/-- Let `S` be an `R`-algebra, `M` a submonoid of `R`, and `S' = M⁻¹S`.
If the image of some `x : S` falls in the span of some finite `s ⊆ S'` over `R`,
then there exists some `m : M` such that `m • x` falls in the
span of `IsLocalization.finsetIntegerMultiple _ s` over `R`.
-/
theorem IsLocalization.smul_mem_finsetIntegerMultiple_span [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')
    (hx : algebraMap S S' x ∈ Submodule.span R (s : Set S')) :
    ∃ m : M, m • x ∈
      Submodule.span R
        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S) := by
  let g : S →ₐ[R] S' :=
    AlgHom.mk' (algebraMap S S') fun c x => by simp [Algebra.algebraMap_eq_smul_one]
  -- We first obtain the `y' ∈ M` such that `s' = y' • s` is falls in the image of `S` in `S'`.
  let y := IsLocalization.commonDenomOfFinset (M.map (algebraMap R S)) s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨y', hy', e : algebraMap R S y' = y⟩ := y.prop
  have : algebraMap R S y' • (s : Set S') = y' • (s : Set S') := by
    simp_rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
  rw [← e, this] at hx₁
  replace hx₁ := congr_arg (Submodule.span R) hx₁
  rw [Submodule.span_smul] at hx₁
  replace hx : _ ∈ y' • Submodule.span R (s : Set S') := Set.smul_mem_smul_set hx
  rw [hx₁] at hx
  erw [← g.map_smul, ← Submodule.map_span (g : S →ₗ[R] S')] at hx
  -- Since `x` falls in the span of `s` in `S'`, `y' • x : S` falls in the span of `s'` in `S'`.
  -- That is, there exists some `x' : S` in the span of `s'` in `S` and `x' = y' • x` in `S'`.
  -- Thus `a • (y' • x) = a • x' ∈ span s'` in `S` for some `a ∈ M`.
  obtain ⟨x', hx', hx'' : algebraMap _ _ _ = _⟩ := hx
  obtain ⟨⟨_, a, ha₁, rfl⟩, ha₂⟩ :=
    (IsLocalization.eq_iff_exists (M.map (algebraMap R S)) S').mp hx''
  use (⟨a, ha₁⟩ : M) * (⟨y', hy'⟩ : M)
  convert (Submodule.span R
    (IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s : Set S)).smul_mem
      a hx' using 1
  convert ha₂.symm using 1
  · 

Proof state:
case h.e'_2
R S : Type u
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝⁷ : CommRing R'
inst✝⁶ : CommRing S'
f : R →+* S
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R S
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R S S'
inst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) S'
x : S
s : Finset S'
g : S →ₐ[R] S' :=
  AlgHom.mk' (algebraMap S S') (_ : ∀ (c : R) (x : S), (algebraMap S S') (c • x) = c • (algebraMap S S') x)
y : ↥(Submonoid.map (algebraMap R S) M) := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s
y' : R
hy' : y' ∈ ↑M
e : (algebraMap R S) y' = ↑y
this : (algebraMap R S) y' • ↑s = y' • ↑s
hx₁ : y' • Submodule.span R ↑s = Submodule.span R (⇑g '' ↑(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))
x' : S
hx' : x' ∈ ↑(Submodule.span R ↑(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))
hx'' : (algebraMap S S') x' = g (y' • x)
a : R
ha₁ : a ∈ ↑M
ha₂ :
  ↑{ val := (algebraMap R S) a, property := (_ : ∃ a_1 ∈ ↑M, (algebraMap R S) a_1 = (algebraMap R S) a) } * x' =
    ↑{ val := (algebraMap R S) a, property := (_ : ∃ a_1 ∈ ↑M, (algebraMap R S) a_1 = (algebraMap R S) a) } * y' • x
⊢ ({ val := a, property := ha₁ } * { val := y', property := hy' }) • x =
    ↑{ val := (algebraMap R S) a, property := (_ : ∃ a_1 ∈ ↑M, (algebraMap R S) a_1 = (algebraMap R S) a) } * y' • x

Next tactic:
rw [Subtype.coe_mk, Submonoid.smul_def, Submonoid.coe_mul, ← smul_smul]