-- Mathlib/RingTheory/Localization/LocalizationLocalization.lean
theorem localization_localization_exists_of_eq [IsLocalization N T] (x y : R) :
    algebraMap R T x = algebraMap R T y →
      ∃ c : localizationLocalizationSubmodule M N, ↑c * x = ↑c * y

Proof state:
case intro.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑s = (algebraMap R S) z'
⊢ (algebraMap R S) (x * z') = (algebraMap R S) (y * z')

Tactic:
rw [map_mul, map_mul, ← eq₂, ← mul_assoc, ← mul_assoc, mul_comm _ (z : S), eq₁,
  mul_comm _ (z : S)]

Result:
no goals

Proof state:
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
⊢ (algebraMap R T) x = (algebraMap R T) y → ∃ c, ↑c * x = ↑c * y

Tactic:
rw [IsScalarTower.algebraMap_apply R S T, IsScalarTower.algebraMap_apply R S T,
  IsLocalization.eq_iff_exists N T]

Result:
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
⊢ (∃ c, ↑c * (algebraMap R S) x = ↑c * (algebraMap R S) y) → ∃ c, ↑c * x = ↑c * y

Proof state:
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
⊢ (∃ c, ↑c * (algebraMap R S) x = ↑c * (algebraMap R S) y) → ∃ c, ↑c * x = ↑c * y

Tactic:
rintro ⟨z, eq₁⟩

Result:
case intro
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
⊢ ∃ c, ↑c * x = ↑c * y

Proof state:
case intro.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑(z', s).2 = (algebraMap R S) (z', s).1
⊢ ∃ c, ↑c * x = ↑c * y

Tactic:
dsimp only at eq₂

Result:
case intro.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑s = (algebraMap R S) z'
⊢ ∃ c, ↑c * x = ↑c * y

Proof state:
case intro.refine_2
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑s = (algebraMap R S) z'
this : (algebraMap R S) (x * z') = (algebraMap R S) (y * z')
c : ↥M
eq₃ : ↑c * (x * z') = ↑c * (y * z')
⊢ ↑⟨↑c * z', ⋯⟩ * x = ↑⟨↑c * z', ⋯⟩ * y

Tactic:
rwa [mul_comm _ z', mul_comm _ z', ← mul_assoc, ← mul_assoc] at eq₃

Result:
no goals

Proof state:
case intro
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑s = (algebraMap R S) z'
this : (algebraMap R S) (x * z') = (algebraMap R S) (y * z')
c : ↥M
eq₃ : ↑c * (x * z') = ↑c * (y * z')
⊢ ∃ c, ↑c * x = ↑c * y

Tactic:
refine ⟨⟨c * z', ?_⟩, ?_⟩

Result:
case intro.refine_1
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑s = (algebraMap R S) z'
this : (algebraMap R S) (x * z') = (algebraMap R S) (y * z')
c : ↥M
eq₃ : ↑c * (x * z') = ↑c * (y * z')
⊢ ↑c * z' ∈ localizationLocalizationSubmodule M N

case intro.refine_2
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑s = (algebraMap R S) z'
this : (algebraMap R S) (x * z') = (algebraMap R S) (y * z')
c : ↥M
eq₃ : ↑c * (x * z') = ↑c * (y * z')
⊢ ↑⟨↑c * z', ?intro.refine_1⟩ * x = ↑⟨↑c * z', ?intro.refine_1⟩ * y

Proof state:
case intro
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
⊢ ∃ c, ↑c * x = ↑c * y

Tactic:
rcases IsLocalization.surj M (z : S) with ⟨⟨z', s⟩, eq₂⟩

Result:
case intro.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
x y : R
z : ↥N
eq₁ : ↑z * (algebraMap R S) x = ↑z * (algebraMap R S) y
z' : R
s : ↥M
eq₂ : ↑z * (algebraMap R S) ↑(z', s).2 = (algebraMap R S) (z', s).1
⊢ ∃ c, ↑c * x = ↑c * y