Theorem: ModelWithCorners.symm_continuousWithinAt_comp_right_iff
File: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean

tactic: refine ⟨fun h => ?_, fun h => ?_⟩
state_before: 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
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)
⊢ ContinuousWithinAt f s x

case refine_2
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 s x
⊢ ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)

tactic: have := h.comp I.continuousWithinAt (mapsTo_preimage _ _)
state_before: 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)
⊢ ContinuousWithinAt f s x
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

tactic: simp_rw [preimage_inter, preimage_preimage, I.left_inv, preimage_id', preimage_range,
  inter_univ] at this
state_before: 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
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) s x
⊢ ContinuousWithinAt f s x

tactic: rwa [Function.comp.assoc, I.symm_comp_self] at this
state_before: 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) s x
⊢ ContinuousWithinAt f s x
state_after: no goals

tactic: rw [← I.left_inv x] at h
state_before: case refine_2
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 s x
⊢ ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
state_after: case refine_2
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 s (↑I.symm (↑I x))
⊢ ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)

tactic: exact h.comp I.continuousWithinAt_symm inter_subset_left
state_before: case refine_2
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 s (↑I.symm (↑I x))
⊢ ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
state_after: no goals