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

Goal state before:
pp: H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
x : H
hx : x ∈ e.source
s : Set H
open_s : IsOpen[inst✝] s
xs : x ∈ s
x's : x ∈ (e.restr s).source
hs : e.restr s = OpenPartialHomeomorph.refl H
this✝ : (e.restr s).source = univ
this : e.source ∩ interior s = univ
⊢ univ ⊆ interior s

Tactic:
rw [← this]

Goal state after:
H✝ : Type u_1
inst✝¹ : TopologicalSpace H✝
H : Type u_2
inst✝ : TopologicalSpace H
e : OpenPartialHomeomorph H H
he : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}
x : H
hx : x ∈ e.source
s : Set H
open_s : IsOpen[inst✝] s
xs : x ∈ s
x's : x ∈ (e.restr s).source
hs : e.restr s = OpenPartialHomeomorph.refl H
this✝ : (e.restr s).source = univ
this : e.source ∩ interior s = univ
⊢ e.source ∩ interior s ⊆ interior s
Module: Mathlib.Geometry.Manifold.StructureGroupoid

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

Tactic:
constructor
  · intro _i
    rw [StructureGroupoid.le_iff]
    rintro e ⟨s, hs, hes⟩
    refine G.mem_of_eqOnSource ?_ hes
    convert! closedUnderRestriction' G.id_mem hs
    ext <;> simp [hs.interior_eq]
  · intro h
    constructor
    intro e he s hs
    rw [← ofSet_trans (e : OpenPartialHomeomorph H H) hs]
    refine G.trans ?_ he
    apply StructureGroupoid.le_iff.mp h
    exact idRestrGroupoid_mem hs

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
o : Set H
ho : IsOpen[inst✝¹] o
hxo : x ∈ o
host : s ∩ o = t ∩ o
⊢ P f s x ↔ P f t x

Tactic:
hG.is_local ho hxo,

Goal state after:
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
o : Set H
ho : IsOpen[inst✝¹] o
hxo : x ∈ o
host : s ∩ o = t ∩ o
⊢ P f (s ∩ o) x ↔ P f t x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case h
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
⊢ (g ∘ ↑(chartAt H x).symm) ∘ ↑((chartAt H x).symm ≫ₕ e).symm =ᶠ[𝓝 (↑e x)] g ∘ ↑e.symm

Tactic:
· refine ((e.eventually_nhds' _ xe).mpr <| (chartAt H x).eventually_left_inverse
      (mem_chart_source H x)).mono fun y hy ↦ ?_
    simp [hy]

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'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
hG : G.LocalInvariantProp G' P
hu : s =ᶠ[𝓝 x] t
⊢ LiftPropWithinAt P g (s ∩ {x | (fun x ↦ s x = t x) x}) x ↔ LiftPropWithinAt P g t x

Tactic:
← hG.liftPropWithinAt_inter (s := t) hu,

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 t : Set M
x : M
hG : G.LocalInvariantProp G' P
hu : s =ᶠ[𝓝 x] t
⊢ LiftPropWithinAt P g (s ∩ {x | (fun x ↦ s x = t x) x}) x ↔ LiftPropWithinAt P g (t ∩ {x | (fun x ↦ s x = t x) x}) x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case left
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e ∘ ↑I.symm) (↑I' ∘ ↑e' ∘ ↑I'.symm))
    ((fun p ↦ (↑I p.1, ↑I' p.2)) '' e.source ×ˢ e'.source)
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e') ∘ ↑(I.prod I').symm) (↑(I.prod I') '' (e.prod e'.toPartialEquiv).source)

Tactic:
exact h3

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

Goal state before:
pp: case right
n : ℕ∞ω
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
E' : Type u_5
H' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : TopologicalSpace H'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
e : OpenPartialHomeomorph H H
e' : OpenPartialHomeomorph H' H'
he : (contDiffPregroupoid n I).property (↑e) e.source
he_symm : (contDiffPregroupoid n I).property (↑e.symm) e.target
he' : (contDiffPregroupoid n I').property (↑e') e'.source
he'_symm : (contDiffPregroupoid n I').property (↑e'.symm) e'.target
h3 :
  ContDiffOn 𝕜 n (Prod.map (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I' ∘ ↑e'.symm ∘ ↑I'.symm))
    ((fun p ↦ (↑I p.1, ↑I' p.2)) '' e.target ×ˢ e'.target)
⊢ ContDiffOn 𝕜 n (↑(I.prod I') ∘ ↑(e.prod e').symm ∘ ↑(I.prod I').symm)
    (↑(I.prod I') '' (e.prod e'.toPartialEquiv).target)

Tactic:
exact h3

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
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
t : Set E := ↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I
hx : x ∈ t
this : (e.symm ≫ₕ e').source = ∅
⊢ t = ↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range ↑I

Tactic:
by
      rw [← Subtype.preimage_val_eq_preimage_val_iff]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
x : M
n : WithTop ℕ∞
f : M → F₁ × F₂
⊢ ContMDiffAt I 𝓘(𝕜, F₁ × F₂) n f x ↔
    ContMDiffAt I 𝓘(𝕜, F₁) n (Prod.fst ∘ f) x ∧ ContMDiffAt I 𝓘(𝕜, F₂) n (Prod.snd ∘ f) x

Tactic:
rw [modelWithCornersSelf_prod, ← chartedSpaceSelf_prod]
  exact contMDiffAt_prod_iff f

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
hst : s ∪ t = univ
hs : IsOpen[inst✝⁶] s
ht : IsOpen[inst✝⁶] t
⊢ ContMDiff I I' n f

Tactic:
rw [← contMDiffOn_univ, ← hst]

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
hst : s ∪ t = univ
hs : IsOpen[inst✝⁶] s
ht : IsOpen[inst✝⁶] t
⊢ ContMDiffOn I I' n f (s ∪ t)
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

Goal state before:
pp: case inr.hx
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
M' : Type u_16
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H M'
n : WithTop ℕ∞
E' : Type u_17
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝² : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H' N
f : M → N
g : M' → N
hf : ContMDiff I J n f
hg : ContMDiff I J n g
x : M'
this :
  ContDiffWithinAt 𝕜 n (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (range ↑I)
    (↑(extChartAt I (Sum.inr x)) (Sum.inr x))
⊢ (↑(extChartAt J (Sum.elim f g (Sum.inr x))) ∘ Sum.elim f g ∘ ↑(extChartAt I (Sum.inr x)).symm)
      (↑(extChartAt I (Sum.inr x)) (Sum.inr x)) =
    (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I (Sum.inr x)) (Sum.inr x))

Tactic:
simp only [extChartAt, ChartedSpace.sum_chartAt_inr, Sum.elim_inr]
      congr

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

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
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
f : M → N
h : ContMDiff I J n (Sum.inl ∘ f)
a✝ : Nontrivial N
inhabited_h : Inhabited N
aux : N ⊕ N' → N := Sum.elim id fun x ↦ default
this : aux ∘ Sum.inl ∘ f = f
⊢ ContMDiff I J n f

Tactic:
rw [← this]

Goal state after:
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
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
f : M → N
h : ContMDiff I J n (Sum.inl ∘ f)
a✝ : Nontrivial N
inhabited_h : Inhabited N
aux : N ⊕ N' → N := Sum.elim id fun x ↦ default
this : aux ∘ Sum.inl ∘ f = f
⊢ ContMDiff I J n (aux ∘ Sum.inl ∘ f)
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

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_16
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H M'
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
g : M' → N'
h : ContMDiff I J n (Sum.inr ∘ g)
a✝ : Nontrivial N'
inhabited_h : Inhabited N'
aux : N ⊕ N' → N' := Sum.elim (fun x ↦ default) id
this : aux ∘ Sum.inr ∘ g = g
⊢ ContMDiff I J n g

Tactic:
rw [← this]

Goal state after:
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_16
inst✝⁸ : TopologicalSpace M'
inst✝⁷ : ChartedSpace H M'
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
g : M' → N'
h : ContMDiff I J n (Sum.inr ∘ g)
a✝ : Nontrivial N'
inhabited_h : Inhabited N'
aux : N ⊕ N' → N' := Sum.elim (fun x ↦ default) id
this : aux ∘ Sum.inr ∘ g = g
⊢ ContMDiff I J n (aux ∘ Sum.inr ∘ g)
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'
e : OpenPartialHomeomorph M H
f : M → M'
s : Set M
x : M
n : ℕ∞ω
he : e ∈ maximalAtlas I n M
hx h2x : x ∈ e.source
⊢ ContMDiffWithinAt I I' n f s x ↔
    ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) x)

Tactic:
← e.extend_source (I := I)

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
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'
e : OpenPartialHomeomorph M H
f : M → M'
s : Set M
x : M
n : ℕ∞ω
he : e ∈ maximalAtlas I n M
hx : x ∈ e.source
h2x : x ∈ (e.extend I).source
⊢ ContMDiffWithinAt I I' n f s x ↔
    ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) 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
u : Set H
f : H → H'
⊢ IsOpen[inst✝³] u → x ∈ u → (DifferentiableWithinAtProp I I' f s x ↔ DifferentiableWithinAtProp I I' f (s ∩ u) x)

Tactic:
u_open

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
u : Set H
f : H → H'
u_open : IsOpen[inst✝³] u
⊢ x ∈ u → (DifferentiableWithinAtProp I I' f s x ↔ DifferentiableWithinAtProp I I' f (s ∩ u) 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
g : E → ℝ≥0∞
⊢ ∫⁻ (a : E) in s, ((fun x ↦ ENNReal.ofReal |(f' x).det|) * g ∘ f) a ∂μ =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * (g ∘ f) x ∂μ

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
g : E → ℝ≥0∞
⊢ ∀ᵐ (x : E) ∂μ.restrict s, ENNReal.ofReal |(f' x).det| < ∞

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
g : E → ℝ≥0∞
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal |(f' x).det|) (μ.restrict s)

Tactic:
· simp only [Pi.mul_apply]

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
g : E → ℝ≥0∞
⊢ ∀ᵐ (x : E) ∂μ.restrict s, ENNReal.ofReal |(f' x).det| < ∞

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
g : E → ℝ≥0∞
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal |(f' x).det|) (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : OpenPartialHomeomorph E E
hf' : ∀ x ∈ f.source, HasFDerivAt (↑f) (f' x) x
g : E → F
this : ↑f '' f.source = f.target
⊢ ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ

Tactic:
rw [← this]

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : OpenPartialHomeomorph E E
hf' : ∀ x ∈ f.source, HasFDerivAt (↑f) (f' x) x
g : E → F
this : ↑f '' f.source = f.target
⊢ ∫ (x : E) in ↑f '' f.source, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ
Module: Mathlib.Geometry.Manifold.MFDeriv.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
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'
x : M
s : Set M
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I 1 M
hx h2x : x ∈ e.source
⊢ MDiffAt[s] f x ↔ MDiffAt[↑(e.extend I).symm ⁻¹' s ∩ range ↑I] (f ∘ ↑(e.extend I).symm) (↑(e.extend I) x)

Tactic:
← e.extend_source (I := I)

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
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'
x : M
s : Set M
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I 1 M
hx : x ∈ e.source
h2x : x ∈ (e.extend I).source
⊢ MDiffAt[s] f x ↔ MDiffAt[↑(e.extend I).symm ⁻¹' s ∩ range ↑I] (f ∘ ↑(e.extend I).symm) (↑(e.extend I) x)
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u_1
inst✝⁶ : TopologicalSpace B
F₁ : Type u_2
inst✝⁵ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝³ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : (x : B) → Zero (E₁ x)
inst✝ : (x : B) → Zero (E₂ x)
x : B
w₁ : F₁
w₂ : F₂
h : (x, w₁, w₂) ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ
⊢ toFun' e₁ e₂ (invFun' e₁ e₂ (x, w₁, w₂)) = (x, w₁, w₂)

Tactic:
obtain ⟨⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩, -⟩ := h

Goal state after:
B : Type u_1
inst✝⁶ : TopologicalSpace B
F₁ : Type u_2
inst✝⁵ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝³ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : (x : B) → Zero (E₁ x)
inst✝ : (x : B) → Zero (E₂ x)
x : B
w₁ : F₁
w₂ : F₂
h₁ : x ∈ e₁.baseSet
h₂ : x ∈ e₂.baseSet
⊢ toFun' e₁ e₂ (invFun' e₁ e₂ (x, w₁, w₂)) = (x, w₁, w₂)
Module: Mathlib.Geometry.Manifold.MFDeriv.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
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'
x : M
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
h : HasMFDerivAt% f x f'
h₁ : f₁ =ᶠ[𝓝 x] f
⊢ HasMFDerivAt% f₁ x f'

Tactic:
by
  rw [← hasMFDerivWithinAt_univ] at h ⊢
  apply h.congr_of_eventuallyEq _ (mem_of_mem_nhds h₁ :)
  rwa [nhdsWithin_univ]

Goal state after: