-- Mathlib/RingTheory/Localization/Basic.lean
/-- If `S₁` is the localization of `R` at `M₁` and `S₂` is the localization of
`R` at `M₂`, then every localization `T` of `S₂` at `M₁` is also a localization of
`S₁` at `M₂`, in other words `M₁⁻¹M₂⁻¹R` can be identified with `M₂⁻¹M₁⁻¹R`. -/
lemma commutes (S₁ S₂ T : Type*) [CommSemiring S₁]
    [CommSemiring S₂] [CommSemiring T] [Algebra R S₁] [Algebra R S₂] [Algebra R T] [Algebra S₁ T]
    [Algebra S₂ T] [IsScalarTower R S₁ T] [IsScalarTower R S₂ T] (M₁ M₂ : Submonoid R)
    [IsLocalization M₁ S₁] [IsLocalization M₂ S₂]
    [IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T] :
    IsLocalization (Algebra.algebraMapSubmonoid S₁ M₂) T where
  map_units'

Proof state:
case h
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
a : T
y : S₂
m : R
hm : m ∈ ↑M₁
hy : a * (algebraMap S₁ T) ((algebraMap R S₁) m) = (algebraMap S₂ T) (y, ⟨(algebraMap R S₂) m, ⋯⟩).1
z n : R
hn : n ∈ M₂
hz : y * (algebraMap R S₂) ↑(z, ⟨n, hn⟩).2 = (algebraMap R S₂) (z, ⟨n, hn⟩).1
hunit : IsUnit ((algebraMap R S₁) m)
⊢ a * (algebraMap S₂ T) ((algebraMap R S₂) n) =
    (algebraMap S₁ T) ((algebraMap R S₁) z) * (algebraMap S₁ T) ↑hunit.unit⁻¹

Tactic:
conv_rhs => rw [← IsScalarTower.algebraMap_apply]

Result:
case h
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
a : T
y : S₂
m : R
hm : m ∈ ↑M₁
hy : a * (algebraMap S₁ T) ((algebraMap R S₁) m) = (algebraMap S₂ T) (y, ⟨(algebraMap R S₂) m, ⋯⟩).1
z n : R
hn : n ∈ M₂
hz : y * (algebraMap R S₂) ↑(z, ⟨n, hn⟩).2 = (algebraMap R S₂) (z, ⟨n, hn⟩).1
hunit : IsUnit ((algebraMap R S₁) m)
⊢ a * (algebraMap S₂ T) ((algebraMap R S₂) n) = (algebraMap R T) z * (algebraMap S₁ T) ↑hunit.unit⁻¹

Proof state:
case h
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
x y : S₁
r s : R
d : ↥M₁
hr : x * (algebraMap R S₁) ↑d = (algebraMap R S₁) r
hs : y * (algebraMap R S₁) ↑d = (algebraMap R S₁) s
hxy : (algebraMap S₂ T) ((algebraMap R S₂) r) = (algebraMap S₂ T) ((algebraMap R S₂) s)
c : R
hmc : c ∈ ↑M₁
hc : (algebraMap R S₂) (c * r) = (algebraMap R S₂) (c * s)
a : ↥M₂
ha : ↑a * (c * r) = ↑a * (c * s)
⊢ ↑⟨(algebraMap R S₁) ↑a, ⋯⟩ * x = ↑⟨(algebraMap R S₁) ↑a, ⋯⟩ * y

Tactic:
apply (map_units S₁ d).mul_right_cancel

Result:
case h
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
x y : S₁
r s : R
d : ↥M₁
hr : x * (algebraMap R S₁) ↑d = (algebraMap R S₁) r
hs : y * (algebraMap R S₁) ↑d = (algebraMap R S₁) s
hxy : (algebraMap S₂ T) ((algebraMap R S₂) r) = (algebraMap S₂ T) ((algebraMap R S₂) s)
c : R
hmc : c ∈ ↑M₁
hc : (algebraMap R S₂) (c * r) = (algebraMap R S₂) (c * s)
a : ↥M₂
ha : ↑a * (c * r) = ↑a * (c * s)
⊢ ↑⟨(algebraMap R S₁) ↑a, ⋯⟩ * x * (algebraMap R S₁) ↑d = ↑⟨(algebraMap R S₁) ↑a, ⋯⟩ * y * (algebraMap R S₁) ↑d

Proof state:
case intro.mk.mk.intro.intro
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
a : T
y : S₂
m : R
hm : m ∈ ↑M₁
hy : a * (algebraMap S₁ T) ((algebraMap R S₁) m) = (algebraMap S₂ T) (y, ⟨(algebraMap R S₂) m, ⋯⟩).1
⊢ ∃ x, a * (algebraMap S₁ T) ↑x.2 = (algebraMap S₁ T) x.1

Tactic:
obtain ⟨⟨z, n, hn⟩, hz⟩ := IsLocalization.surj (M := M₂) y

Result:
case intro.mk.mk.intro.intro.intro.mk.mk
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
a : T
y : S₂
m : R
hm : m ∈ ↑M₁
hy : a * (algebraMap S₁ T) ((algebraMap R S₁) m) = (algebraMap S₂ T) (y, ⟨(algebraMap R S₂) m, ⋯⟩).1
z n : R
hn : n ∈ M₂
hz : y * (algebraMap R S₂) ↑(z, ⟨n, hn⟩).2 = (algebraMap R S₂) (z, ⟨n, hn⟩).1
⊢ ∃ x, a * (algebraMap S₁ T) ↑x.2 = (algebraMap S₁ T) x.1

Proof state:
case intro.intro.intro.intro.intro.mk.intro.intro
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
x y : S₁
r s : R
d : ↥M₁
hr : x * (algebraMap R S₁) ↑d = (algebraMap R S₁) r
hs : y * (algebraMap R S₁) ↑d = (algebraMap R S₁) s
hxy : (algebraMap S₂ T) ((algebraMap R S₂) r) = (algebraMap S₂ T) ((algebraMap R S₂) s)
c : R
hmc : c ∈ ↑M₁
hc : (algebraMap R S₂) (c * r) = (algebraMap R S₂) (c * s)
⊢ ∃ c, ↑c * x = ↑c * y

Tactic:
obtain ⟨a, ha⟩ := IsLocalization.exists_of_eq (M := M₂) hc

Result:
case intro.intro.intro.intro.intro.mk.intro.intro.intro
R : Type u_1
inst✝¹³ : CommSemiring R
S₁ : Type u_4
S₂ : Type u_5
T : Type u_6
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
M₁ M₂ : Submonoid R
inst✝² : IsLocalization M₁ S₁
inst✝¹ : IsLocalization M₂ S₂
inst✝ : IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T
x y : S₁
r s : R
d : ↥M₁
hr : x * (algebraMap R S₁) ↑d = (algebraMap R S₁) r
hs : y * (algebraMap R S₁) ↑d = (algebraMap R S₁) s
hxy : (algebraMap S₂ T) ((algebraMap R S₂) r) = (algebraMap S₂ T) ((algebraMap R S₂) s)
c : R
hmc : c ∈ ↑M₁
hc : (algebraMap R S₂) (c * r) = (algebraMap R S₂) (c * s)
a : ↥M₂
ha : ↑a * (c * r) = ↑a * (c * s)
⊢ ∃ c, ↑c * x = ↑c * y