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

Goal state before:
pp: case refine_1
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' ↑e.symm ⁻¹' s) x
⊢ (f ∘ ↑e.symm) ∘ ↑e =ᶠ[𝓝 x] f

pp: case refine_2
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' ↑e.symm ⁻¹' s) x
⊢ ↑e ⁻¹' ↑e.symm ⁻¹' s =ᶠ[𝓝 x] s

Tactic:
· refine eventually_of_mem (e.open_source.mem_nhds hxe) fun x' hx' ↦ ?_
    simp_rw [Function.comp_apply, e.left_inv hx']

Goal state after:
case refine_2
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' ↑e.symm ⁻¹' s) x
⊢ ↑e ⁻¹' ↑e.symm ⁻¹' s =ᶠ[𝓝 x] s
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ P (g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔ P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

Tactic:
rw [← hG.right_invariance (compatible_of_mem_maximalAtlas_right (x := x) he)]

Goal state after:
H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ P ((g ∘ ↑(chartAt H x).symm) ∘ ↑((chartAt H x).symm ≫ₕ e).symm)
      (↑((chartAt H x).symm ≫ₕ e).symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s) (↑((chartAt H x).symm ≫ₕ e) (↑(chartAt H x) x)) ↔
    P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ ↑(chartAt H x) x ∈ ((chartAt H x).symm ≫ₕ e).source
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case refine_2
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
s : Set H
⊢ ↑I.symm ⁻¹' s ∩ I.target = ↑I.symm ⁻¹' s ∩ range ↑I

Tactic:
I.target_eq,

Goal state after:
case refine_2
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
s : Set H
⊢ ↑I.symm ⁻¹' s ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
⊢ x ∈ s → LiftPropWithinAt P g s x

Tactic:
hx

Goal state after:
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
hG : G.LocalInvariantProp G' P
h : ∀ x ∈ s, ∃ u, IsOpen[inst✝⁴] u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)
x : M
hx : x ∈ s
⊢ LiftPropWithinAt P g s x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : TopologicalSpace H'
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g g' : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
h : LiftPropWithinAt P g s x
h₁ : g' =ᶠ[𝓝[s] x] g
hx : g' x = g x
⊢ ↑(chartAt H' (g' x)) (g' x) = ↑(chartAt H' (g x)) (g x)

Tactic:
hx

Goal state after:
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
x : M
⊢ x ∈ e.source → LiftPropWithinAt Q (↑e) e.source x

Tactic:
hx

Goal state after:
H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
x : M
hx : x ∈ e.source
⊢ LiftPropWithinAt Q (↑e) e.source x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
x : H
⊢ x ∈ e.target → LiftPropWithinAt Q (↑e.symm) e.target x

Tactic:
hx

Goal state after:
H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
x : H
hx : x ∈ e.target
⊢ LiftPropWithinAt Q (↑e.symm) e.target x
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
n : ℕ
x : E
⊢ ‖f' x - A n‖₊ ≤ δ (A n) → ENNReal.ofReal |(f' x).det| ≤ ENNReal.ofReal |(A n).det| + ↑ε

Tactic:
hx

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
n : ℕ
x : E
hx : ‖f' x - A n‖₊ ≤ δ (A n)
⊢ ENNReal.ofReal |(f' x).det| ≤ ENNReal.ofReal |(A n).det| + ↑ε
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
⊢ G.IsLocalStructomorphWithinAt (f ∘ ↑e'.symm) (↑e'.symm ⁻¹' s) (↑e' x)

Tactic:
hx

Goal state after:
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
he'x : x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : ↑e' x ∈ ↑e'.symm ⁻¹' s
⊢ ∃ e ∈ G, EqOn (f ∘ ↑e'.symm) (↑e.toPartialEquiv) (↑e'.symm ⁻¹' s ∩ e.source) ∧ ↑e' x ∈ e.source
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f g : H → H
hfgs : ∀ y ∈ s, f y = g y
a✝ : f x = g x
h : G.IsLocalStructomorphWithinAt f s x
⊢ G.IsLocalStructomorphWithinAt g s x

Tactic:
hx

Goal state after:
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f g : H → H
hfgs : ∀ y ∈ s, f y = g y
a✝ : f x = g x
h : G.IsLocalStructomorphWithinAt f s x
hx : x ∈ s
⊢ ∃ e ∈ G, EqOn g (↑e.toPartialEquiv) (s ∩ e.source) ∧ x ∈ e.source
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
a✝ : s ⊆ f ⁻¹' e'.source
hfx : f x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
⊢ G.IsLocalStructomorphWithinAt (↑e' ∘ f) s x

Tactic:
hx

Goal state after:
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
f : H → H
e' : OpenPartialHomeomorph H H
he'G : e' ∈ G
a✝ : s ⊆ f ⁻¹' e'.source
hfx : f x ∈ e'.source
h : G.IsLocalStructomorphWithinAt f s x
hx : x ∈ s
⊢ ∃ e ∈ G, EqOn (↑e' ∘ f) (↑e.toPartialEquiv) (s ∩ e.source) ∧ x ∈ e.source
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

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
n : ℕ∞ω
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsEmpty M
e e' : OpenPartialHomeomorph M H
a✝¹ : e ∈ atlas H M
a✝ : e' ∈ atlas H M
x : E
⊢ x ∈ ↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I →
    ContDiffWithinAt 𝕜 n (↑I ∘ ↑(e.symm ≫ₕ e') ∘ ↑I.symm) (↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I) x

Tactic:
hx

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
n : ℕ∞ω
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsEmpty M
e e' : OpenPartialHomeomorph M H
a✝¹ : e ∈ atlas H M
a✝ : e' ∈ atlas H M
x : E
hx : x ∈ ↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑(e.symm ≫ₕ e') ∘ ↑I.symm) (↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I) x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case mpr
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : OpenPartialHomeomorph H H
s : Set H
x : H
hx : x ∈ f.source ∪ sᶜ
hf : x ∈ s → ∃ e ∈ G, e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source
⊢ G.IsLocalStructomorphWithinAt (↑f) s x

Tactic:
hx

Goal state after:
case mpr
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : OpenPartialHomeomorph H H
s : Set H
x : H
hx✝ : x ∈ f.source ∪ sᶜ
hf : x ∈ s → ∃ e ∈ G, e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source
hx : x ∈ s
⊢ ∃ e ∈ G, EqOn (↑f) (↑e.toPartialEquiv) (s ∩ e.source) ∧ x ∈ e.source
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H₁ : Type u_6
inst✝⁷ : TopologicalSpace H₁
H₂ : Type u_7
inst✝⁶ : TopologicalSpace H₂
H₃ : Type u_8
inst✝⁵ : TopologicalSpace H₃
inst✝⁴ : ChartedSpace H₁ H₂
inst✝³ : ChartedSpace H₂ H₃
G₁ : StructureGroupoid H₁
inst✝² : HasGroupoid H₂ G₁
inst✝¹ : ClosedUnderRestriction G₁
G₂ : StructureGroupoid H₂
inst✝ : HasGroupoid H₃ G₂
H : ∀ e ∈ G₂, LiftPropOn G₁.IsLocalStructomorphWithinAt (↑e) e.source
x✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃
e : OpenPartialHomeomorph H₃ H₂
he : e ∈ atlas H₂ H₃
f : OpenPartialHomeomorph H₂ H₁
hf : f ∈ atlas H₁ H₂
e' : OpenPartialHomeomorph H₃ H₂
he' : e' ∈ atlas H₂ H₃
f' : OpenPartialHomeomorph H₂ H₁
hf' : f' ∈ atlas H₁ H₂
x : H₁
⊢ x ∈ ((e ≫ₕ f).symm ≫ₕ e' ≫ₕ f').source → ∃ s, IsOpen[inst✝⁷] s ∧ x ∈ s ∧ ((e ≫ₕ f).symm ≫ₕ e' ≫ₕ f').restr s ∈ G₁

Tactic:
hx

Goal state after:
H₁ : Type u_6
inst✝⁷ : TopologicalSpace H₁
H₂ : Type u_7
inst✝⁶ : TopologicalSpace H₂
H₃ : Type u_8
inst✝⁵ : TopologicalSpace H₃
inst✝⁴ : ChartedSpace H₁ H₂
inst✝³ : ChartedSpace H₂ H₃
G₁ : StructureGroupoid H₁
inst✝² : HasGroupoid H₂ G₁
inst✝¹ : ClosedUnderRestriction G₁
G₂ : StructureGroupoid H₂
inst✝ : HasGroupoid H₃ G₂
H : ∀ e ∈ G₂, LiftPropOn G₁.IsLocalStructomorphWithinAt (↑e) e.source
x✝ : ChartedSpace H₁ H₃ := ChartedSpace.comp H₁ H₂ H₃
e : OpenPartialHomeomorph H₃ H₂
he : e ∈ atlas H₂ H₃
f : OpenPartialHomeomorph H₂ H₁
hf : f ∈ atlas H₁ H₂
e' : OpenPartialHomeomorph H₃ H₂
he' : e' ∈ atlas H₂ H₃
f' : OpenPartialHomeomorph H₂ H₁
hf' : f' ∈ atlas H₁ H₂
x : H₁
hx : x ∈ ((e ≫ₕ f).symm ≫ₕ e' ≫ₕ f').source
⊢ ∃ s, IsOpen[inst✝⁷] s ∧ x ∈ s ∧ ((e ≫ₕ f).symm ≫ₕ e' ≫ₕ f').restr s ∈ G₁
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
s t : Set M
inst✝ : ChartedSpace H M
x x' : M
h : x' ∈ (extChartAt I x).source
ht : t ∈ 𝓝[s] x'
⊢ ↑(extChartAt I x).symm ⁻¹' t ∈ 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x'

Tactic:
by
  rwa [← map_extChartAt_symm_nhdsWithin' h, mem_map] at ht

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
x : M × M'
⊢ (chartAt H x.1).prod (chartAt H' x.2).toPartialEquiv ≫ I.prod I'.toPartialEquiv =
    ((chartAt H x.1).toPartialEquiv ≫ I.toPartialEquiv).prod ((chartAt H' x.2).toPartialEquiv ≫ I'.toPartialEquiv)

Tactic:
rw [PartialEquiv.prod_trans]

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 : M → M'
n : ℕ∞ω
⊢ ContMDiffOn I I' n f univ ↔ ContMDiff I I' n f

Tactic:
simp only [ContMDiffOn, ContMDiff, contMDiffWithinAt_univ, forall_prop_of_true, mem_univ]

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

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
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✝ : ChartedSpace H' M'
f : M → M'
n : ℕ∞ω
s t : Set M
hf : ContMDiffOn I I' n f s
hf' : ContMDiffOn I I' n f t
hs : IsOpen[inst✝⁶] s
ht : IsOpen[inst✝⁶] t
x : M
⊢ x ∈ s ∪ t → ContMDiffWithinAt I I' n f (s ∪ t) x

Tactic:
hx

Goal state after:
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
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✝ : ChartedSpace H' M'
f : M → M'
n : ℕ∞ω
s t : Set M
hf : ContMDiffOn I I' n f s
hf' : ContMDiffOn I I' n f t
hs : IsOpen[inst✝⁶] s
ht : IsOpen[inst✝⁶] t
x : M
hx : x ∈ s ∪ t
⊢ ContMDiffWithinAt I I' n f (s ∪ t) x
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
t : Set E
ts : t ⊆ s
ht : MeasurableSet t
t_eq_s : t =ᵐ[μ] s
A : μ (f '' s) = μ (f '' t)
B : ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = ∫⁻ (x : E) in t, ENNReal.ofReal |(f' x).det| ∂μ
x : E
⊢ x ∈ t → HasFDerivWithinAt f (f' x) t x

Tactic:
hx

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
t : Set E
ts : t ⊆ s
ht : MeasurableSet t
t_eq_s : t =ᵐ[μ] s
A : μ (f '' s) = μ (f '' t)
B : ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = ∫⁻ (x : E) in t, ENNReal.ofReal |(f' x).det| ∂μ
x : E
hx : x ∈ t
⊢ HasFDerivWithinAt f (f' x) t x
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
h'f : AEMeasurable f (μ.restrict s)
h''f : AEMeasurable f ((μ.restrict s).withDensity fun x ↦ ENNReal.ofReal |(f' x).det|)
t : Set E
ht : MeasurableSet t
h't : NullMeasurableSet (f ⁻¹' t) (μ.restrict s)
⊢ ∫⁻ (a : E) in f ⁻¹' t, ENNReal.ofReal |(f' a).det| ∂μ.restrict s = μ (t ∩ f '' s)

Tactic:
restrict_restrict₀ h't,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
h'f : AEMeasurable f (μ.restrict s)
h''f : AEMeasurable f ((μ.restrict s).withDensity fun x ↦ ENNReal.ofReal |(f' x).det|)
t : Set E
ht : MeasurableSet t
h't : NullMeasurableSet (f ⁻¹' t) (μ.restrict s)
⊢ ∫⁻ (a : E) in f ⁻¹' t ∩ s, ENNReal.ofReal |(f' a).det| ∂μ = μ (t ∩ f '' s)
Module: Mathlib.Geometry.Manifold.MFDeriv.Defs

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
⊢ x ∈ e.source →
    DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

Tactic:
hx

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
⊢ DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
Module: Mathlib.Geometry.Manifold.MFDeriv.Defs

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
⊢ f x = g x → DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' g s x

Tactic:
hx

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
⊢ DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' g s x
Module: Mathlib.Geometry.Manifold.MFDeriv.Defs

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
⊢ f x ∈ e'.source → DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' (↑e' ∘ f) s x

Tactic:
hx

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
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
⊢ DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' (↑e' ∘ f) s x