/--
An invertible fractional ideal of a commutative ring with finitely many maximal ideals is principal.

https://math.stackexchange.com/a/95857 -/
theorem FractionalIdeal.isPrincipal.of_finite_maximals_of_inv {A : Type*} [CommRing A]
    [Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S ≤ R⁰)
    (hf : {I : Ideal R | I.IsMaximal}.Finite) (I I' : FractionalIdeal S A) (hinv : I * I' = 1) :
    Submodule.IsPrincipal (I : Submodule R A) := by
  have hinv' := hinv
  rw [Subtype.ext_iff, val_eq_coe, coe_mul] at hinv
  let s := hf.toFinset
  haveI := Classical.decEq (Ideal R)
  have coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤ := by
    simp_rw [Finset.mem_erase, hf.mem_toFinset]
    rintro M hM M' ⟨hne, hM'⟩
    exact Ideal.IsMaximal.coprime_of_ne hM hM' hne.symm
  have nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M := fun M hM =>
    left_lt_sup.1
      ((hf.mem_toFinset.1 hM).ne_top.lt_top.trans_eq (Ideal.sup_iInf_eq_top <| coprime M hM).symm)
  have : ∀ M ∈ s, ∃ a ∈ I, ∃ b ∈ I', a * b ∉ IsLocalization.coeSubmodule A M := by
    intro M hM; by_contra! h
    obtain ⟨x, hx, hxM⟩ :=
      SetLike.exists_of_lt
        ((IsLocalization.coeSubmodule_strictMono hS (hf.mem_toFinset.1 hM).ne_top.lt_top).trans_eq
          hinv.symm)
    refine' hxM (Submodule.map₂_le.2 _ hx); exact h
  choose! a ha b hb hm using this
  choose! u hu hum using fun M hM => SetLike.not_le_iff_exists.1 (nle M hM)
  let v := ∑ M in s, u M • b M
  have hv : v ∈ I' := Submodule.sum_mem _ fun M hM => Submodule.smul_mem _ _ <| hb M hM
  refine'
    FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top
      (Units.mkOfMulEqOne I I' hinv') hv (of_not_not fun h => _)
  obtain ⟨M, hM, hJM⟩ := Ideal.exists_le_maximal _ h
  replace hM := hf.mem_toFinset.2 hM
  have : ∀ a ∈ I, ∀ b ∈ I', ∃ c, algebraMap R _ c = a * b := by
    intro a ha b hb; have hi := hinv.le
    obtain ⟨c, -, hc⟩ := hi (Submodule.mul_mem_mul ha hb)
    exact ⟨c, hc⟩
  have hmem : a M * v ∈ IsLocalization.coeSubmodule A M := by
    obtain ⟨c, hc⟩ := this _ (ha M hM) v hv
    refine' IsLocalization.coeSubmodule_mono _ hJM ⟨c, _, hc⟩
    have := Submodule.mul_mem_mul (ha M hM) (Submodule.mem_span_singleton_self v)
    rwa [← hc] at this
  simp_rw [Finset.mul_sum, mul_smul_comm] at hmem
  rw [← s.add_sum_erase _ hM, Submodule.add_mem_iff_left] at hmem
  · refine' hm M hM _
    obtain ⟨c, hc : algebraMap R A c = a M * b M⟩ := this _ (ha M hM) _ (hb M hM)
    rw [← hc] at hmem ⊢
    rw [Algebra.smul_def, ← _root_.map_mul] at hmem
    obtain ⟨d, hdM, he⟩ := hmem
    rw [IsLocalization.injective _ hS he] at hdM
    exact
      Submodule.mem_map_of_mem
        (((hf.mem_toFinset.1 hM).isPrime.mem_or_mem hdM).resolve_left <| hum M hM)
  · refine' Submodule.sum_mem _ fun M' hM' => _
    rw [Finset.mem_erase] at hM'
    obtain ⟨c, hc⟩ := this _ (ha M hM) _ (hb M' hM'.2)
    rw [← hc, Algebra.smul_def, ← _root_.map_mul]
    specialize hu M' hM'.2
    simp_rw [Ideal.mem_iInf, Finset.mem_erase] at hu
    

Proof state:
case intro.intro.a.intro
R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : Set.Finite {I | IsMaximal I}
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := Set.Finite.toFinset hf
this✝ : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ Finset.erase s M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ Finset.erase s M, M' ≤ M
a : Ideal R → A
ha : ∀ M ∈ s, a M ∈ I
b : Ideal R → A
hb : ∀ M ∈ s, b M ∈ I'
hm : ∀ M ∈ s, a M * b M ∉ IsLocalization.coeSubmodule A M
u : Ideal R → R
hum : ∀ M ∈ s, u M ∉ M
v : A := ∑ M in s, u M • b M
hv : v ∈ I'
h : ¬Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * Submodule.span R {v}) = ⊤
M : Ideal R
hJM : Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * Submodule.span R {v}) ≤ M
hM : M ∈ Set.Finite.toFinset hf
this : ∀ a ∈ I, ∀ b ∈ I', ∃ c, (algebraMap R A) c = a * b
hmem : u M • (a M * b M) + ∑ x in Finset.erase s M, u x • (a M * b x) ∈ IsLocalization.coeSubmodule A M
M' : Ideal R
hM' : M' ≠ M ∧ M' ∈ s
c : R
hc : (algebraMap R A) c = a M * b M'
hu : ∀ (i : Ideal R), i ≠ M' ∧ i ∈ Set.Finite.toFinset hf → u M' ∈ i
⊢ (algebraMap R A) (u M' * c) ∈ IsLocalization.coeSubmodule A M

Next tactic:
exact Submodule.mem_map_of_mem (M.mul_mem_right _ <| hu M ⟨hM'.1.symm, hM⟩)