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
m n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
h : ContMDiffWithinAt I I' n f s x
hm : ω ≤ n
h' : ω = ∞ → n = ω
this✝ : IsManifold I ω M
this : IsManifold I' ω M'
⊢ ω ≠ ∞

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
hf : ContMDiffOn I' I n f s
hg : ContMDiffOn I' I n g s
h₀ : ∀ x ∈ s, g x ≠ 0
⊢ ContMDiffOn I' I n (f / g) s

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
a : M
hf : ContMDiffAt I' I n f a
hg : ContMDiffAt I' I n g a
h₀ : g a ≠ 0
⊢ ContMDiffAt I' I n (f / g) a

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
hf : ContMDiff I' I n f
hg : ContMDiff I' I n g
h₀ : ∀ (x : M), g x ≠ 0
⊢ ContMDiff I' I n (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
m n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
h : ContMDiffWithinAt I I' n f s x
hm : ∞ ≤ n
h' : ∞ = ∞ → n = ω
this✝ : IsManifold I ∞ M
this : IsManifold I' ∞ M'
⊢ n ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
⊢ UniqueMDiffAt[s] x ↔ UniqueDiffWithinAt 𝕜 s x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ UniqueMDiff[s] ↔ UniqueDiffOn 𝕜 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'
hm : m ≤ n
h' : m = ∞ → n = ω
h : ContMDiffWithinAt I I' n f s x
⊢ ∃ u, IsOpen[inst✝⁸] u ∧ x ∈ u ∧ ContMDiffOn I I' m f (insert x s ∩ u)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
s : Set E
x : E
f' : TangentSpace 𝓘(𝕜, E) x →L[𝕜] TangentSpace 𝓘(𝕜, E') (f x)
⊢ HasMFDerivAt[s] f x f' ↔ HasFDerivWithinAt f f' s 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
m n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hm : m ≤ n
h' : m = ∞ → n = ω
h : ContMDiffWithinAt I I' n f s x
⊢ ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ContMDiffOn I I' m f u

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'
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
⊢ ContMDiffAt I I' n f x ↔ ∃ u ∈ 𝓝 x, ContMDiffOn I I' n f u

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
x : E
f' : TangentSpace 𝓘(𝕜, E) x →L[𝕜] TangentSpace 𝓘(𝕜, E') (f x)
⊢ HasMFDerivAt% f x f' ↔ HasFDerivAt f f' x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
s : Set E
x : E
⊢ MDiffAt[s] f x ↔ DifferentiableWithinAt 𝕜 f s x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
x : E
⊢ MDiffAt f x ↔ DifferentiableAt 𝕜 f x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
s : Set E
⊢ MDiff[s] f ↔ DifferentiableOn 𝕜 f s

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
⊢ MDiff f ↔ Differentiable 𝕜 f

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'
x : M
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
⊢ ContMDiffAt I I' n f x ↔ ∀ᶠ (x' : M) in 𝓝 x, ContMDiffAt I I' n f x'

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
s : Set E
x : E
⊢ mfderiv[s] f x = fderivWithin 𝕜 f s x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
x : E
⊢ mfderiv% f x = fderiv 𝕜 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 ≠ ∞
⊢ ContMDiffWithinAt I I' n f s x ↔ ∀ᶠ (x' : M) in 𝓝[insert x s] x, ContMDiffWithinAt I I' n f s x'

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
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 f₁ : M → M'
n : ℕ∞ω
h : ContMDiff I I' n f
h₁ : ∀ (y : M), f₁ y = f y
⊢ ContMDiff I I' n f₁

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
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 f₁ : M → M'
n : ℕ∞ω
h₁ : ∀ (y : M), f₁ y = f y
⊢ ContMDiff I I' n f₁ ↔ ContMDiff I I' n f

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
x : H
y : E
hy : y ∈ range ↑I
⊢ (↑I ∘ ↑I.symm) y = id y

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
x : H
⊢ (↑I ∘ ↑I.symm) (↑I x) = id (↑I x)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
⊢ ContMDiff I 𝓘(𝕜, E) n ↑I

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
x : E
hx : x ∈ range ↑I
⊢ ContinuousWithinAt (↑I.symm) (range ↑I) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
x : E
hx : x ∈ range ↑I
y : E
hy : y ∈ range ↑I
⊢ (↑I ∘ ↑I.symm) y = id y

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
x : E
hx : x ∈ range ↑I
⊢ (↑I ∘ ↑I.symm) x = id x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
⊢ ContMDiffOn 𝓘(𝕜, E) I n (↑I.symm) (range ↑I)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
x : M
⊢ ↑(extChartAt I x) x ∈ range ↑I

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
x : M
⊢ ContMDiffAt I 𝓘(𝕜, E) n (↑(extChartAt I x)) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
⊢ ContMDiffOn 𝓘(𝕜, E) I n (↑(e.extend I).symm) (↑I '' e.target)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
x : M
⊢ ContMDiffOn 𝓘(𝕜, E) I n (↑(extChartAt I x).symm) (extChartAt I x).target

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
x : M
⊢ ↑I.symm (↑(extChartAt I x) x) ∈ (chartAt H x).symm.source

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
x : M
⊢ ↑(extChartAt I x) x ∈ (extChartAt I x).target

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
x : M
⊢ ContMDiffWithinAt 𝓘(𝕜, E) I n (↑(extChartAt I x).symm) (extChartAt I x).target (↑(extChartAt I x) x)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : TopologicalSpace B
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
φ φ' : B → F ≃L[𝕜] F
U U' : Set B
hU : IsOpen[inst✝³] U
hφ : ContinuousOn (fun x ↦ ↑(φ x)) U
h2φ : ContinuousOn (fun x ↦ ↑(φ x).symm) U
hU' : IsOpen[inst✝³] U'
hφ' : ContinuousOn (fun x ↦ ↑(φ' x)) U'
h2φ' : ContinuousOn (fun x ↦ ↑(φ' x).symm) U'
⊢ (openPartialHomeomorph φ hU hφ h2φ ≫ₕ openPartialHomeomorph φ' hU' hφ' h2φ').source = (U ∩ U') ×ˢ univ

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : TopologicalSpace B
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
φ φ' : B → F ≃L[𝕜] F
U U' : Set B
hU : IsOpen[inst✝³] U
hφ : ContinuousOn (fun x ↦ ↑(φ x)) U
h2φ : ContinuousOn (fun x ↦ ↑(φ x).symm) U
hU' : IsOpen[inst✝³] U'
hφ' : ContinuousOn (fun x ↦ ↑(φ' x)) U'
h2φ' : ContinuousOn (fun x ↦ ↑(φ' x).symm) U'
⊢ (openPartialHomeomorph φ hU hφ h2φ ≫ₕ openPartialHomeomorph φ' hU' hφ' h2φ').target = (U ∩ U') ×ˢ univ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
φ : OpenPartialHomeomorph M H
hφ : ContMDiffOn I I n (↑φ) φ.source
hφ' : ContMDiffOn I I n (↑φ.symm) φ.target
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
he' : ContMDiffOn I I n (↑e) e.source
he'' : ContMDiffOn I I n (↑e.symm) e.target
⊢ ↑I.symm ⁻¹' φ.target ∩ ↑I.symm ⁻¹' ↑φ.symm ⁻¹' e.source ∩ range ↑I ⊆ range ↑I

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
φ : OpenPartialHomeomorph M H
hφ : ContMDiffOn I I n (↑φ) φ.source
hφ' : ContMDiffOn I I n (↑φ.symm) φ.target
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
he' : ContMDiffOn I I n (↑e) e.source
he'' : ContMDiffOn I I n (↑e.symm) e.target
⊢ ↑I.symm ⁻¹' φ.target ∩ ↑I.symm ⁻¹' ↑φ.symm ⁻¹' e.source ∩ range ↑I ⊆ ↑I.symm ⁻¹' φ.target

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
φ : OpenPartialHomeomorph M H
hφ : ContMDiffOn I I n (↑φ) φ.source
hφ' : ContMDiffOn I I n (↑φ.symm) φ.target
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
he' : ContMDiffOn I I n (↑e) e.source
he'' : ContMDiffOn I I n (↑e.symm) e.target
⊢ ↑I.symm ⁻¹' φ.target ∩ ↑I.symm ⁻¹' ↑φ.symm ⁻¹' e.source ∩ range ↑I ⊆ ↑φ.symm ∘ ↑I.symm ⁻¹' e.source

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
φ : OpenPartialHomeomorph M H
hφ : ContMDiffOn I I n (↑φ) φ.source
hφ' : ContMDiffOn I I n (↑φ.symm) φ.target
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
he' : ContMDiffOn I I n (↑e) e.source
he'' : ContMDiffOn I I n (↑e.symm) e.target
⊢ ↑I.symm ⁻¹' e.target ∩ ↑I.symm ⁻¹' ↑e.symm ⁻¹' φ.source ∩ range ↑I ⊆ range ↑I

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
φ : OpenPartialHomeomorph M H
hφ : ContMDiffOn I I n (↑φ) φ.source
hφ' : ContMDiffOn I I n (↑φ.symm) φ.target
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
he' : ContMDiffOn I I n (↑e) e.source
he'' : ContMDiffOn I I n (↑e.symm) e.target
⊢ ↑I.symm ⁻¹' e.target ∩ ↑I.symm ⁻¹' ↑e.symm ⁻¹' φ.source ∩ range ↑I ⊆ ↑I.symm ⁻¹' e.target

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
φ : OpenPartialHomeomorph M H
hφ : ContMDiffOn I I n (↑φ) φ.source
hφ' : ContMDiffOn I I n (↑φ.symm) φ.target
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
he' : ContMDiffOn I I n (↑e) e.source
he'' : ContMDiffOn I I n (↑e.symm) e.target
⊢ ↑I.symm ⁻¹' e.target ∩ ↑I.symm ⁻¹' ↑e.symm ⁻¹' φ.source ∩ range ↑I ⊆ ↑e.symm ∘ ↑I.symm ⁻¹' φ.source

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
φ : OpenPartialHomeomorph M H
hφ : ContMDiffOn I I n (↑φ) φ.source
hφ' : ContMDiffOn I I n (↑φ.symm) φ.target
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
he' : ContMDiffOn I I n (↑e) e.source
he'' : ContMDiffOn I I n (↑e.symm) e.target
⊢ ↑I.symm ⁻¹' e.target ∩ ↑I.symm ⁻¹' ↑e.symm ⁻¹' φ.source ∩ range ↑I ⊆ (fun x ↦ ↑e.symm (↑I.symm x)) ⁻¹' φ.source

Tactic:
by

Goal state after: