l3lab/ntp-mathlib huggingface dataset:
theorem localization_finiteType : RingHom.LocalizationPreserves @RingHom.FiniteType := by
  introv R hf
  -- mirrors the proof of `localization_map_finite`
  letI := f.toAlgebra
  letI := ((algebraMap S S').comp f).toAlgebra
  let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
  letI := f'.toAlgebra
  haveI : IsScalarTower R R' S' :=
    IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp M.le_comap_map).symm
  let fₐ : S →ₐ[R] S' := AlgHom.mk' (algebraMap S S') fun c x => RingHom.map_mul _ _ _
  obtain ⟨T, hT⟩ := id hf
  use T.image (algebraMap S S')
  rw [eq_top_iff]
  rintro x -
  obtain ⟨y, ⟨_, ⟨r, hr, rfl⟩⟩, rfl⟩ := IsLocalization.mk'_surjective (M.map f) x
  rw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, Finset.coe_image]
  have hy : y ∈ Algebra.adjoin R (T : Set S) := by rw [hT]; trivial
  replace hy : algebraMap S S' y ∈ (Algebra.adjoin R (T : Set S)).map fₐ :=
    Subalgebra.mem_map.mpr ⟨_, hy, rfl⟩
  rw [fₐ.map_adjoin T] at hy
  have H : Algebra.adjoin R (algebraMap S S' '' T) ≤
      (Algebra.adjoin R' (algebraMap S S' '' T)).restrictScalars R := by
    rw [Algebra.adjoin_le_iff]; exact Algebra.subset_adjoin
  convert (Algebra.adjoin R' (algebraMap S S' '' T)).smul_mem (H hy)
    (IsLocalization.mk' R' (1 : R) ⟨r, hr⟩) using 1
  rw [Algebra.smul_def]
  erw [IsLocalization.map_mk' M.le_comap_map]
  

Proof state:
case h.e'_4
R✝ S✝ : Type u
inst✝¹³ : CommRing R✝
inst✝¹² : CommRing S✝
M✝ : Submonoid R✝
N : Submonoid S✝
R'✝ S'✝ : Type u
inst✝¹¹ : CommRing R'✝
inst✝¹⁰ : CommRing S'✝
f✝ : R✝ →+* S✝
inst✝⁹ : Algebra R✝ R'✝
inst✝⁸ : Algebra S✝ S'✝
R S : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
M : Submonoid R
R' S' : Type u_1
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization M R'
inst✝ : IsLocalization (Submonoid.map f M) S'
hf : RingHom.FiniteType f
this✝² : Algebra R S := RingHom.toAlgebra f
this✝¹ : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)
f' : R' →+* S' := IsLocalization.map S' f (_ : M ≤ Submonoid.comap f (Submonoid.map f M))
this✝ : Algebra R' S' := RingHom.toAlgebra f'
this : IsScalarTower R R' S'
fₐ : S →ₐ[R] S' :=
  AlgHom.mk' (algebraMap S S')
    (_ : ∀ (c : R) (x : S), (algebraMap S S') (f c * x) = (algebraMap S S') (f c) * (algebraMap S S') x)
T : Finset S
hT : Algebra.adjoin R ↑T = ⊤
y : S
r : R
hr : r ∈ ↑M
hy : (algebraMap S S') y ∈ Algebra.adjoin R (⇑fₐ '' ↑T)
H :
  Algebra.adjoin R (⇑(algebraMap S S') '' ↑T) ≤
    Subalgebra.restrictScalars R (Algebra.adjoin R' (⇑(algebraMap S S') '' ↑T))
⊢ IsLocalization.mk' S' 1 { val := f r, property := (_ : ∃ a ∈ ↑M, f a = f r) } * (algebraMap S S') y =
    IsLocalization.mk' S' (f 1)
        { val := f ↑{ val := r, property := hr },
          property := (_ : ↑{ val := r, property := hr } ∈ Submonoid.comap f (Submonoid.map f M)) } *
      (algebraMap S S') y

Next tactic:
rw [map_one]
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  

Proof state:
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
⊢ ∃ m, m • x ∈ A

Next tactic:
let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  

Proof state:
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
⊢ ∃ m, m • x ∈ A

Next tactic:
let y := IsLocalization.commonDenomOfFinset M s
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  

Proof state:
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
⊢ ∃ m, m • x ∈ A

Next tactic:
have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  

Proof state:
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
⊢ ∃ m, m • x ∈ A

Next tactic:
obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by 

Proof state:
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
⊢ ↑y • ↑s ⊆ ↑(Subalgebra.map g A)

Next tactic:
rw [hx₁]
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by rw [hx₁]; 

Proof state:
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
⊢ ⇑g '' ↑(finsetIntegerMultiple M s) ⊆ ↑(Subalgebra.map g A)

Next tactic:
exact Set.image_subset _ hA₁
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
  

Proof state:
case intro
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
n : ℕ
hn : ∀ n_1 ≥ n, ↑y ^ n_1 • (algebraMap S S') x ∈ Subalgebra.map g A
⊢ ∃ m, m • x ∈ A

Next tactic:
obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl)
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
  obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl)
  

Proof state:
case intro.intro.intro
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
n : ℕ
hn : ∀ n_1 ≥ n, ↑y ^ n_1 • (algebraMap S S') x ∈ Subalgebra.map g A
x' : S
hx' : x' ∈ ↑A.toSubsemiring
hx'' : ↑g x' = ↑y ^ n • (algebraMap S S') x
⊢ ∃ m, m • x ∈ A

Next tactic:
rw [Algebra.smul_def, ← _root_.map_mul] at hx''
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
  obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl)
  rw [Algebra.smul_def, ← _root_.map_mul] at hx''
  

Proof state:
case intro.intro.intro
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
n : ℕ
hn : ∀ n_1 ≥ n, ↑y ^ n_1 • (algebraMap S S') x ∈ Subalgebra.map g A
x' : S
hx' : x' ∈ ↑A.toSubsemiring
hx'' : ↑g x' = (algebraMap S S') (↑y ^ n * x)
⊢ ∃ m, m • x ∈ A

Next tactic:
obtain ⟨a, ha₂⟩ := (IsLocalization.eq_iff_exists M S').mp hx''
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
  obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl)
  rw [Algebra.smul_def, ← _root_.map_mul] at hx''
  obtain ⟨a, ha₂⟩ := (IsLocalization.eq_iff_exists M S').mp hx''
  

Proof state:
case intro.intro.intro.intro
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
n : ℕ
hn : ∀ n_1 ≥ n, ↑y ^ n_1 • (algebraMap S S') x ∈ Subalgebra.map g A
x' : S
hx' : x' ∈ ↑A.toSubsemiring
hx'' : ↑g x' = (algebraMap S S') (↑y ^ n * x)
a : ↥M
ha₂ : ↑a * x' = ↑a * (↑y ^ n * x)
⊢ ∃ m, m • x ∈ A

Next tactic:
use a * y ^ n
/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`.
-/
theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
    ∃ m : M, m • x ∈ A := by
  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
  let y := IsLocalization.commonDenomOfFinset M s
  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
    (IsLocalization.finsetIntegerMultiple_image _ s).symm
  obtain ⟨n, hn⟩ :=
    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
  obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl)
  rw [Algebra.smul_def, ← _root_.map_mul] at hx''
  obtain ⟨a, ha₂⟩ := (IsLocalization.eq_iff_exists M S').mp hx''
  use a * y ^ n
  

Proof state:
case h
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'
M : Submonoid S
inst✝ : IsLocalization M S'
x : S
s : Finset S'
A : Subalgebra R S
hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A
hA₂ : M ≤ A.toSubmonoid
hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s
g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
y : ↥M := commonDenomOfFinset M s
hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)
n : ℕ
hn : ∀ n_1 ≥ n, ↑y ^ n_1 • (algebraMap S S') x ∈ Subalgebra.map g A
x' : S
hx' : x' ∈ ↑A.toSubsemiring
hx'' : ↑g x' = (algebraMap S S') (↑y ^ n * x)
a : ↥M
ha₂ : ↑a * x' = ↑a * (↑y ^ n * x)
⊢ (a * y ^ n) • x ∈ A

Next tactic:
convert A.mul_mem hx' (hA₂ a.prop) using 1