-- Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
theorem contDiffWithinAtProp_id (x : H) : ContDiffWithinAtProp I I n id univ x

Proof state:
case refine_2
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
n : WithTop ℕ∞
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
⊢ (↑I ∘ ↑I.symm) (↑I x) = id (↑I x)

Tactic:
simp only [mfld_simps]

Result:
no goals

Proof state:
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
n : WithTop ℕ∞
x : H
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑I.symm) (range ↑I) (↑I x)

Tactic:
have : ContDiffWithinAt 𝕜 n id (range I) (I x) := contDiff_id.contDiffAt.contDiffWithinAt

Result:
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
n : WithTop ℕ∞
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑I.symm) (range ↑I) (↑I x)

Proof state:
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
n : WithTop ℕ∞
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑I.symm) (range ↑I) (↑I x)

Tactic:
refine this.congr (fun y hy => ?_) ?_

Result:
case refine_1
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
n : WithTop ℕ∞
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
y : E
hy : y ∈ range ↑I
⊢ (↑I ∘ ↑I.symm) y = id y

case refine_2
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
n : WithTop ℕ∞
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
⊢ (↑I ∘ ↑I.symm) (↑I x) = id (↑I x)

Proof state:
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
n : WithTop ℕ∞
x : H
⊢ ContDiffWithinAtProp I I n id univ x

Tactic:
simp only [ContDiffWithinAtProp, id_comp, preimage_univ, univ_inter]

Result:
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
n : WithTop ℕ∞
x : H
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑I.symm) (range ↑I) (↑I x)

Proof state:
case refine_1
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
n : WithTop ℕ∞
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
y : E
hy : y ∈ range ↑I
⊢ (↑I ∘ ↑I.symm) y = id y

Tactic:
simp only [ModelWithCorners.right_inv I hy, mfld_simps]

Result:
no goals