-- Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
theorem contDiffWithinAt_localInvariantProp_of_le (n m : WithTop ℕ∞) (hmn : m ≤ n) :
    (contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
      (ContDiffWithinAtProp I I' m) where
  is_local {s x u f} u_open xu

Proof state:
case convert_2
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 n (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑I.symm ⁻¹' e.target ∩ (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) ⊆
    ↑I.symm ⁻¹' e.target ∩ range ↑I ∩ ↑I ∘ ↑e.symm ∘ ↑I.symm ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I)

Tactic:
mfld_set_tac

Result:
no goals

Proof state:
case h₁
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : ContDiffWithinAtProp I I' m f s x
y : E
hy : ↑I.symm y ∈ s ∧ y ∈ range ↑I
⊢ (↑I' ∘ g ∘ ↑I.symm) y = (↑I' ∘ f ∘ ↑I.symm) y

Tactic:
simp only [h, hy, mfld_simps]

Result:
no goals

Proof state:
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
⊢ ContDiffWithinAt 𝕜 m (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) (↑I (↑e x))

Tactic:
have : I x = (I ∘ e.symm ∘ I.symm) (I (e x)) := by simp only [hx, mfld_simps]

Result:
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
this : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
⊢ ContDiffWithinAt 𝕜 m (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) (↑I (↑e x))

Proof state:
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid n I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 n (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
y : E
hy : ↑I.symm y ∈ s ∧ y ∈ range ↑I
⊢ (↑I' ∘ f ∘ ↑I.symm) y ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'

Tactic:
simpa only [hy, mfld_simps] using hs hy.1

Result:
no goals

Proof state:
case h.e'_10.h
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid n I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 n (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
y : E
⊢ (↑I' ∘ (↑e' ∘ f) ∘ ↑I.symm) y = ((↑I' ∘ ↑e' ∘ ↑I'.symm) ∘ ↑I' ∘ f ∘ ↑I.symm) y

Tactic:
simp only [mfld_simps]

Result:
no goals

Proof state:
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
⊢ ContDiffWithinAt 𝕜 m (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) (↑I (↑e x))

Tactic:
have := (mem_groupoid_of_pregroupoid.2 he).2.contDiffWithinAt this

Result:
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'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 n (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ContDiffWithinAt 𝕜 m (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) (↑I (↑e x))