mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q : M
hmem : ↑I.symm ⁻¹' (chartAt H q).target ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I q) q
z : H
hyT : ↑I z ∈ ↑I.symm ⁻¹' (chartAt H q).target
⊢ ?m.212 ∈ (chartAt H q).target

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q : M
hmem : ↑I.symm ⁻¹' (chartAt H q).target ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I q) q
z : H
hyT : ↑I z ∈ ↑I.symm ⁻¹' (chartAt H q).target
⊢ ?m.212 ∈ (chartAt H q).target
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q' : M'
hmem : ↑I.symm ⁻¹' (chartAt H q').target ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I q') q'
z : H
hyT : ↑I z ∈ ↑I.symm ⁻¹' (chartAt H q').target
⊢ ?m.212 ∈ (chartAt H q').target

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q' : M'
hmem : ↑I.symm ⁻¹' (chartAt H q').target ∩ range ↑I ∈ 𝓝[range ↑I] ↑(extChartAt I q') q'
z : H
hyT : ↑I z ∈ ↑I.symm ⁻¹' (chartAt H q').target
⊢ ?m.212 ∈ (chartAt H q').target
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
q : M
⊢ mfderiv% Sum.inl q = ContinuousLinearMap.id 𝕜 (TangentSpace I p)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
q : M
⊢ mfderiv% Sum.inl q = ContinuousLinearMap.id 𝕜 (TangentSpace I p)
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q' : M'
⊢ mfderiv% Sum.inr q' = ContinuousLinearMap.id 𝕜 (TangentSpace I q')

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q' : M'
⊢ mfderiv% Sum.inr q' = ContinuousLinearMap.id 𝕜 (TangentSpace I q')
Module: Mathlib.Geometry.Manifold.Diffeomorph

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁸ : TopologicalSpace H
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
G : Type u_7
inst✝⁶ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
M' : Type u_10
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n m : ℕ∞ω
h : M ≃ₘ^n⟮I, J⟯ N
f : N → M'
s : Set M
x : M
hm : m ≤ n
Hfh : ContMDiffWithinAt I I' m (f ∘ ⇑h) s (h.symm (h x))
⊢ ContMDiffWithinAt J I' m f (⇑h.symm ⁻¹' s) (h x)

Tactic:
simpa using

Goal state after:
case mp
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁸ : TopologicalSpace H
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
G : Type u_7
inst✝⁶ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
M' : Type u_10
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n m : ℕ∞ω
h : M ≃ₘ^n⟮I, J⟯ N
f : N → M'
s : Set M
x : M
hm : m ≤ n
Hfh : ContMDiffWithinAt I I' m (f ∘ ⇑h) s (h.symm (h x))
⊢ ContMDiffWithinAt J I' m f (⇑h.symm ⁻¹' s) (h x)
Module: Mathlib.Geometry.Manifold.Diffeomorph

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁸ : TopologicalSpace H
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
G : Type u_7
inst✝⁶ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
M' : Type u_10
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n m : ℕ∞ω
h : M ≃ₘ^n⟮I, J⟯ N
f : M' → M
hm : m ≤ n
s : Set M'
x : M'
Hhf : ContMDiffWithinAt I' J m (⇑h ∘ f) s x
⊢ ContMDiffWithinAt I' I m f s x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
H : Type u_5
inst✝⁸ : TopologicalSpace H
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
G : Type u_7
inst✝⁶ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
M : Type u_9
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
M' : Type u_10
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
N : Type u_11
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n m : ℕ∞ω
h : M ≃ₘ^n⟮I, J⟯ N
f : M' → M
hm : m ≤ n
s : Set M'
x : M'
Hhf : ContMDiffWithinAt I' J m (⇑h ∘ f) s x
⊢ ContMDiffWithinAt I' I m f s x
Module: Mathlib.Geometry.Manifold.Diffeomorph

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
H : Type u_5
inst✝¹¹ : TopologicalSpace H
H' : Type u_6
inst✝¹⁰ : TopologicalSpace H'
G : Type u_7
inst✝⁹ : TopologicalSpace G
G' : Type u_8
inst✝⁸ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F G'
M : Type u_9
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_10
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
N : Type u_11
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace G N
N' : Type u_12
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : ℕ∞ω
e : E ≃L[𝕜] E'
⊢ (⇑e '' interior (range ↑I)).Nonempty

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
H : Type u_5
inst✝¹¹ : TopologicalSpace H
H' : Type u_6
inst✝¹⁰ : TopologicalSpace H'
G : Type u_7
inst✝⁹ : TopologicalSpace G
G' : Type u_8
inst✝⁸ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F G'
M : Type u_9
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_10
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
N : Type u_11
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace G N
N' : Type u_12
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : ℕ∞ω
e : E ≃L[𝕜] E'
⊢ (interior (range ↑I)).Nonempty
Module: Mathlib.Geometry.Manifold.MFDeriv.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
inst✝ : IsManifold I 1 M
x y : M
hy : y ∈ (extChartAt I x).source
this : y = ↑(extChartAt I x).symm (↑(extChartAt I x) y)
⊢ y ∈ (extChartAt I x).source

Tactic:
simpa using

Goal state after:
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
inst✝ : IsManifold I 1 M
x y : M
hy : y ∈ (extChartAt I x).source
this : y = ↑(extChartAt I x).symm (↑(extChartAt I x) y)
⊢ y ∈ (chartAt H x).source
Module: Mathlib.Geometry.Manifold.MFDeriv.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
inst✝ : IsManifold I 1 M
x₀ x : M
hx : x ∈ (chartAt H x₀).source
⊢ MDiffAt[range ↑I] (↑(chartAt H x₀).symm ∘ ↑I.symm) (↑I (↑(chartAt H x₀) x))

Tactic:
simpa using

Goal state after:
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
inst✝ : IsManifold I 1 M
x₀ x : M
hx : x ∈ (chartAt H x₀).source
⊢ MDiffAt[range ↑I] (↑(chartAt H x₀).symm ∘ ↑I.symm) (↑I (↑(chartAt H x₀) x))
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
apply Subset.antisymm

Goal state after:
case h₁
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s ⊆ closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

case h₂
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s ⊆ ↑(extChartAt I c) '' s
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiffAt[s] p z
hq : MDiffAt[s] q z
hq_ne : q z ≠ 0
⊢ MDiffAt[s] (p / q) z

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiffAt[s] p z
hq : MDiffAt[s] q z
hq_ne : q z ≠ 0
⊢ MDiffAt[s] (p * q⁻¹) z
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
z : M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiffAt p z
hq : MDiffAt q z
hq_ne : q z ≠ 0
⊢ MDiffAt (p / q) z

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
z : M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiffAt p z
hq : MDiffAt q z
hq_ne : q z ≠ 0
⊢ MDiffAt (p * q⁻¹) z
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hsc : IsClosed[inst✝²] s
hs : s ⊆ support ↑f
⊢ IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑(extChartAt I c) '' s)

Tactic:
by
  rw [f.image_eq_inter_preimage_of_subset_support hs]
  refine ContinuousOn.preimage_isClosed_of_isClosed
    ((continuousOn_extChartAt_symm _).mono f.closedBall_subset) ?_ hsc
  exact IsClosed.inter isClosed_closedBall I.isClosed_range

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiff[s] p
hq : MDiff[s] q
hq_ne : ∀ z ∈ s, q z ≠ 0
⊢ MDiff[s] (p / q)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiff[s] p
hq : MDiff[s] q
hq_ne : ∀ z ∈ s, q z ≠ 0
⊢ MDiff[s] (p * q⁻¹)
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiff p
hq : MDiff q
hq_ne : ∀ (z : M), q z ≠ 0
⊢ MDiff (p / q)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F' : Type u_21
inst✝¹ : NormedDivisionRing F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
hp : MDiff p
hq : MDiff q
hq_ne : ∀ (z : M), q z ≠ 0
⊢ MDiff (p * q⁻¹)
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
p' q' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hq : HasMFDerivAt[s] q z q'
hq_ne : q z ≠ 0
⊢ HasMFDerivAt[s] (p / q) z ((1 / q z) • p' - (p z / q z ^ 2) • q')

Tactic:
convert! hp.mul (hq.inv hq_ne) using 1

Goal state after:
case e'_21
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
p' q' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hq : HasMFDerivAt[s] q z q'
hq_ne : q z ≠ 0
⊢ p / q = p * q⁻¹

case e'_24
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
p' q' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hq : HasMFDerivAt[s] q z q'
hq_ne : q z ≠ 0
e_21✝ : p / q = p * q⁻¹
⊢ (1 / q z) • p' - (p z / q z ^ 2) • q' = p z • -(q z ^ 2)⁻¹ • q' + q⁻¹ z • p'
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝¹ : FiniteDimensional ℝ E
inst✝ : T2Space M
⊢ tsupport ↑f ⊆ (chartAt H c).source

Tactic:
simpa using

Goal state after:
E : Type uE
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝¹ : FiniteDimensional ℝ E
inst✝ : T2Space M
⊢ tsupport ↑f ⊆ (chartAt H c).source
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
x : X
⊢ x ∈ Kdiff (K'.find x + 1)

Tactic:
simpa using

Goal state after:
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
x : X
⊢ x ∈ Kdiff (K'.find x + 1)
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a ↦ B (c a) (r a)

Tactic:
set K' : CompactExhaustion X := CompactExhaustion.choice X
  set K : CompactExhaustion X := K'.shiftr.shiftr
  set Kdiff := fun n ↦ K (n + 1) \ interior (K n)
  -- Now we restate some properties of `CompactExhaustion` for `K`/`Kdiff`
  have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) := fun x ↦ by
    simpa only [K'.find_shiftr] using
      sdiff_subset_sdiff_right interior_subset (K'.shiftr.mem_sdiff_shiftr_find x)
  have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) :=
    fun n ↦ ((K.isCompact _).diff isOpen_interior).inter_right hs
  -- Next we choose a finite covering `B (c n i) (r n i)` of each
  -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
  have : ∀ (n) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 (x : X) :=
    fun n x ↦ (K.isClosed n).compl_mem_nhds fun hx' ↦ x.2.1.2 <| K.subset_interior_succ _ hx'
  choose! r hrp hr using fun n (x : ↑(Kdiff (n + 1) ∩ s)) ↦ (hB x x.2.2).mem_iff.1 (this n x)
  have hxr : ∀ (n x) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x := fun n x hx ↦
    (hB x hx.2).mem_of_mem (hrp _ ⟨x, hx⟩)
  choose T hT using fun n ↦ (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n)
  set T' : ∀ n, Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ T n
  -- Finally, we take the union of all these coverings
  refine ⟨Σ n, T' n, fun a ↦ a.2, fun a ↦ r a.1 a.2, ?_, ?_, ?_⟩
  · rintro ⟨n, x, hx⟩
    exact ⟨x.2.2, hrp _ _⟩
  · refine fun x hx ↦ mem_iUnion.2 ?_
    rcases mem_iUnion₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
    exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩
  · intro x
    refine
      ⟨interior (K (K'.find x + 3)),
        IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), ?_⟩
    have : (⋃ k ≤ K'.find x + 2, range (Sigma.mk k) : Set (Σ n, T' n)).Finite :=
      (finite_le_nat _).biUnion fun k _ ↦ finite_range _
    apply this.subset
    rintro ⟨k, c, hc⟩
    simp only [mem_iUnion, mem_ofPred_eq, Subtype.coe_mk]
    rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
    refine ⟨k, ?_, ⟨c, hc⟩, rfl⟩
    have := (mem_compl_iff _ _).1 (hr k c hxB)
    contrapose! this with hnk
    exact K.subset hnk (interior_subset hxK)

Goal state after: