-- Mathlib/Topology/Algebra/InfiniteSum/Module.lean
/-- Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
`R`-scalar multiplication. -/
lemma AddAction.automorphize_smul_left [AddGroup α] [AddAction α β]  (f : β → M)
    (g : Quotient (AddAction.orbitRel α β) → R) :
    AddAction.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
      = g • (AddAction.automorphize f : Quotient (AddAction.orbitRel α β) → M)

Proof state:
case h
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
⊢ automorphize (g ∘ Quotient.mk' • f) x = (g • automorphize f) x

Tactic:
apply @Quotient.inductionOn' β (AddAction.orbitRel α β) _ x _

Result:
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
⊢ ∀ (a : β), automorphize (g ∘ Quotient.mk' • f) (Quotient.mk'' a) = (g • automorphize f) (Quotient.mk'' a)

Proof state:
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
⊢ ∀ (a : β), automorphize (g ∘ Quotient.mk' • f) (Quotient.mk'' a) = (g • automorphize f) (Quotient.mk'' a)

Tactic:
intro b

Result:
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
b : β
⊢ automorphize (g ∘ Quotient.mk' • f) (Quotient.mk'' b) = (g • automorphize f) (Quotient.mk'' b)

Proof state:
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
⊢ automorphize (g ∘ Quotient.mk' • f) = g • automorphize f

Tactic:
ext x

Result:
case h
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
⊢ automorphize (g ∘ Quotient.mk' • f) x = (g • automorphize f) x

Proof state:
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
b : β
⊢ Quotient.lift (fun b => ∑' (a : α), g (Quotient.mk' (a +ᵥ b)) • f (a +ᵥ b)) ⋯ (Quotient.mk'' b) =
    g (Quotient.mk'' b) • Quotient.lift (fun b => ∑' (a : α), f (a +ᵥ b)) ⋯ (Quotient.mk'' b)

Tactic:
set π : β → Quotient (AddAction.orbitRel α β) := Quotient.mk (AddAction.orbitRel α β)

Result:
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
b : β
π : β → Quotient (orbitRel α β) := Quotient.mk (orbitRel α β)
⊢ Quotient.lift (fun b => ∑' (a : α), g (Quotient.mk' (a +ᵥ b)) • f (a +ᵥ b)) ⋯ (Quotient.mk'' b) =
    g (Quotient.mk'' b) • Quotient.lift (fun b => ∑' (a : α), f (a +ᵥ b)) ⋯ (Quotient.mk'' b)