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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := e.toOpenPartialHomeomorph
f₂ : OpenPartialHomeomorph M' M'' := e'.toOpenPartialHomeomorph
f : OpenPartialHomeomorph M M'' := (e.trans e'.toHomeomorph).toOpenPartialHomeomorph
feq : f = f₁ ≫ₕ f₂
y : M' := ↑(c.symm ≫ₕ f₁) x
g : OpenPartialHomeomorph M' H := chartAt H y
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := (c.symm ≫ₕ f₁).source ∩ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source
open_s : IsOpen[inst✝⁶] s
this : x ∈ s
F₁ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f₁ ≫ₕ g) ≫ₕ g.symm ≫ₕ f₂ ≫ₕ c'
A : F₁ ∈ G
F₂ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f ≫ₕ c').restr s
⊢ (c.symm ≫ₕ (f₁ ≫ₕ f₂) ≫ₕ c').restr s ≈ F₂

Tactic:
by

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

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
⊢ ∀ {e : OpenPartialHomeomorph H H}, e ∈ idRestrGroupoid → ∀ (s : Set H), IsOpen[inst✝] s → e.restr s ∈ idRestrGroupoid

Tactic:
by

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

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
G : StructureGroupoid H
⊢ ClosedUnderRestriction G ↔ idRestrGroupoid ≤ G

Tactic:
by

Goal state after:
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
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⟨δ' / 2, ⋯⟩
⊢ ∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ↑ε

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
H : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ range ↑I.toPartialEquiv = I.target

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.extend I).source = f.source

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ IsOpen[inst✝] (f.extend I).source

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.extend I).target = ↑I.symm ⁻¹' f.target ∩ range ↑I

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.extend I).target = ↑I '' f.target

Tactic:
by

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

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
φ : PartialEquiv H E
hsource : φ.source = univ
htarget : φ.target = univ
hcont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑φ
hcont_inv : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝] ↑φ.symm
⊢ range ↑φ = φ.target

Tactic:
by

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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
⊢ ∀ (c : OpenPartialHomeomorph M H) (c' : OpenPartialHomeomorph M'' H),
    c ∈ atlas H M → c' ∈ atlas H M'' → c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c' ∈ G

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.extend I).target = ↑(f.extend I) '' f.source

Tactic:
by

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
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝ : I.Boundaryless
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f.extend I).target

Tactic:
by

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

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
φ : PartialEquiv H E
hsource : φ.source = univ
htarget : φ.target = univ
hcont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑φ
hcont_inv : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝] ↑φ.symm
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (range ↑φ) else range ↑φ = univ

Tactic:
by

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
hs : Nonempty ↑e.source
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
s : Opens M := { carrier := e.source, is_open' := ⋯ }
t : Opens H := { carrier := e.target, is_open' := ⋯ }
c' : OpenPartialHomeomorph (↥t) H
hc'✝ : c' ∈ atlas H ↥t
this : Nonempty ↥t
x : ↥t
hc' : c' = (chartAt H ↑x).subtypeRestr this
goal : OpenPartialHomeomorph (↑e.source) H :=
  e.toHomeomorphSourceTarget.toOpenPartialHomeomorph ≫ₕ t.openPartialHomeomorphSubtypeCoe this
⊢ goal.source = (e.subtypeRestr hs).source

Tactic:
by

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
hs : Nonempty ↑e.source
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
s : Opens M := { carrier := e.source, is_open' := ⋯ }
t : Opens H := { carrier := e.target, is_open' := ⋯ }
c' : OpenPartialHomeomorph (↥t) H
hc'✝ : c' ∈ atlas H ↥t
this : Nonempty ↥t
x : ↥t
hc' : c' = (chartAt H ↑x).subtypeRestr this
goal : OpenPartialHomeomorph (↑e.source) H :=
  e.toHomeomorphSourceTarget.toOpenPartialHomeomorph ≫ₕ t.openPartialHomeomorphSubtypeCoe this
⊢ EqOn (↑goal.toPartialEquiv) (↑(e.subtypeRestr hs).toPartialEquiv) goal.source

Tactic:
by

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
hs : Nonempty ↑e.source
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
⊢ let s := { carrier := e.source, is_open' := ⋯ };
  let t := { carrier := e.target, is_open' := ⋯ };
  ∀ c' ∈ atlas H ↥t, e.toHomeomorphSourceTarget.toOpenPartialHomeomorph ≫ₕ c' ∈ maximalAtlas (↥s) G

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
hs : s ⊆ f.source
⊢ MapsTo (↑(f.extend I)) s (↑(f.extend I).symm ⁻¹' s ∩ range ↑I)

Tactic:
by

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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁸ : TopologicalSpace H
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : TopologicalSpace M''
G : StructureGroupoid H
inst✝³ : ChartedSpace H M'
inst✝² : ChartedSpace H M''
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
s : Opens M := { carrier := e.source, is_open' := ⋯ }
t : Opens H := { carrier := e.target, is_open' := ⋯ }
h : IsEmpty ↑e.source
⊢ ↑t = ∅

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hxf : x ∈ f.source
⊢ x ∈ (f.extend I).source

Tactic:
by

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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁸ : TopologicalSpace H
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : TopologicalSpace M''
G : StructureGroupoid H
inst✝³ : ChartedSpace H M'
inst✝² : ChartedSpace H M''
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
⊢ let s := { carrier := e.source, is_open' := ⋯ };
  let t := { carrier := e.target, is_open' := ⋯ };
  Structomorph G ↥s ↥t

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
h : x ∈ f.source
⊢ x ∈ (f.extend I).source

Tactic:
by

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

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
φ : PartialEquiv H E
hsource : φ.source = univ
htarget : φ.target = univ
hcont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑φ
hcont_inv : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝] ↑φ.symm
⊢ (interior (range ↑φ)).Nonempty

Tactic:
by

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

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
⊢ I.target = range ↑I

Tactic:
by

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

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
h : ¬IsRCLikeNormedField 𝕜
⊢ range ↑I = univ

Tactic:
by

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

Goal state before:
pp: 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 t : Set H
x : H
f : H → H'
hu : s =ᶠ[𝓝 x] t
⊢ P f s x ↔ P f t x

Tactic:
by

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

Goal state before:
pp: 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 g : H → H'
h1 : f =ᶠ[𝓝[s] x] g
h2 : f x = g x
⊢ P f s x ↔ P g s x

Tactic:
by

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

Goal state before:
pp: 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 t : Set H
x : H
f g : H → H'
hu : s =ᶠ[𝓝 x] t
h : f =ᶠ[𝓝 x] g
⊢ P f s x ↔ P g t x

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ ContinuousOn (↑(f.extend I)) (f.extend I).source

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hy : x ∈ f.source
⊢ map (↑(f.extend I)) (𝓝 x) = 𝓝[range ↑I] ↑(f.extend I) x

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
h'x : ↑(f.extend I) x ∈ interior (range ↑I)
⊢ map (↑(f.extend I)) (𝓝 x) = 𝓝 (↑(f.extend I) x)

Tactic:
by

Goal state after:
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
A : E →L[ℝ] E
δ' : ℝ
δ'pos : 0 < δ'
hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist B.det A.det < ↑ε
δ'' : ℝ≥0 := ⟨δ' / 2, ⋯⟩
I'' : ∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ↑ε
hA : A.det ≠ 0
m : ℝ≥0 := |A.det|.toNNReal - ε
⊢ ↑m < ENNReal.ofReal |A.det|

Tactic:
by

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
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝ : I.Boundaryless
x : M
hx : x ∈ f.source
⊢ map (↑(f.extend I)) (𝓝 x) = 𝓝 (↑(f.extend I) x)

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
y : M
hy : y ∈ f.source
⊢ (f.extend I).target ∈ 𝓝[range ↑I] ↑(f.extend I) y

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
⊢ ↑I '' f.target ∈ 𝓝[range ↑I] ↑(f.extend I) x

Tactic:
by

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

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
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : Convex ℝ s
⊢ Convex ℝ s

Tactic:
by

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
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝ : I.Boundaryless
x : M
hx : x ∈ f.source
s : Set M
h : s ∈ 𝓝 x
⊢ ↑(f.extend I) '' s ∈ 𝓝 (↑(f.extend I) x)

Tactic:
by

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

Goal state before:
pp: 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'
hfs : ContinuousWithinAt f s x
hxe' : f x ∈ e'.source
⊢ P (↑e' ∘ f) s x ↔ P f s x

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
h'x : ↑(f.extend I) x ∈ interior (range ↑I)
s : Set M
h : s ∈ 𝓝 x
⊢ ↑(f.extend I) '' s ∈ 𝓝 (↑(f.extend I) x)

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.extend I).target ⊆ range ↑I

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ interior (f.extend I).target ⊆ interior (range ↑I)

Tactic:
by

Goal state after:
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✝
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
H : Type u_5
inst✝ : TopologicalSpace H
φ : PartialEquiv H E
hsource : φ.source = univ
htarget : Convex ℝ φ.target
hcont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑φ
hcont_inv : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝] ↑φ.symm
hint : (interior φ.target).Nonempty
⊢ range ↑φ = φ.target

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
y : H
hy : y ∈ f.target
hy' : ↑I y ∈ interior (range ↑I)
⊢ ↑I y ∈ interior (f.extend I).target

Tactic:
by

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
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
X : Type u_8
inst✝ : TopologicalSpace X
g : M → X
s : Set M
x : M
⊢ ContinuousWithinAt (g ∘ ↑(f.extend I).symm) (↑(f.extend I).symm ⁻¹' s ∩ range ↑I) (↑(f.extend I) x) ↔
    ContinuousWithinAt (g ∘ ↑f.symm) (↑f.symm ⁻¹' s) (↑f x)

Tactic:
by

Goal state after:
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✝
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
H : Type u_5
inst✝ : TopologicalSpace H
φ : PartialEquiv H E
hsource : φ.source = univ
htarget : Convex ℝ φ.target
hcont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑φ
hcont_inv : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝] ↑φ.symm
hint : (interior φ.target).Nonempty
⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range ↑φ) else range ↑φ = univ

Tactic:
by

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

Goal state before:
pp: 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
⊢ P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P f s x

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ IsOpen[inst✝] (f.source ∩ ↑(f.extend I) ⁻¹' s)

Tactic:
by

Goal state after:
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'
P : (H → H') → Set H → H → Prop
f : M → M'
x : M
⊢ LiftPropAt P f x ↔ ContinuousAt f x ∧ P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) univ (↑(chartAt H x) x)

Tactic:
by

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
y : M
hy : y ∈ f.source
e : PartialEquiv M E := f.extend I
⊢ y ∈ (f.extend I).source

Tactic:
by

Goal state after: