Theorem: mem_analyticGroupoid_of_boundaryless
File: Mathlib/Geometry/Manifold/AnalyticManifold.lean

tactic: apply Iff.intro
state_before: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
⊢ e ∈ analyticGroupoid I ↔
    AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
state_after: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
⊢ e ∈ analyticGroupoid I →
    AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)

case mpr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target) →
    e ∈ analyticGroupoid I

tactic: intro he
state_before: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
⊢ e ∈ analyticGroupoid I →
    AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
state_after: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : e ∈ analyticGroupoid I
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)

tactic: have := mem_groupoid_of_pregroupoid.mp he.right
state_before: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : e ∈ analyticGroupoid I
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
state_after: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : e ∈ analyticGroupoid I
this :
  {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e) e.source ∧
    {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e.symm) e.target
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)

tactic: simp only [I.image_eq, I.range_eq_univ, interior_univ, subset_univ, and_true] at this ⊢
state_before: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : e ∈ analyticGroupoid I
this :
  {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e) e.source ∧
    {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e.symm) e.target
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
state_after: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : e ∈ analyticGroupoid I
this :
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)

tactic: exact this
state_before: case mp
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : e ∈ analyticGroupoid I
this :
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
state_after: no goals

tactic: intro he
state_before: case mpr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target) →
    e ∈ analyticGroupoid I
state_after: case mpr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ e ∈ analyticGroupoid I

tactic: apply And.intro
state_before: case mpr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ e ∈ analyticGroupoid I
state_after: case mpr.left
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ e ∈ (contDiffGroupoid ⊤ I).members

case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ e ∈
    {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.groupoid.members

tactic: all_goals apply mem_groupoid_of_pregroupoid.mpr; simp only [I.image_eq, I.range_eq_univ,
  interior_univ, subset_univ, and_true, contDiffPregroupoid] at he ⊢
state_before: case mpr.left
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ e ∈ (contDiffGroupoid ⊤ I).members

case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ e ∈
    {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.groupoid.members
state_after: case mpr.left
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he :
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
⊢ ContDiffOn 𝕜 ⊤ (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    ContDiffOn 𝕜 ⊤ (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)

case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he :
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)

tactic: apply mem_groupoid_of_pregroupoid.mpr
state_before: case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ e ∈
    {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.groupoid.members
state_after: case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e) e.source ∧
    {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e.symm) e.target

tactic: simp only [I.image_eq, I.range_eq_univ,
interior_univ, subset_univ, and_true, contDiffPregroupoid] at he ⊢
state_before: case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he : AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source) ∧ AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I '' e.target)
⊢ {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e) e.source ∧
    {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑I.symm '' (↑I.symm ⁻¹' s ∩ interior (range ↑I)) ⊆ interior (range ↑I),
          comp := ⋯, id_mem := ⋯, locality := ⋯, congr := ⋯ }.property
      (↑e.symm) e.target
state_after: case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he :
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)

tactic: exact ⟨he.left.contDiffOn, he.right.contDiffOn⟩
state_before: case mpr.left
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he :
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
⊢ ContDiffOn 𝕜 ⊤ (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    ContDiffOn 𝕜 ⊤ (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
state_after: no goals

tactic: exact he
state_before: case mpr.right
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : CompleteSpace E
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
he :
  AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source ∩ univ) ∧
    AnalyticOn 𝕜 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ univ)
state_after: no goals