-- Mathlib/Geometry/Manifold/MFDeriv/Defs.lean
theorem differentiableWithinAt_localInvariantProp :
    (contDiffGroupoid 1 I).LocalInvariantProp (contDiffGroupoid 1 I')
      (DifferentiableWithinAtProp I I') :=
  { is_local

Proof state:
case hx
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'
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : DifferentiableWithinAtProp I I' f s x
⊢ (↑I' ∘ g ∘ ↑I.symm) (↑I x) = (↑I' ∘ f ∘ ↑I.symm) (↑I x)

Tactic:
simp only [hx, 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'
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑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 1 (↑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 ht
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'
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : DifferentiableWithinAtProp I I' f s x
⊢ ∀ x ∈ ↑I.symm ⁻¹' s ∩ range ↑I, (↑I' ∘ g ∘ ↑I.symm) x = (↑I' ∘ f ∘ ↑I.symm) x

Tactic:
intro y hy

Result:
case ht
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'
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : DifferentiableWithinAtProp I I' f s x
y : E
hy : y ∈ ↑I.symm ⁻¹' s ∩ range ↑I
⊢ (↑I' ∘ g ∘ ↑I.symm) y = (↑I' ∘ f ∘ ↑I.symm) y

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'
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑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))
⊢ ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I

Tactic:
simp only [hx, 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'
s : Set H
x : H
u : Set H
f : H → H'
u_open : IsOpen u
xu : x ∈ u
this : ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u) (↑I x) ↔
    DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)

Tactic:
apply differentiableWithinAt_inter

Result:
case ht
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'
s : Set H
x : H
u : Set H
f : H → H'
u_open : IsOpen u
xu : x ∈ u
this : ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u
⊢ ↑I.symm ⁻¹' u ∈ 𝓝 (↑I 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'
s : Set H
x : H
u : Set H
f : H → H'
u_open : IsOpen u
xu : x ∈ u
⊢ ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u

Tactic:
simp only [Set.inter_right_comm, Set.preimage_inter]

Result:
no goals

Proof state:
case h.e'_11
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'
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑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 1 (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
⊢ ↑I' ∘ (↑e' ∘ f) ∘ ↑I.symm = (↑I' ∘ ↑e' ∘ ↑I'.symm) ∘ ↑I' ∘ f ∘ ↑I.symm

Tactic:
ext y

Result:
case h.e'_11.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'
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑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 1 (↑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