-- Mathlib/RingTheory/Localization/LocalizationLocalization.lean
theorem isLocalization_of_submonoid_le (M N : Submonoid R) (h : M ≤ N) [IsLocalization M S]
    [IsLocalization N T] [Algebra S T] [IsScalarTower R S T] :
    IsLocalization (N.map (algebraMap R S)) T :=
  { map_units'

Proof state:
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
⊢ ∀ (y : ↥(Submonoid.map (algebraMap R S) N)), IsUnit ((algebraMap S T) ↑y)

Tactic:
rintro ⟨_, ⟨y, hy, rfl⟩⟩

Result:
case mk.intro.intro
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
y : R
hy : y ∈ ↑N
⊢ IsUnit ((algebraMap S T) ↑⟨(algebraMap R S) y, ⋯⟩)

Proof state:
case intro.mk.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
x₁ x₂ : S
y₁ : R
s₁ : ↥M
e₁ : x₁ * (algebraMap R S) ↑s₁ = (algebraMap R S) y₁
y₂ : R
s₂ : ↥M
e₂ : x₂ * (algebraMap R S) ↑s₂ = (algebraMap R S) y₂
a : ↥N
e : ↑a * (y₁ * ↑s₂) = ↑a * (y₂ * ↑s₁)
⊢ ∃ a c, ↑c * (↑a * (y₁ * ↑s₂)) = ↑c * (↑a * (y₂ * ↑s₁))

Tactic:
exact ⟨a, 1, by convert e using 1 <;> simp⟩

Result:
no goals

Proof state:
case intro.mk.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
x₁ x₂ : S
y₁ : R
s₁ : ↥M
e₁ : x₁ * (algebraMap R S) ↑s₁ = (algebraMap R S) y₁
y₂ : R
s₂ : ↥M
e₂ : x₂ * (algebraMap R S) ↑s₂ = (algebraMap R S) y₂
⊢ (algebraMap S T) x₁ = (algebraMap S T) x₂ → ∃ a, (algebraMap R S) ↑a * x₁ = (algebraMap R S) ↑a * x₂

Tactic:
suffices algebraMap R T (y₁ * s₂) = algebraMap R T (y₂ * s₁) →
    ∃ a : N, algebraMap R S (a * (y₁ * s₂)) = algebraMap R S (a * (y₂ * s₁)) by
  have h₁ := @IsUnit.mul_left_inj T _ _ (algebraMap S T x₁) (algebraMap S T x₂)
    (IsLocalization.map_units T ⟨(s₁ : R), h s₁.prop⟩)
  have h₂ := @IsUnit.mul_left_inj T _ _ ((algebraMap S T x₁) * (algebraMap R T s₁))
    ((algebraMap S T x₂) * (algebraMap R T s₁))
    (IsLocalization.map_units T ⟨(s₂ : R), h s₂.prop⟩)
  simp only [IsScalarTower.algebraMap_apply R S T, Subtype.coe_mk] at h₁ h₂
  simp only [IsScalarTower.algebraMap_apply R S T, map_mul, ← e₁, ← e₂, ← mul_assoc,
    mul_right_comm _ (algebraMap R S s₂),
    mul_right_comm _ (algebraMap S T (algebraMap R S s₂)),
    (IsLocalization.map_units S s₁).mul_left_inj,
    (IsLocalization.map_units S s₂).mul_left_inj] at this
  rw [h₂, h₁] at this
  simpa only [mul_comm] using this

Result:
case intro.mk.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
x₁ x₂ : S
y₁ : R
s₁ : ↥M
e₁ : x₁ * (algebraMap R S) ↑s₁ = (algebraMap R S) y₁
y₂ : R
s₂ : ↥M
e₂ : x₂ * (algebraMap R S) ↑s₂ = (algebraMap R S) y₂
⊢ (algebraMap R T) (y₁ * ↑s₂) = (algebraMap R T) (y₂ * ↑s₁) →
    ∃ a, (algebraMap R S) (↑a * (y₁ * ↑s₂)) = (algebraMap R S) (↑a * (y₂ * ↑s₁))

Proof state:
case intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
x₁ x₂ : S
y₁ : R
s₁ : ↥M
e₁ : x₁ * (algebraMap R S) ↑(y₁, s₁).2 = (algebraMap R S) (y₁, s₁).1
⊢ (algebraMap S T) x₁ = (algebraMap S T) x₂ → ∃ c, ↑c * x₁ = ↑c * x₂

Tactic:
obtain ⟨⟨y₂, s₂⟩, e₂⟩ := IsLocalization.surj M x₂

Result:
case intro.mk.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
x₁ x₂ : S
y₁ : R
s₁ : ↥M
e₁ : x₁ * (algebraMap R S) ↑(y₁, s₁).2 = (algebraMap R S) (y₁, s₁).1
y₂ : R
s₂ : ↥M
e₂ : x₂ * (algebraMap R S) ↑(y₂, s₂).2 = (algebraMap R S) (y₂, s₂).1
⊢ (algebraMap S T) x₁ = (algebraMap S T) x₂ → ∃ c, ↑c * x₁ = ↑c * x₂

Proof state:
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
x₁ x₂ : S
y₁ : R
s₁ : ↥M
e₁ : x₁ * (algebraMap R S) ↑s₁ = (algebraMap R S) y₁
y₂ : R
s₂ : ↥M
e₂ : x₂ * (algebraMap R S) ↑s₂ = (algebraMap R S) y₂
this :
  (algebraMap R T) (y₁ * ↑s₂) = (algebraMap R T) (y₂ * ↑s₁) →
    ∃ a, (algebraMap R S) (↑a * (y₁ * ↑s₂)) = (algebraMap R S) (↑a * (y₂ * ↑s₁))
h₁ :
  (algebraMap S T) x₁ * (algebraMap S T) ((algebraMap R S) ↑s₁) =
      (algebraMap S T) x₂ * (algebraMap S T) ((algebraMap R S) ↑s₁) ↔
    (algebraMap S T) x₁ = (algebraMap S T) x₂
h₂ :
  (algebraMap S T) x₁ * (algebraMap S T) ((algebraMap R S) ↑s₁) * (algebraMap S T) ((algebraMap R S) ↑s₂) =
      (algebraMap S T) x₂ * (algebraMap S T) ((algebraMap R S) ↑s₁) * (algebraMap S T) ((algebraMap R S) ↑s₂) ↔
    (algebraMap S T) x₁ * (algebraMap S T) ((algebraMap R S) ↑s₁) =
      (algebraMap S T) x₂ * (algebraMap S T) ((algebraMap R S) ↑s₁)
⊢ (algebraMap S T) x₁ = (algebraMap S T) x₂ → ∃ a, (algebraMap R S) ↑a * x₁ = (algebraMap R S) ↑a * x₂

Tactic:
simp only [IsScalarTower.algebraMap_apply R S T, map_mul, ← e₁, ← e₂, ← mul_assoc,
  mul_right_comm _ (algebraMap R S s₂),
  mul_right_comm _ (algebraMap S T (algebraMap R S s₂)),
  (IsLocalization.map_units S s₁).mul_left_inj,
  (IsLocalization.map_units S s₂).mul_left_inj] at this

Result:
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
M N : Submonoid R
h : M ≤ N
inst✝³ : IsLocalization M S
inst✝² : IsLocalization N T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
x₁ x₂ : S
y₁ : R
s₁ : ↥M
e₁ : x₁ * (algebraMap R S) ↑s₁ = (algebraMap R S) y₁
y₂ : R
s₂ : ↥M
e₂ : x₂ * (algebraMap R S) ↑s₂ = (algebraMap R S) y₂
h₁ :
  (algebraMap S T) x₁ * (algebraMap S T) ((algebraMap R S) ↑s₁) =
      (algebraMap S T) x₂ * (algebraMap S T) ((algebraMap R S) ↑s₁) ↔
    (algebraMap S T) x₁ = (algebraMap S T) x₂
h₂ :
  (algebraMap S T) x₁ * (algebraMap S T) ((algebraMap R S) ↑s₁) * (algebraMap S T) ((algebraMap R S) ↑s₂) =
      (algebraMap S T) x₂ * (algebraMap S T) ((algebraMap R S) ↑s₁) * (algebraMap S T) ((algebraMap R S) ↑s₂) ↔
    (algebraMap S T) x₁ * (algebraMap S T) ((algebraMap R S) ↑s₁) =
      (algebraMap S T) x₂ * (algebraMap S T) ((algebraMap R S) ↑s₁)
this :
  (algebraMap S T) x₁ * (algebraMap S T) ((algebraMap R S) ↑s₁) * (algebraMap S T) ((algebraMap R S) ↑s₂) =
      (algebraMap S T) x₂ * (algebraMap S T) ((algebraMap R S) ↑s₁) * (algebraMap S T) ((algebraMap R S) ↑s₂) →
    ∃ a, (algebraMap R S) ↑a * x₁ = (algebraMap R S) ↑a * x₂
⊢ (algebraMap S T) x₁ = (algebraMap S T) x₂ → ∃ a, (algebraMap R S) ↑a * x₁ = (algebraMap R S) ↑a * x₂