Theorem: IsLocalization.localization_localization_exists_of_eq
File: Mathlib/RingTheory/Localization/LocalizationLocalization.lean

tactic: rw [IsScalarTower.algebraMap_apply R S T, IsScalarTower.algebraMap_apply R S T,
  IsLocalization.eq_iff_exists N T]
state_before: R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_after: R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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⟩
state_before: R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_after: case intro
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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⟩
state_before: case intro
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_after: case intro.intro.mk
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_before: case intro.intro.mk
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_after: case intro.intro.mk
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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

tactic: rw [map_mul, map_mul, ← eq₂, ← mul_assoc, ← mul_assoc, mul_comm _ (z : S), eq₁,
  mul_comm _ (z : S)]
state_before: case intro.intro.mk
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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')
state_after: no goals

tactic: obtain ⟨c, eq₃ : ↑c * (x * z') = ↑c * (y * z')⟩ := (IsLocalization.eq_iff_exists M S).mp this
state_before: R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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, ↑c * x = ↑c * y
state_after: case intro
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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', ?_⟩, ?_⟩
state_before: case intro
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_after: case intro.refine_1
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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

tactic: rw [mem_localizationLocalizationSubmodule]
state_before: case intro.refine_1
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_after: case intro.refine_1
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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')
⊢ ∃ y z, (algebraMap R S) (↑c * z') = ↑y * (algebraMap R S) ↑z

tactic: refine ⟨z, c * s, ?_⟩
state_before: case intro.refine_1
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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')
⊢ ∃ y z, (algebraMap R S) (↑c * z') = ↑y * (algebraMap R S) ↑z
state_after: case intro.refine_1
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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')
⊢ (algebraMap R S) (↑c * z') = ↑z * (algebraMap R S) ↑(c * s)

tactic: rw [map_mul, ← eq₂, Submonoid.coe_mul, map_mul, mul_left_comm]
state_before: case intro.refine_1
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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')
⊢ (algebraMap R S) (↑c * z') = ↑z * (algebraMap R S) ↑(c * s)
state_after: no goals

tactic: rwa [mul_comm _ z', mul_comm _ z', ← mul_assoc, ← mul_assoc] at eq₃
state_before: case intro.refine_2
R : Type u_1
inst✝⁹ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommSemiring P
N : Submonoid S
T : Type u_4
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
state_after: no goals