mathlib-initiative/mathlib-tactics huggingface dataset:
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'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
hg : ContinuousWithinAt g s x
⊢ P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔
    ContinuousWithinAt (↑f ∘ g) s x ∧ P ((↑f ∘ g) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)

Tactic:
[

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'
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
hg : ContinuousWithinAt g s x
⊢ P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔
    ContinuousWithinAt (↑f ∘ g) s x ∧ P ((↑f ∘ g) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) 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'
e : OpenPartialHomeomorph M H
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
h : ContinuousWithinAt g s x ∧ P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
h1 : ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e.symm.symm x)
⊢ ContinuousAt (↑f) ((g ∘ ↑e.symm) (↑e x))

Tactic:
[

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'
e : OpenPartialHomeomorph M H
f : OpenPartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g : M → M'
s : Set M
x : M
hG : G.LocalInvariantProp G' P
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
hf : f ∈ maximalAtlas M' G'
xf : g x ∈ f.source
h : ContinuousWithinAt g s x ∧ P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
h1 : ContinuousWithinAt (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e.symm.symm x)
⊢ ContinuousAt (↑f) ((g ∘ ↑e.symm) (↑e x))
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
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ (extendCoordChange e e').source = ↑I '' (e.symm ≫ₕ e').source

Tactic:
[

Goal state after:
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
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ (extendCoordChange e e').source = ↑I '' (e.symm ≫ₕ e').source
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
X : Type u_4
inst✝ : TopologicalSpace X
f : H → X
s : Set H
x : H
h : ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
this : ContinuousWithinAt ((f ∘ ↑I.symm) ∘ ↑I) (↑I ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I)) x
⊢ ContinuousWithinAt f s x

Tactic:
[

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
X : Type u_4
inst✝ : TopologicalSpace X
f : H → X
s : Set H
x : H
h : ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
this : ContinuousWithinAt ((f ∘ ↑I.symm) ∘ ↑I) (↑I ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I)) x
⊢ ContinuousWithinAt f s x
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 f' : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ ↑(f.extend I) '' (f.source ∩ f'.source) = (extendCoordChange f f').source

Tactic:
[

Goal state after:
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 f' : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ ↑(f.extend I) '' (f.source ∩ f'.source) = (extendCoordChange f f').source
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
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
x : H
hx : x ∈ (e.symm ≫ₕ e').source
⊢ (e.symm ≫ₕ e').source ∈ 𝓝 x

Tactic:
exact (OpenPartialHomeomorph.open_source _).mem_nhds hx

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
X : Type u_4
inst✝ : TopologicalSpace X
f : H → X
s : Set H
x : H
⊢ ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x) ↔ ContinuousWithinAt f s x

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩
  · have := h.comp I.continuousWithinAt (mapsTo_preimage _ _)
    simp_rw [preimage_inter, preimage_preimage, I.left_inv, preimage_id', preimage_range,
      inter_univ] at this
    rwa [Function.comp_assoc, I.symm_comp_self] at this
  · rw [← I.left_inv x] at h; exact h.comp I.continuousWithinAt_symm inter_subset_left

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
ht : t ∈ 𝓝[s] x
⊢ ↑(chartAt H x).symm ⁻¹' (s ∩ t) =ᶠ[𝓝 (↑(chartAt H x) x)] ↑(chartAt H x).symm ⁻¹' s

Tactic:
[

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
ht : t ∈ 𝓝[s] x
⊢ ↑(chartAt H x).symm ⁻¹' (s ∩ t) =ᶠ[𝓝 (↑(chartAt H x) x)] ↑(chartAt H x).symm ⁻¹' s
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' ∘ ↑(chartAt H x).symm) (↑(chartAt H x) x) =
    (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x) x)

Tactic:
[

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 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' ∘ ↑(chartAt H x).symm) (↑(chartAt H x) x) =
    (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x) 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' ∘ ↑(chartAt H x).symm =ᶠ[𝓝[↑(chartAt H x).symm ⁻¹' s] ↑(chartAt H x) x]
    ↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm

Tactic:
[

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 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' ∘ ↑(chartAt H x).symm =ᶠ[𝓝[↑(chartAt H x).symm ⁻¹' s] ↑(chartAt H x) x]
    ↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm
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'
x : M
hG : G.LocalInvariantProp G' P
h₁ : g' =ᶠ[𝓝 x] g
⊢ g' =ᶠ[𝓝[univ] x] g

Tactic:
[

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 g' : M → M'
x : M
hG : G.LocalInvariantProp G' P
h₁ : g' =ᶠ[𝓝 x] g
⊢ g' =ᶠ[𝓝[univ] x] g
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
g : M → M'
s t : Set M
x : M
mono_of_mem_nhdsWithin : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, s ∈ 𝓝[t] x → P f s x → P f t x
hst : s ∈ 𝓝[t] x
h :
  ContinuousWithinAt g s x ∧
    P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
⊢ ↑(chartAt H x).symm ⁻¹' s ∈ 𝓝[↑(chartAt H x).symm ⁻¹' t] ↑(chartAt H x) x

Tactic:
[

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'
P : (H → H') → Set H → H → Prop
g : M → M'
s t : Set M
x : M
mono_of_mem_nhdsWithin : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, s ∈ 𝓝[t] x → P f s x → P f t x
hst : s ∈ 𝓝[t] x
h :
  ContinuousWithinAt g s x ∧
    P (↑(chartAt H' (g x)) ∘ g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
⊢ ↑(chartAt H x).symm ⁻¹' s ∈ 𝓝[↑(chartAt H x).symm ⁻¹' t] ↑(chartAt H x) x
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
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
H : Type u_5
inst✝¹ : TopologicalSpace H
G : Type u_7
inst✝ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 F G
⊢ range ↑(I.prod J) = range ↑I ×ˢ range ↑J

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
H : Type u_5
inst✝¹ : TopologicalSpace H
G : Type u_7
inst✝ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 F G
⊢ range ↑(I.prod J) = range ↑I ×ˢ range ↑J
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
x : M
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.source
⊢ LiftPropAt Q (↑e) x

Tactic:
[

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
x : M
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.source
⊢ LiftPropAt Q (↑e) 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
x : H
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.target
h : Q (↑e ∘ ↑e.symm) univ x
this : ↑e.symm x ∈ e.source
⊢ LiftPropAt Q (↑e.symm) x

Tactic:
rw [LiftPropAt, hG.liftPropWithinAt_indep_chart G.id_mem_maximalAtlas (mem_univ _) he this]

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
x : H
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.target
h : Q (↑e ∘ ↑e.symm) univ x
this : ↑e.symm x ∈ e.source
⊢ ContinuousWithinAt (↑e.symm) univ x ∧
    Q (↑e ∘ ↑e.symm ∘ ↑(OpenPartialHomeomorph.refl H).symm) (↑(OpenPartialHomeomorph.refl H).symm ⁻¹' univ)
      (↑(OpenPartialHomeomorph.refl H) 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
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
⊢ LiftProp Q id

Tactic:
[

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
Q : (H → H) → Set H → H → Prop
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
⊢ LiftProp Q id
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case refine_1
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
hG : G.LocalInvariantProp G' P
U : Opens M
f : M → M'
x : ↥U
⊢ ContinuousWithinAt f univ ↑x ↔ ContinuousWithinAt (f ∘ Subtype.val) univ x

Tactic:
[

Goal state after:
case refine_1
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
hG : G.LocalInvariantProp G' P
U : Opens M
f : M → M'
x : ↥U
⊢ ContinuousWithinAt f univ ↑x ↔ ContinuousWithinAt (f ∘ Subtype.val) univ x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case refine_1
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
hG : G.LocalInvariantProp G' P
U V : Opens M
hUV : U ≤ V
f : ↥V → M'
x : ↥U
⊢ ContinuousWithinAt f univ (inclusion hUV x) ↔ ContinuousWithinAt (f ∘ inclusion hUV) univ x

Tactic:
[

Goal state after:
case refine_1
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
hG : G.LocalInvariantProp G' P
U V : Opens M
hUV : U ≤ V
f : ↥V → M'
x : ↥U
⊢ ContinuousWithinAt f univ (inclusion hUV x) ↔ ContinuousWithinAt (f ∘ inclusion hUV) univ x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: m n : ℕ∞ω
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝¹ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝ : TopologicalSpace M
f : H✝ → H✝
u : Set H✝
x✝ : IsOpen[inst✝¹] u
H : ∀ x ∈ u, ∃ v, IsOpen[inst✝¹] v ∧ x ∈ v ∧ ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I)
y : E
hy2 : y ∈ range ↑I
x : H✝
hx : ↑I x = y
hy1 : x ∈ u
v : Set H✝
v_open : IsOpen[inst✝¹] v
xv : x ∈ v
hv : ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I)
this : ↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I = ↑I.symm ⁻¹' u ∩ range ↑I ∩ ↑I.symm ⁻¹' v
⊢ ∃ u_1,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u_1 ∧
      ↑I x ∈ u_1 ∧ ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' u ∩ range ↑I ∩ u_1)

Tactic:
this

Goal state after:
m n : ℕ∞ω
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝¹ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝ : TopologicalSpace M
f : H✝ → H✝
u : Set H✝
x✝ : IsOpen[inst✝¹] u
H : ∀ x ∈ u, ∃ v, IsOpen[inst✝¹] v ∧ x ∈ v ∧ ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I)
y : E
hy2 : y ∈ range ↑I
x : H✝
hx : ↑I x = y
hy1 : x ∈ u
v : Set H✝
v_open : IsOpen[inst✝¹] v
xv : x ∈ v
hv : ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' u ∩ range ↑I ∩ ↑I.symm ⁻¹' v)
this : ↑I.symm ⁻¹' (u ∩ v) ∩ range ↑I = ↑I.symm ⁻¹' u ∩ range ↑I ∩ ↑I.symm ⁻¹' v
⊢ ∃ u_1,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u_1 ∧
      ↑I x ∈ u_1 ∧ ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' u ∩ range ↑I ∩ u_1)
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✝⁶ : NontriviallyNormedField 𝕜
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✝ : I.Boundaryless
x : M
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (extChartAt I x).target

Tactic:
[

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
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✝ : I.Boundaryless
x : M
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (extChartAt I x).target
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✝⁶ : NontriviallyNormedField 𝕜
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✝ : I.Boundaryless
x : M
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I)

Tactic:
I.range_eq_univ,

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
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✝ : I.Boundaryless
x : M
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑I.symm ⁻¹' (chartAt H x).target ∩ univ)
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: case mp.refine_1
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 : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : OpenPartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartialEquiv) (s ∩ e.source)
hxe : x ∈ e.source
⊢ (e.restr f.source).source ⊆ f.source

Tactic:
[

Goal state after:
case mp.refine_1
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 : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : OpenPartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartialEquiv) (s ∩ e.source)
hxe : x ∈ e.source
⊢ (e.restr f.source).source ⊆ f.source
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 : Set M
inst✝ : ChartedSpace H M
x₀ x : M
hx : x ∈ closure[inst✝¹] (interior s)
h'x : x ∈ (extChartAt I x₀).source
⊢ ↑(extChartAt I x₀) x ∈
    closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (interior (↑(extChartAt I x₀).symm ⁻¹' s ∩ (extChartAt I x₀).target))

Tactic:
[

Goal state after:
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 : Set M
inst✝ : ChartedSpace H M
x₀ x : M
hx : x ∈ closure[inst✝¹] (interior s)
h'x : x ∈ (extChartAt I x₀).source
⊢ ↑(extChartAt I x₀) x ∈
    closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (interior (↑(extChartAt I x₀).symm ⁻¹' s ∩ (extChartAt I x₀).target))
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : OpenPartialHomeomorph H H
x : H
⊢ x ∈ f.source ∪ f.sourceᶜ

Tactic:
[

Goal state after:
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : OpenPartialHomeomorph H H
x : H
⊢ x ∈ f.source ∪ f.sourceᶜ