Theorem: IsLocalization.bijective
File: Mathlib/RingTheory/Localization/Basic.lean

Proof steps:
state_before: R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
Q : Type u_4
inst✝² : CommSemiring Q
inst✝¹ : Algebra R Q
inst✝ : IsLocalization M Q
f : S →+* Q
hf : f.comp (algebraMap R S) = algebraMap R Q
⊢ f = ↑(algEquiv M S Q)
state_after: case h
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
Q : Type u_4
inst✝² : CommSemiring Q
inst✝¹ : Algebra R Q
inst✝ : IsLocalization M Q
f : S →+* Q
hf : f.comp (algebraMap R S) = algebraMap R Q
⊢ f.comp (algebraMap R S) = (↑(algEquiv M S Q)).comp (algebraMap R S)
tactic: apply IsLocalization.ringHom_ext M
state_before: case h
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
Q : Type u_4
inst✝² : CommSemiring Q
inst✝¹ : Algebra R Q
inst✝ : IsLocalization M Q
f : S →+* Q
hf : f.comp (algebraMap R S) = algebraMap R Q
⊢ f.comp (algebraMap R S) = (↑(algEquiv M S Q)).comp (algebraMap R S)
state_after: case h
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
Q : Type u_4
inst✝² : CommSemiring Q
inst✝¹ : Algebra R Q
inst✝ : IsLocalization M Q
f : S →+* Q
hf : f.comp (algebraMap R S) = algebraMap R Q
⊢ algebraMap R Q = (↑(algEquiv M S Q)).comp (algebraMap R S)
tactic: rw [hf]
state_before: case h
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
Q : Type u_4
inst✝² : CommSemiring Q
inst✝¹ : Algebra R Q
inst✝ : IsLocalization M Q
f : S →+* Q
hf : f.comp (algebraMap R S) = algebraMap R Q
⊢ algebraMap R Q = (↑(algEquiv M S Q)).comp (algebraMap R S)
state_after: case h.a
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
Q : Type u_4
inst✝² : CommSemiring Q
inst✝¹ : Algebra R Q
inst✝ : IsLocalization M Q
f : S →+* Q
hf : f.comp (algebraMap R S) = algebraMap R Q
x✝ : R
⊢ (algebraMap R Q) x✝ = ((↑(algEquiv M S Q)).comp (algebraMap R S)) x✝
tactic: ext
state_before: case h.a
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
Q : Type u_4
inst✝² : CommSemiring Q
inst✝¹ : Algebra R Q
inst✝ : IsLocalization M Q
f : S →+* Q
hf : f.comp (algebraMap R S) = algebraMap R Q
x✝ : R
⊢ (algebraMap R Q) x✝ = ((↑(algEquiv M S Q)).comp (algebraMap R S)) x✝
state_after: no goals
tactic: simp