/-- If `b` mod `p` spans `S/p` as `R/p`-space, then `b` itself spans `Frac(S)` as `K`-space.

Here,
 * `p` is an ideal of `R` such that `R / p` is nontrivial
 * `K` is a field that has an embedding of `R` (in particular we can take `K = Frac(R)`)
 * `L` is a field extension of `K`
 * `S` is the integral closure of `R` in `L`

More precisely, we avoid quotients in this statement and instead require that `b ∪ pS` spans `S`.
-/
theorem FinrankQuotientMap.span_eq_top [IsDomain R] [IsDomain S] [Algebra K L] [IsNoetherian R S]
    [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [IsIntegralClosure S R L]
    [NoZeroSMulDivisors R K] (hp : p ≠ ⊤) (b : Set S)
    (hb' : Submodule.span R b ⊔ (p.map (algebraMap R S)).restrictScalars R = ⊤) :
    Submodule.span K (algebraMap S L '' b) = ⊤ := by
  have hRL : Function.Injective (algebraMap R L) := by
    rw [IsScalarTower.algebraMap_eq R K L]
    exact (algebraMap K L).injective.comp (NoZeroSMulDivisors.algebraMap_injective R K)
  -- Let `M` be the `R`-module spanned by the proposed basis elements.
  let M : Submodule R S := Submodule.span R b
  -- Then `S / M` is generated by some finite set of `n` vectors `a`.
  letI h : Module.Finite R (S ⧸ M) :=
    Module.Finite.of_surjective (Submodule.mkQ _) (Submodule.Quotient.mk_surjective _)
  obtain ⟨n, a, ha⟩ := @Module.Finite.exists_fin _ _ _ _ _ h
  -- Because the image of `p` in `S / M` is `⊤`,
  have smul_top_eq : p • (⊤ : Submodule R (S ⧸ M)) = ⊤ := by
    calc
      p • ⊤ = Submodule.map M.mkQ (p • ⊤) := by
        rw [Submodule.map_smul'', Submodule.map_top, M.range_mkQ]
      _ = ⊤ := by rw [Ideal.smul_top_eq_map, (Submodule.map_mkQ_eq_top M _).mpr hb']
  -- we can write the elements of `a` as `p`-linear combinations of other elements of `a`.
  have exists_sum : ∀ x : S ⧸ M, ∃ a' : Fin n → R, (∀ i, a' i ∈ p) ∧ ∑ i, a' i • a i = x := by
    intro x
    obtain ⟨a'', ha'', hx⟩ := (Submodule.mem_ideal_smul_span_iff_exists_sum p a x).1
      (by { rw [ha, smul_top_eq]; exact Submodule.mem_top } :
        x ∈ p • Submodule.span R (Set.range a))
    · refine ⟨fun i => a'' i, fun i => ha'' _, ?_⟩
      rw [← hx, Finsupp.sum_fintype]
      exact fun _ => zero_smul _ _
  choose A' hA'p hA' using fun i => exists_sum (a i)
  -- This gives us a(n invertible) matrix `A` such that `det A ∈ (M = span R b)`,
  let A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
  let B := A.adjugate
  have A_smul : ∀ i, ∑ j, A i j • a j = 0 := by
    intros
    simp [Matrix.sub_apply, Matrix.of_apply, ne_eq, Matrix.one_apply, sub_smul,
      Finset.sum_sub_distrib, hA', sub_self]
  -- since `span S {det A} / M = 0`.
  have d_smul : ∀ i, A.det • a i = 0 := by
    intro i
    calc
      A.det • a i = ∑ j, (B * A) i j • a j := ?_
      _ = ∑ k, B i k • ∑ j, A k j • a j := ?_
      _ = 0 := Finset.sum_eq_zero fun k _ => ?_
    · simp only [Matrix.adjugate_mul, Matrix.smul_apply, Matrix.one_apply, smul_eq_mul, ite_true,
        mul_ite, mul_one, mul_zero, ite_smul, zero_smul, Finset.sum_ite_eq, Finset.mem_univ]
    · simp only [Matrix.mul_apply, Finset.smul_sum, Finset.sum_smul, smul_smul]
      rw [Finset.sum_comm]
    · rw [A_smul, smul_zero]
  -- In the rings of integers we have the desired inclusion.
  have span_d : (Submodule.span S ({algebraMap R S A.det} : Set S)).restrictScalars R ≤ M := by
    intro x hx
    rw [Submodule.restrictScalars_mem] at hx
    obtain ⟨x', rfl⟩ := Submodule.mem_span_singleton.mp hx
    rw [smul_eq_mul, mul_comm, ← Algebra.smul_def] at hx ⊢
    rw [← Submodule.Quotient.mk_eq_zero, Submodule.Quotient.mk_smul]
    obtain ⟨a', _, quot_x_eq⟩ := exists_sum (Submodule.Quotient.mk x')
    rw [← quot_x_eq, Finset.smul_sum]
    conv =>
      lhs; congr; next => skip
      intro x; rw [smul_comm A.det, d_smul, smul_zero]
    exact Finset.sum_const_zero
  refine top_le_iff.mp
      (calc
        ⊤ = (Ideal.span {algebraMap R L A.det}).restrictScalars K := ?_
        _ ≤ Submodule.span K (algebraMap S L '' b) := ?_)
  -- Because `det A ≠ 0`, we have `span L {det A} = ⊤`.
  · rw [eq_comm, Submodule.restrictScalars_eq_top_iff, Ideal.span_singleton_eq_top]
    refine IsUnit.mk0 _ ((map_ne_zero_iff (algebraMap R L) hRL).mpr ?_)
    refine @ne_zero_of_map _ _ _ _ _ _ (Ideal.Quotient.mk p) _ ?_
    haveI := Ideal.Quotient.nontrivial hp
    calc
      Ideal.Quotient.mk p A.det = Matrix.det ((Ideal.Quotient.mk p).mapMatrix A) := by
        rw [RingHom.map_det, RingHom.mapMatrix_apply]
      _ = Matrix.det ((Ideal.Quotient.mk p).mapMatrix (Matrix.of A' - 1)) := rfl
      _ = Matrix.det fun i j =>
          (Ideal.Quotient.mk p) (A' i j) - (1 : Matrix (Fin n) (Fin n) (R ⧸ p)) i j := ?_
      _ = Matrix.det (-1 : Matrix (Fin n) (Fin n) (R ⧸ p)) := ?_
      _ = (-1 : R ⧸ p) ^ n := by rw [Matrix.det_neg, Fintype.card_fin, Matrix.det_one, mul_one]
      _ ≠ 0 := IsUnit.ne_zero (isUnit_one.neg.pow _)
    · refine congr_arg Matrix.det (Matrix.ext fun i j => ?_)
      rw [map_sub, RingHom.mapMatrix_apply, map_one]
      rfl
    · refine congr_arg Matrix.det (Matrix.ext fun i j => ?_)
      rw [Ideal.Quotient.eq_zero_iff_mem.mpr (hA'p i j), zero_sub]
      rfl
  -- And we conclude `L = span L {det A} ≤ span K b`, so `span K b` spans everything.
  · intro x hx
    rw [Submodule.restrictScalars_mem, IsScalarTower.algebraMap_apply R S L] at hx
    refine' IsFractionRing.ideal_span_singleton_map_subset R _ hRL span_d hx
    haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.of_algebraMap_injective hRL
    

Proof state:
case intro.intro.refine_2
R : Type u
inst✝²⁶ : CommRing R
S : Type v
inst✝²⁵ : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
inst✝²⁴ : Algebra R S
K : Type u_1
inst✝²³ : Field K
inst✝²² : Algebra R K
hRK : IsFractionRing R K
L : Type u_2
inst✝²¹ : Field L
inst✝²⁰ : Algebra S L
inst✝¹⁹ : IsFractionRing S L
V : Type u_3
V' : Type u_4
V'' : Type u_5
inst✝¹⁸ : AddCommGroup V
inst✝¹⁷ : Module R V
inst✝¹⁶ : Module K V
inst✝¹⁵ : IsScalarTower R K V
inst✝¹⁴ : AddCommGroup V'
inst✝¹³ : Module R V'
inst✝¹² : Module S V'
inst✝¹¹ : IsScalarTower R S V'
inst✝¹⁰ : AddCommGroup V''
inst✝⁹ : Module R V''
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : IsNoetherian R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : IsIntegralClosure S R L
inst✝ : NoZeroSMulDivisors R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map (algebraMap R S) p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
h : Module.Finite R (S ⧸ M) := Module.Finite.of_surjective (Submodule.mkQ M) (Submodule.Quotient.mk_surjective M)
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i : Fin n, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1 : Fin n, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := Matrix.adjugate A
A_smul : ∀ (i : Fin n), ∑ j : Fin n, A i j • a j = 0
d_smul : ∀ (i : Fin n), Matrix.det A • a i = 0
span_d : Submodule.restrictScalars R (Submodule.span S {(algebraMap R S) (Matrix.det A)}) ≤ M
x : L
hx : x ∈ span {(algebraMap S L) ((algebraMap R S) (Matrix.det A))}
this : NoZeroSMulDivisors R L
⊢ Algebra.IsAlgebraic R L

Next tactic:
rw [← IsFractionRing.isAlgebraic_iff' R S]