Theorem: RingHom.HoldsForLocalizationAway.of_bijective
File: Mathlib/RingTheory/LocalProperties.lean

Proof steps:
state_before: R S : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝³ : CommRing R'
inst✝² : CommRing S'
f : R →+* S
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P
hf : Function.Bijective ⇑f
⊢ P f
state_after: R S : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝³ : CommRing R'
inst✝² : CommRing S'
f : R →+* S
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P
hf : Function.Bijective ⇑f
this : Algebra R S := f.toAlgebra
⊢ P f
tactic: letI := f.toAlgebra
state_before: R S : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝³ : CommRing R'
inst✝² : CommRing S'
f : R →+* S
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P
hf : Function.Bijective ⇑f
this : Algebra R S := f.toAlgebra
⊢ P f
state_after: R S : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝³ : CommRing R'
inst✝² : CommRing S'
f : R →+* S
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P
hf : Function.Bijective ⇑f
this✝ : Algebra R S := f.toAlgebra
this : IsLocalization (Submonoid.powers 1) R
⊢ P f
tactic: have := IsLocalization.at_units (.powers (1 : R)) (by simp)
state_before: R S : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝³ : CommRing R'
inst✝² : CommRing S'
f : R →+* S
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P
hf : Function.Bijective ⇑f
this✝ : Algebra R S := f.toAlgebra
this : IsLocalization (Submonoid.powers 1) R
⊢ P f
state_after: R S : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝³ : CommRing R'
inst✝² : CommRing S'
f : R →+* S
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P
hf : Function.Bijective ⇑f
this✝¹ : Algebra R S := f.toAlgebra
this✝ : IsLocalization (Submonoid.powers 1) R
this : IsLocalization (Submonoid.powers 1) S
⊢ P f
tactic: have := IsLocalization.isLocalization_of_algEquiv (.powers (1 : R))
    (AlgEquiv.ofBijective (Algebra.ofId R S) hf)
state_before: R S : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Submonoid R
N : Submonoid S
R' S' : Type u
inst✝³ : CommRing R'
inst✝² : CommRing S'
f : R →+* S
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P
hf : Function.Bijective ⇑f
this✝¹ : Algebra R S := f.toAlgebra
this✝ : IsLocalization (Submonoid.powers 1) R
this : IsLocalization (Submonoid.powers 1) S
⊢ P f
state_after: no goals
tactic: exact H _ 1