜 2 + 1 = minSmoothness 𝕜 3
fg : G → TangentBundle I G := fun g ↦ ⟨g, 0⟩
sfg : ContMDiff I I.tangent (minSmoothness 𝕜 2) fg
fv : G → TangentBundle I G := fun x ↦ ⟨1, v⟩
sfv : ContMDiff I I.tangent (minSmoothness 𝕜 2) fv
F₁ : G → TangentBundle I G × TangentBundle I G := fun g ↦ (fg g, fv g)
S₁ : ContMDiff I (I.tangent.prod I.tangent) (minSmoothness 𝕜 2) F₁
F₂ : TangentBundle I G × TangentBundle I G → TangentBundle (I.prod I) (G × G) := ⇑(equivTangentBundleProd I G I G).symm
S₂ : ContMDiff (I.tangent.prod I.tangent) (I.prod I).tangent (minSmoothness 𝕜 2) F₂
F₃ : TangentBundle (I.prod I) (G × G) → TangentBundle I G := tangentMap (I.prod I) I fun p ↦ p.1 * p.2
S₃ : ContMDiff (I.prod I).tangent I.tangent (minSmoothness 𝕜 2) F₃
S : ContMDiff I I.tangent (minSmoothness 𝕜 2) ((F₃ ∘ F₂) ∘ F₁) := ContMDiff.comp (ContMDiff.comp S₃ S₂) S₁
⊢ ContMDiff I I.tangent (minSmoothness 𝕜 2) fun g ↦ ⟨g, mulInvariantVectorField v g⟩

Tactic:
convert! S with g

Goal state after:
case e'_4
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
A : minSmoothness 𝕜 2 + 1 = minSmoothness 𝕜 3
fg : G → TangentBundle I G := fun g ↦ ⟨g, 0⟩
sfg : ContMDiff I I.tangent (minSmoothness 𝕜 2) fg
fv : G → TangentBundle I G := fun x ↦ ⟨1, v⟩
sfv : ContMDiff I I.tangent (minSmoothness 𝕜 2) fv
F₁ : G → TangentBundle I G × TangentBundle I G := fun g ↦ (fg g, fv g)
S₁ : ContMDiff I (I.tangent.prod I.tangent) (minSmoothness 𝕜 2) F₁
F₂ : TangentBundle I G × TangentBundle I G → TangentBundle (I.prod I) (G × G) := ⇑(equivTangentBundleProd I G I G).symm
S₂ : ContMDiff (I.tangent.prod I.tangent) (I.prod I).tangent (minSmoothness 𝕜 2) F₂
F₃ : TangentBundle (I.prod I) (G × G) → TangentBundle I G := tangentMap (I.prod I) I fun p ↦ p.1 * p.2
S₃ : ContMDiff (I.prod I).tangent I.tangent (minSmoothness 𝕜 2) F₃
S : ContMDiff I I.tangent (minSmoothness 𝕜 2) ((F₃ ∘ F₂) ∘ F₁) := ContMDiff.comp (ContMDiff.comp S₃ S₂) S₁
g : G
⊢ g = (((F₃ ∘ F₂) ∘ F₁) g).proj

case e'_5
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Group G
inst✝ : LieGroup I (minSmoothness 𝕜 3) G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
A : minSmoothness 𝕜 2 + 1 = minSmoothness 𝕜 3
fg : G → TangentBundle I G := ⋯
sfg : ContMDiff I I.tangent (minSmoothness 𝕜 2) fg
fv : G → TangentBundle I G := ⋯
sfv : ContMDiff I I.tangent (minSmoothness 𝕜 2) fv
F₁ : G → TangentBundle I G × TangentBundle I G := ⋯
S₁ : ContMDiff I (I.tangent.prod I.tangent) (minSmoothness 𝕜 2) F₁
F₂ : TangentBundle I G × TangentBundle I G → TangentBundle (I.prod I) (G × G) := ⋯
S₂ : ContMDiff (I.tangent.prod I.tangent) (I.prod I).tangent (minSmoothness 𝕜 2) F₂
F₃ : TangentBundle (I.prod I) (G × G) → TangentBundle I G := ⋯
S₃ : ContMDiff (I.prod I).tangent I.tangent (minSmoothness 𝕜 2) F₃
S : ContMDiff I I.tangent (minSmoothness 𝕜 2) ((F₃ ∘ F₂) ∘ F₁) := ⋯
g : G
e_4✝ : g = (((F₃ ∘ F₂) ∘ F₁) g).proj
⊢ mulInvariantVectorField v g = (((F₃ ∘ F₂) ∘ F₁) g).snd