mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s✝ : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
h : ContMDiffWithinAt I I' n f s x
⊢ n = ∞ → n = ω

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
n : WithTop ℕ∞
f : E → E'
s : Set E
x : E
⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiffWithinAt 𝕜 n f s x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
n : WithTop ℕ∞
f : E → E'
x : E
⊢ ContMDiffAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f x ↔ ContDiffAt 𝕜 n f x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s✝ : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
h : ContMDiffWithinAt I I' n f s x
v : Set E
hmem :
  v ∈
    𝓝[insert (↑(extChartAt I x) x)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))]
      ↑(extChartAt I x) x
hsub :
  v ⊆
    insert (↑(extChartAt I x) x)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
hv : ContDiffOn 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) v
⊢ ↑(extChartAt I x).symm (↑(extChartAt I x) x) ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
n : WithTop ℕ∞
f : E → E'
s : Set E
⊢ ∀ (a : E), a ∈ s → ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f s a ↔ a ∈ s → ContDiffWithinAt 𝕜 n f s a

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
n : WithTop ℕ∞
f : E → E'
⊢ ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, E') n f ↔ ContDiff 𝕜 n f

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝² : Mul G
inst✝¹ : TopologicalSpace G
inst✝ : ChartedSpace H G
m n : ℕ∞ω
hmn : m ≤ n
h : ContMDiffMul I n G
⊢ ContMDiffMul I m G

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
G : Type u_4
inst✝⁸ : Mul G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
inst✝ : ContinuousMul G
⊢ ContMDiffMul I 0 G

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_8
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_9
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
n : WithTop ℕ∞
g : M → F₁ →L[𝕜] F₂
f : M → F₁
s : Set M
x : M
hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g s x
hf : ContMDiffWithinAt I 𝓘(𝕜, F₁) n f s x
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ x.1 x.2) univ (g x, f x)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

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_8
inst✝³ : Semigroup G
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : ContMDiffMul I ∞ G
g h : G
⊢ 𝑳 I (g * h) = (𝑳 I g).comp (𝑳 I h)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

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_8
inst✝³ : Semigroup G
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : ContMDiffMul I ∞ G
g h : G
⊢ 𝑹 I (g * h) = (𝑹 I h).comp (𝑹 I g)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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 : Monoid G
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : ContMDiffMul I n G
⊢ ContMDiff I I n fun a ↦ a ^ 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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 : Monoid G
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : ContMDiffMul I n G
k : ℕ
⊢ ContMDiff I I n fun a ↦ a ^ (k + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_8
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_9
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
n : WithTop ℕ∞
g : M → F₁ →L[𝕜] F₂
f : M → F₁
s : Set M
x : M
hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g s x
hf : ContMDiffWithinAt I 𝓘(𝕜, F₁) n f s x
⊢ s ⊆ (fun x ↦ (g x, f x)) ⁻¹' univ

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
n : ℕ∞ω
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝¹⁰ : Monoid G
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : ContMDiffMul I n G
H' : Type u_5
inst✝⁶ : TopologicalSpace H'
E' : Type u_6
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
G' : Type u_7
inst✝³ : Monoid G'
inst✝² : TopologicalSpace G'
inst✝¹ : ChartedSpace H' G'
inst✝ : ContMDiffMul I' n G'
f g : ContMDiffMonoidMorphism I I' n G G'
h : (fun a ↦ (↑a.toMonoidHom).toFun) f = (fun a ↦ (↑a.toMonoidHom).toFun) g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s✝ : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
h : ContMDiffWithinAt I I' n f s x
v : Set E
hmem :
  v ∈
    𝓝[insert (↑(extChartAt I x) x)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))]
      ↑(extChartAt I x) x
hsub :
  v ⊆
    insert (↑(extChartAt I x) x)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
hv : ContDiffOn 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) v
⊢ ↑(extChartAt I x).symm (↑(extChartAt I x) x) ∈ f ⁻¹' (extChartAt I' (f x)).source

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
n : ℕ∞ω
H : Type u_3
inst✝¹¹ : TopologicalSpace H
E : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
inst✝⁸ : CommMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : ContMDiffMul I n G
E' : Type u_6
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
s : Set M
x₀ : M
t : Finset ι
f : ι → M → G
h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x₀
⊢ ContMDiffWithinAt I' I n (fun x ↦ ∏ i ∈ t, f i x) s x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
n : ℕ∞ω
H : Type u_3
inst✝¹¹ : TopologicalSpace H
E : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
inst✝⁸ : CommMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : ContMDiffMul I n G
E' : Type u_6
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
s : Set M
x : M
t : Finset ι
f : ι → M → G
h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x
⊢ ContMDiffWithinAt I' I n (fun x ↦ ∏ i ∈ t, f i x) s x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
n : ℕ∞ω
H : Type u_3
inst✝¹¹ : TopologicalSpace H
E : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
inst✝⁸ : CommMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : ContMDiffMul I n G
E' : Type u_6
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
x₀ : M
t : Finset ι
f : ι → M → G
h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀
⊢ ContMDiffAt I' I n (fun x ↦ ∏ i ∈ t, f i x) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
n : ℕ∞ω
H : Type u_3
inst✝¹¹ : TopologicalSpace H
E : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
inst✝⁸ : CommMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : ContMDiffMul I n G
E' : Type u_6
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
f : ι → M → G
p : ι → Prop
hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)
hf : LocallyFinite fun i ↦ mulSupport (f i)
⊢ ContMDiff I' I n fun x ↦ ∏ᶠ (i : ι) (_ : p i), f i x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n : ℕ∞ω
⊢ ContMDiffAdd 𝓘(𝕜, E) n E

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
n : ℕ∞ω
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝⁸ : DivInvMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : ContMDiffMul I n G
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
f : M → G
s : Set M
x : M
c : G
hf : ContMDiffWithinAt I' I n f s x
⊢ ContMDiffWithinAt I' I n (fun x ↦ f x / c) s x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Structures

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
n : ℕ∞ω
I : ModelWithCorners 𝕜 E H
R : Type u_4
inst✝³ : Ring R
inst✝² : TopologicalSpace R
inst✝¹ : ChartedSpace H R
inst✝ : ContMDiffRing I n R
⊢ ContMDiff I I n fun a ↦ -a

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Structures

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ∞ω
⊢ ContMDiff (𝓘(𝕜, 𝕜).prod 𝓘(𝕜, 𝕜)) 𝓘(𝕜, 𝕜) n fun p ↦ p.1 * p.2

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s✝ : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
h : ContMDiffWithinAt I I' n f s x
v : Set E
hmem :
  v ∈
    𝓝[(extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)] ↑(extChartAt I x) x
hsub : v ⊆ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)
hv : ContDiffOn 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) v
hxs' :
  ↑(extChartAt I x) x ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)
hv₁ : ↑(extChartAt I x).symm '' v ⊆ (extChartAt I x).source
⊢ MapsTo f (↑(extChartAt I x).symm '' v) (extChartAt I' (f x)).source

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : Group G
m n : ℕ∞ω
hmn : m ≤ n
h : LieGroup I n G
⊢ LieGroup I m G

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
G : Type u_4
inst✝⁸ : TopologicalSpace G
inst✝⁷ : ChartedSpace H G
inst✝⁶ : Group G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
inst✝ : IsTopologicalGroup G
⊢ LieGroup I 0 G

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
G : Type u_4
inst✝⁸ : TopologicalSpace G
inst✝⁷ : ChartedSpace H G
inst✝⁶ : Group G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
inst✝ : LieGroup I n G
f g : M → G
s : Set M
x₀ : M
hf : ContMDiffWithinAt I' I n f s x₀
hg : ContMDiffWithinAt I' I n g s x₀
⊢ ContMDiffWithinAt I' I n (fun x ↦ f x / g x) s x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
G : Type u_4
inst✝⁸ : TopologicalSpace G
inst✝⁷ : ChartedSpace H G
inst✝⁶ : Group G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
inst✝ : LieGroup I n G
f g : M → G
x₀ : M
hf : ContMDiffAt I' I n f x₀
hg : ContMDiffAt I' I n g x₀
⊢ ContMDiffAt I' I n (fun x ↦ f x / g x) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
⊢ ContMDiffWithinAt I I' n f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ContMDiffOn I I' n f u

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
G : Type u_4
inst✝⁸ : TopologicalSpace G
inst✝⁷ : ChartedSpace H G
inst✝⁶ : Group G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
inst✝ : LieGroup I n G
f g : M → G
s : Set M
hf : ContMDiffOn I' I n f s
hg : ContMDiffOn I' I n g s
⊢ ContMDiffOn I' I n (fun x ↦ f x / g x) s

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
m✝ n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
h : ContMDiffWithinAt I I' n f s x
m : ℕ
hm : ↑m ≤ n
h' : ↑m = ∞ → n = ω
this✝ : IsManifold I (↑m) M
this : IsManifold I' (↑m) M'
⊢ ↑m ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹¹ : TopologicalSpace H
E : Type u_3
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
G : Type u_4
inst✝⁸ : TopologicalSpace G
inst✝⁷ : ChartedSpace H G
inst✝⁶ : Group G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
inst✝ : LieGroup I n G
f g : M → G
hf : ContMDiff I' I n f
hg : ContMDiff I' I n g
⊢ ContMDiff I' I n fun x ↦ f x / g x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ∞ω
x : 𝕜
hx : x ≠ 0
⊢ ContMDiffAt 𝓘(𝕜, 𝕜) 𝓘(𝕜, 𝕜) n (fun y ↦ y⁻¹) x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

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 : Inv G
inst✝ : Zero G
m n : ℕ∞ω
hmn : m ≤ n
h : ContMDiffInv₀ I n G
⊢ ContMDiffInv₀ I m G

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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
G : Type u_4
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : Inv G
inst✝⁶ : Zero G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
f : M → G
inst✝ : ContinuousInv₀ G
this : T1Space G
⊢ ContMDiffOn I I 0 (fun x ↦ x⁻¹) {0}ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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
G : Type u_4
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : Inv G
inst✝⁶ : Zero G
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H' M
f : M → G
inst✝ : ContinuousInv₀ G
⊢ ContMDiffInv₀ I 0 G

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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
G : Type u_4
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : GroupWithZero G
inst✝⁶ : ContMDiffInv₀ I n G
inst✝⁵ : ContMDiffMul I n G
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
f g : M → G
s : Set M
a : M
hf : ContMDiffWithinAt I' I n f s a
hg : ContMDiffWithinAt I' I n g s a
h₀ : g a ≠ 0
⊢ ContMDiffWithinAt I' I n (f / g) s a

Tactic:
by

Goal state after: