-- Mathlib/Geometry/Manifold/VectorField.lean
lemma mlieBracketWithin_smul_right
    (hW : MDifferentiableWithinAt I I.tangent (fun x ↦ (W x : TangentBundle I M)) s x)
    (hs : UniqueMDiffWithinAt I s x) :
    mlieBracketWithin I V (c • W) s x = c • mlieBracketWithin I V W s x

Proof state:
case hW
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDifferentiableWithinAt I I.tangent (fun x => { proj := x, snd := W x }) s x
hs : UniqueMDiffWithinAt I s x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

Tactic:
exact hW.differentiableWithinAt_mpullbackWithin_vectorField

Result:
no goals

Proof state:
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDifferentiableWithinAt I I.tangent (fun x => { proj := x, snd := W x }) s x
hs : UniqueMDiffWithinAt I s x
⊢ mlieBracketWithin I V (c • W) s x = c • mlieBracketWithin I V W s x

Tactic:
simp only [mlieBracketWithin_apply]

Result:
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDifferentiableWithinAt I I.tangent (fun x => { proj := x, snd := W x }) s x
hs : UniqueMDiffWithinAt I s x
⊢ (mfderiv I 𝓘(𝕜, E) (↑(extChartAt I x)) x).inverse
      (lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
        (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) (c • W) (range ↑I))
        (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)) =
    c •
      (mfderiv I 𝓘(𝕜, E) (↑(extChartAt I x)) x).inverse
        (lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
          (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
          (↑(extChartAt I x) x))

Proof state:
case hs
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDifferentiableWithinAt I I.tangent (fun x => { proj := x, snd := W x }) s x
hs : UniqueMDiffWithinAt I s x
⊢ UniqueDiffWithinAt 𝕜 (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

Tactic:
exact uniqueMDiffWithinAt_iff_inter_range.1 hs

Result:
no goals

Proof state:
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDifferentiableWithinAt I I.tangent (fun x => { proj := x, snd := W x }) s x
hs : UniqueMDiffWithinAt I s x
⊢ (mfderiv I 𝓘(𝕜, E) (↑(extChartAt I x)) x).inverse
      (lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
        (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) (c • W) (range ↑I))
        (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)) =
    c •
      (mfderiv I 𝓘(𝕜, E) (↑(extChartAt I x)) x).inverse
        (lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
          (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
          (↑(extChartAt I x) x))

Tactic:
rw [← ContinuousLinearMap.map_smul, mpullbackWithin_smul, lieBracketWithin_smul_right]

Result:
case hW
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDifferentiableWithinAt I I.tangent (fun x => { proj := x, snd := W x }) s x
hs : UniqueMDiffWithinAt I s x
⊢ DifferentiableWithinAt 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I))
    (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

case hs
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDifferentiableWithinAt I I.tangent (fun x => { proj := x, snd := W x }) s x
hs : UniqueMDiffWithinAt I s x
⊢ UniqueDiffWithinAt 𝕜 (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)