mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
⊢ mlieBracketWithin I V W s x = -mlieBracketWithin I W V s x

Tactic:
by

Goal state after:
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
g h : G
M : minSmoothness 𝕜 3 ≠ 0
⊢ ((fun x ↦ g⁻¹ * x) ∘ fun x ↦ g * x) = id

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt[t] (T% V) (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : 2 ≤ n
b₁ : M → M' := f
b₂ : M → M := id
v : (x : M) → TangentSpace I' (f x) := V ∘ f
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := fun x ↦ (mfderiv[s] f x).inverse
hv : (MDiffAt[s ∩ f ⁻¹' t] fun x ↦ ⟨f x, v x⟩) x₀
this✝ : (MDiffAt[s] fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x)) x₀
this :
  MDiffAt[s]
      (inverse ∘ fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x))
    x₀
A : (trivializationAt E (TangentSpace I) x₀).baseSet ∈ 𝓝[s] x₀
B : f ⁻¹' (trivializationAt E' (TangentSpace I') (f x₀)).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (trivializationAt E (TangentSpace I) x₀).baseSet
h'x : x ∈ f ⁻¹' (trivializationAt E' (TangentSpace I') (f x₀)).baseSet
⊢ b₂ x ∈ (trivializationAt E (TangentSpace I) (b₂ x₀)).baseSet

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
V W : (x : M) → TangentSpace I x
⊢ mlieBracketWithin I V W s = -mlieBracketWithin I W V s

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
V : (x : M) → TangentSpace I x
⊢ mlieBracketWithin I V V = 0

Tactic:
by

Goal state after:
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
g h : G
M : minSmoothness 𝕜 3 ≠ 0
⊢ mfderiv% ((fun x ↦ g⁻¹ * x) ∘ fun x ↦ g * x) h = ContinuousLinearMap.id 𝕜 (TangentSpace I h)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
V : (x : M) → TangentSpace I x
⊢ mlieBracket I V V = 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
W : (x : M) → TangentSpace I x
⊢ mlieBracketWithin I 0 W s = 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
W : (x : M) → TangentSpace I x
⊢ mlieBracketWithin I W 0 s = 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
W : (x : M) → TangentSpace I x
⊢ mlieBracket I 0 W = 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
W : (x : M) → TangentSpace I x
⊢ mlieBracket I W 0 = 0

Tactic:
by

Goal state after:
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
g h : G
M : minSmoothness 𝕜 3 ≠ 0
A : (mfderiv% fun x ↦ g⁻¹ * x) (g * h) ∘SL (mfderiv% fun x ↦ g * x) h = ContinuousLinearMap.id 𝕜 (TangentSpace I h)
⊢ ((fun x ↦ g * x) ∘ fun x ↦ g⁻¹ * x) = id

Tactic:
by

Goal state after:
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
g h : G
M : minSmoothness 𝕜 3 ≠ 0
A : (mfderiv% fun x ↦ g⁻¹ * x) (g * h) ∘SL (mfderiv% fun x ↦ g * x) h = ContinuousLinearMap.id 𝕜 (TangentSpace I h)
⊢ mfderiv% ((fun x ↦ g * x) ∘ fun x ↦ g⁻¹ * x) (g * h) = ContinuousLinearMap.id 𝕜 (TangentSpace I (g * h))

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s t : Set M
x : M
V W : (x : M) → TangentSpace I x
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
A : ↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I =ᶠ[𝓝[≠] ↑(extChartAt I x) x] ↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I
⊢ 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) =
    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' t ∩ range ↑I)
      (↑(extChartAt I x) x)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s t : Set M
x : M
V W : (x : M) → TangentSpace I x
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
⊢ mlieBracketWithin I V W s x = mlieBracketWithin I V W t x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt[t] (T% V) (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : 2 ≤ n
⊢ (MDiffAt[s ∩ f ⁻¹' t] fun x ↦ ⟨x, mpullbackWithin I I' f V s x⟩) x₀

Tactic:
by

Goal state after:
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
g h : G
⊢ ((mfderiv% fun b ↦ g * b) h).inverse = (mfderiv% fun b ↦ g⁻¹ * b) (g * h)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s t : Set M
x : M
V W : (x : M) → TangentSpace I x
ht : t ∈ 𝓝 x
⊢ mlieBracketWithin I V W (s ∩ t) x = mlieBracketWithin I V W s x

Tactic:
by

Goal state after:
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
g : G
v : GroupLieAlgebra I G
⊢ 0 < 3

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
h : s ∈ 𝓝 x
⊢ mlieBracketWithin I V W s x = mlieBracket I V W x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
y₀ : M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt[t] (T% V) y₀
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : 2 ≤ n
h : y₀ = f x₀
⊢ (MDiffAt[s ∩ f ⁻¹' t] fun x ↦ ⟨x, mpullbackWithin I I' f V s x⟩) x₀

Tactic:
by

Goal state after:
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
g : G
v : GroupLieAlgebra I G
M : minSmoothness 𝕜 3 ≠ 0
h : G
⊢ (fun x ↦ h * x) = (fun b ↦ g⁻¹ * b) ∘ fun x ↦ g * h * x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt[t] (T% V) (f x₀)
hf : ContMDiffAt I I' n f x₀
hf' : (mfderiv% f x₀).IsInvertible
hmn : 2 ≤ n
⊢ (MDiffAt[f ⁻¹' t] fun x ↦ ⟨x, mpullback I I' f V x⟩) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
y₀ : M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt[t] (T% V) y₀
hf : ContMDiffAt I I' n f x₀
hf' : (mfderiv% f x₀).IsInvertible
hmn : 2 ≤ n
hy₀ : y₀ = f x₀
⊢ (MDiffAt[f ⁻¹' t] fun x ↦ ⟨x, mpullback I I' f V x⟩) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt (T% V) (f x₀)
hf : ContMDiffAt I I' n f x₀
hf' : (mfderiv% f x₀).IsInvertible
hmn : 2 ≤ n
⊢ (MDiffAt fun x ↦ ⟨x, mpullback I I' f V x⟩) x₀

Tactic:
by

Goal state after:
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
g : G
v : GroupLieAlgebra I G
⊢ mpullback I I (fun x ↦ g * x) (mulInvariantVectorField v) = mulInvariantVectorField v

Tactic:
by

Goal state after:
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
g : G
V : (g : G) → TangentSpace I g
⊢ 1 = g⁻¹ * g

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) :=
  { toNorm := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1,
    toAddCommGroup := instAddCommGroupTangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x),
    toPseudoMetricSpace := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_13, eq_of_dist_eq_zero := ⋯,
    dist_eq := ⋯ }
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
⊢ x = ↑(extChartAt I x).symm (↑(extChartAt I x) x)

Tactic:
by

Goal state after:
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
g : G
V : (g : G) → TangentSpace I g
⊢ mulInvariantVectorField (V 1) g = mpullback I I (fun x ↦ g⁻¹ * x) V g

Tactic:
by

Goal state after:
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
⊢ 0 < 3

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
m n : ℕ∞ω
t : Set M'
inst✝² : CompleteSpace E
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
hV : ContMDiffWithinAt I' I'.tangent m (fun x ↦ ⟨x, V x⟩) t (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : m + 1 ≤ n
b₁ : M → M' := f
b₂ : M → M := id
v : (x : M) → TangentSpace I' (f x) := V ∘ f
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := fun x ↦ (mfderiv[s] f x).inverse
⊢ ContMDiffWithinAt I I'.tangent m (fun x ↦ ⟨f x, v x⟩) (s ∩ f ⁻¹' t) x₀

Tactic:
by

Goal state after:
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
⊢ minSmoothness 𝕜 2 + 1 = minSmoothness 𝕜 3

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
m n : ℕ∞ω
t : Set M'
inst✝² : CompleteSpace E
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
hV : ContMDiffWithinAt I' I'.tangent m (fun x ↦ ⟨x, V x⟩) t (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : m + 1 ≤ n
b₁ : M → M' := f
b₂ : M → M := id
v : (x : M) → TangentSpace I' (f x) := V ∘ f
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := fun x ↦ (mfderiv[s] f x).inverse
hv : ContMDiffWithinAt I I'.tangent m (fun x ↦ ⟨f x, v x⟩) (s ∩ f ⁻¹' t) x₀
this :
  ContMDiffWithinAt I 𝓘(𝕜, E →L[𝕜] E') m
    (fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x)) s x₀
⊢ ContMDiffWithinAt I 𝓘(𝕜, E' →L[𝕜] E) m
    (inverse ∘ fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x)) s x₀

Tactic:
by

Goal state after:
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
⊢ ContMDiff (I.prod I).tangent I.tangent (minSmoothness 𝕜 2) F₃

Tactic:
by

Goal state after: