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agdaproofs/Fields/FieldOfFractions.agda
Patrick Stevens 1f26064502 Split out much more structure (#37)
2019-08-18 14:57:41 +01:00

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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Fields.Fields
open import Functions
open import Setoids.Setoids
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields.FieldOfFractions where
fieldOfFractionsSet : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} → {R : Ring S _+_ _*_} → IntegralDomain R → Set (a ⊔ b)
fieldOfFractionsSet {A = A} {S = S} {R = R} I = (A && (Sg A (λ a → (Setoid._∼_ S a (Ring.0R R) → False))))
fieldOfFractionsSetoid : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} → {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Setoid (fieldOfFractionsSet I)
Setoid._∼_ (fieldOfFractionsSetoid {S = S} {_*_ = _*_} I) (a ,, (b , b!=0)) (c ,, (d , d!=0)) = Setoid._∼_ S (a * d) (b * c)
Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq (fieldOfFractionsSetoid {R = R} I))) {a ,, (b , b!=0)} = Ring.multCommutative R
Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq (fieldOfFractionsSetoid {S = S} {R = R} I))) {a ,, (b , b!=0)} {c ,, (d , d!=0)} ad=bc = transitive (Ring.multCommutative R) (transitive (symmetric ad=bc) (Ring.multCommutative R))
where
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
Transitive.transitive (Equivalence.transitiveEq (Setoid.eq (fieldOfFractionsSetoid {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I))) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} ad=bc cf=de = p5
where
open Setoid S
open Ring R
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
p : (a * d) * f ∼ (b * c) * f
p = Ring.multWellDefined R ad=bc reflexive
p2 : (a * f) * d ∼ b * (d * e)
p2 = transitive (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive p (transitive (symmetric multAssoc) (multWellDefined reflexive cf=de)))
p3 : (a * f) * d ∼ (b * e) * d
p3 = transitive p2 (transitive (multWellDefined reflexive multCommutative) multAssoc)
p4 : (d ∼ 0R) || ((a * f) ∼ (b * e))
p4 = cancelIntDom I (transitive multCommutative (transitive p3 multCommutative))
p5 : (a * f) ∼ (b * e)
p5 with p4
p5 | inl d=0 = exFalso (d!=0 d=0)
p5 | inr x = x
fieldOfFractionsPlus : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → fieldOfFractionsSet I → fieldOfFractionsSet I → fieldOfFractionsSet I
fieldOfFractionsPlus {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (((a * d) + (b * c)) ,, ((b * d) , ans))
where
open Setoid S
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
fieldOfFractionsTimes : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → fieldOfFractionsSet I → fieldOfFractionsSet I → fieldOfFractionsSet I
fieldOfFractionsTimes {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (a * c) ,, ((b * d) , ans)
where
open Setoid S
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
fieldOfFractionsRing : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Ring (fieldOfFractionsSetoid I) (fieldOfFractionsPlus I) (fieldOfFractionsTimes I)
Group.wellDefined (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} {g ,, (h , h!=0)} af=be ch=dg = need
where
open Setoid S
open Ring R
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
have1 : (c * h) ∼ (d * g)
have1 = ch=dg
have2 : (a * f) ∼ (b * e)
have2 = af=be
need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
need = transitive (transitive (Ring.multCommutative R) (transitive (Ring.multDistributes R) (Group.wellDefined (Ring.additiveGroup R) (transitive multAssoc (transitive (multWellDefined (multCommutative) reflexive) (transitive (multWellDefined multAssoc reflexive) (transitive (multWellDefined (multWellDefined have2 reflexive) reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined (transitive (transitive (symmetric multAssoc) (multWellDefined reflexive multCommutative)) multAssoc) reflexive) (symmetric multAssoc))))))))) (transitive multCommutative (transitive (transitive (symmetric multAssoc) (multWellDefined reflexive (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined have1 reflexive) (transitive (symmetric multAssoc) (multWellDefined reflexive multCommutative))))))) multAssoc))))) (symmetric (Ring.multDistributes R))
Group.identity (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) = Ring.0R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Group.inverse (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) (a ,, b) = Group.inverse (Ring.additiveGroup R) a ,, b
Group.multAssoc (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : (((a * (d * f)) + (b * ((c * f) + (d * e)))) * ((b * d) * f)) ∼ ((b * (d * f)) * ((((a * d) + (b * c)) * f) + ((b * d) * e)))
need = transitive (Ring.multCommutative R) (Ring.multWellDefined R (symmetric (Ring.multAssoc R)) (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (Ring.multDistributes R)) (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (Group.wellDefined (Ring.additiveGroup R) (Ring.multAssoc R) (Ring.multAssoc R))) (transitive (Group.multAssoc (Ring.additiveGroup R)) (Group.wellDefined (Ring.additiveGroup R) (transitive (transitive (Group.wellDefined (Ring.additiveGroup R) (transitive (Ring.multAssoc R) (Ring.multCommutative R)) (Ring.multCommutative R)) (symmetric (Ring.multDistributes R))) (Ring.multCommutative R)) reflexive)))))
Group.multIdentRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : (((a * Ring.1R R) + (b * Group.identity (Ring.additiveGroup R))) * b) ∼ ((b * Ring.1R R) * a)
need = transitive (transitive (Ring.multWellDefined R (transitive (Group.wellDefined (Ring.additiveGroup R) (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) reflexive) (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (ringTimesZero R)) (Group.multIdentRight (Ring.additiveGroup R)))) reflexive) (Ring.multCommutative R)) (symmetric (Ring.multWellDefined R (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) reflexive))
Group.multIdentLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : (((Group.identity (Ring.additiveGroup R) * b) + (Ring.1R R * a)) * b) ∼ ((Ring.1R R * b) * a)
need = transitive (transitive (Ring.multWellDefined R (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (Ring.multIdentIsIdent R)) (transitive (Group.wellDefined (Ring.additiveGroup R) (transitive (Ring.multCommutative R) (ringTimesZero R)) reflexive) (Group.multIdentLeft (Ring.additiveGroup R)))) reflexive) (Ring.multCommutative R)) (Ring.multWellDefined R (symmetric (Ring.multIdentIsIdent R)) reflexive)
Group.invLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : (((Group.inverse (Ring.additiveGroup R) a * b) + (b * a)) * Ring.1R R) ∼ ((b * b) * Group.identity (Ring.additiveGroup R))
need = transitive (transitive (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) (transitive (Group.wellDefined (Ring.additiveGroup R) (Ring.multCommutative R) reflexive) (transitive (symmetric (Ring.multDistributes R)) (transitive (Ring.multWellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (ringTimesZero R))))) (symmetric (ringTimesZero R))
Group.invRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : (((a * b) + (b * Group.inverse (Ring.additiveGroup R) a)) * Ring.1R R) ∼ ((b * b) * Group.identity (Ring.additiveGroup R))
need = transitive (transitive (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) (transitive (Group.wellDefined (Ring.additiveGroup R) (Ring.multCommutative R) reflexive) (transitive (symmetric (Ring.multDistributes R)) (transitive (Ring.multWellDefined R reflexive (Group.invRight (Ring.additiveGroup R))) (ringTimesZero R))))) (symmetric (ringTimesZero R))
Ring.multWellDefined (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : ((a * c) * (f * h)) ∼ ((b * d) * (e * g))
need = transitive (Ring.multWellDefined R reflexive (Ring.multCommutative R)) (transitive (Ring.multAssoc R) (transitive (Ring.multWellDefined R (symmetric (Ring.multAssoc R)) reflexive) (transitive (Ring.multWellDefined R (Ring.multWellDefined R reflexive ch=dg) reflexive) (transitive (Ring.multCommutative R) (transitive (Ring.multAssoc R) (transitive (Ring.multWellDefined R (Ring.multCommutative R) reflexive) (transitive (Ring.multWellDefined R af=be reflexive) (transitive (Ring.multAssoc R) (transitive (Ring.multWellDefined R (transitive (symmetric (Ring.multAssoc R)) (transitive (Ring.multWellDefined R reflexive (Ring.multCommutative R)) (Ring.multAssoc R))) reflexive) (symmetric (Ring.multAssoc R)))))))))))
Ring.1R (fieldOfFractionsRing {R = R} I) = Ring.1R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Ring.groupIsAbelian (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : (((a * d) + (b * c)) * (d * b)) ∼ ((b * d) * ((c * b) + (d * a)))
need = transitive (Ring.multCommutative R) (Ring.multWellDefined R (Ring.multCommutative R) (transitive (Group.wellDefined (Ring.additiveGroup R) (Ring.multCommutative R) (Ring.multCommutative R)) (Ring.groupIsAbelian R)))
Ring.multAssoc (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : ((a * (c * e)) * ((b * d) * f)) ∼ ((b * (d * f)) * ((a * c) * e))
need = transitive (Ring.multWellDefined R (Ring.multAssoc R) (symmetric (Ring.multAssoc R))) (Ring.multCommutative R)
Ring.multCommutative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : ((a * c) * (d * b)) ∼ ((b * d) * (c * a))
need = transitive (Ring.multCommutative R) (Ring.multWellDefined R (Ring.multCommutative R) (Ring.multCommutative R))
Ring.multDistributes (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where
open Setoid S
open Ring R
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
inter : b * (a * ((c * f) + (d * e))) ∼ (((a * c) * (b * f)) + ((b * d) * (a * e)))
inter = transitive multAssoc (transitive multDistributes (Group.wellDefined additiveGroup (transitive multAssoc (transitive (multWellDefined (transitive (multWellDefined (multCommutative) reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc))) reflexive) (symmetric multAssoc))) (transitive multAssoc (transitive (multWellDefined (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) reflexive) (symmetric multAssoc)))))
need : ((a * ((c * f) + (d * e))) * ((b * d) * (b * f))) ∼ ((b * (d * f)) * (((a * c) * (b * f)) + ((b * d) * (a * e))))
need = transitive (Ring.multWellDefined R reflexive (Ring.multWellDefined R reflexive (Ring.multCommutative R))) (transitive (Ring.multWellDefined R reflexive (Ring.multAssoc R)) (transitive (Ring.multCommutative R) (transitive (Ring.multWellDefined R (Ring.multWellDefined R (symmetric (Ring.multAssoc R)) reflexive) reflexive) (transitive (symmetric (Ring.multAssoc R)) (Ring.multWellDefined R reflexive inter)))))
Ring.multIdentIsIdent (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : (((Ring.1R R) * a) * b) ∼ (((Ring.1R R * b)) * a)
need = transitive (Ring.multWellDefined R (Ring.multIdentIsIdent R) reflexive) (transitive (Ring.multCommutative R) (Ring.multWellDefined R (symmetric (Ring.multIdentIsIdent R)) reflexive))
fieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Field (fieldOfFractionsRing I)
Field.allInvertible (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) (fst ,, (b , _)) prA = (b ,, (fst , ans)) , need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R)
need = Ring.multWellDefined R (Ring.multCommutative R) reflexive
ans : fst ∼ Ring.0R R → False
ans pr = prA need'
where
need' : (fst * Ring.1R R) ∼ (b * Ring.0R R)
need' = transitive (Ring.multWellDefined R pr reflexive) (transitive (transitive (Ring.multCommutative R) (ringTimesZero R)) (symmetric (ringTimesZero R)))
Field.nontrivial (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) pr = IntegralDomain.nontrivial I (symmetric (transitive (symmetric (ringTimesZero R)) (transitive (Ring.multCommutative R) (transitive pr (Ring.multIdentIsIdent R)))))
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
pr' : (Ring.0R R) * (Ring.1R R) ∼ (Ring.1R R) * (Ring.1R R)
pr' = pr
embedIntoFieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → A → fieldOfFractionsSet I
embedIntoFieldOfFractions {R = R} I a = a ,, (Ring.1R R , IntegralDomain.nontrivial I)
homIntoFieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → RingHom R (fieldOfFractionsRing I) (embedIntoFieldOfFractions I)
RingHom.preserves1 (homIntoFieldOfFractions {S = S} I) = Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))
RingHom.ringHom (homIntoFieldOfFractions {S = S} {R = R} I) {a} {b} = Transitive.transitive (Equivalence.transitiveEq (Setoid.eq S)) (Ring.multWellDefined R (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) (Ring.multIdentIsIdent R)) (Ring.multCommutative R)
GroupHom.groupHom (RingHom.groupHom (homIntoFieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {x} {y} = need
where
open Setoid S
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
need : ((x + y) * (Ring.1R R * Ring.1R R)) ∼ (Ring.1R R * ((x * Ring.1R R) + (Ring.1R R * y)))
need = transitive (transitive (Ring.multWellDefined R reflexive (Ring.multIdentIsIdent R)) (transitive (Ring.multCommutative R) (transitive (Ring.multIdentIsIdent R) (Group.wellDefined (Ring.additiveGroup R) (symmetric (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R))) (symmetric (Ring.multIdentIsIdent R)))))) (symmetric (Ring.multIdentIsIdent R))
GroupHom.wellDefined (RingHom.groupHom (homIntoFieldOfFractions {S = S} {R = R} I)) x=y = transitive (Ring.multCommutative R) (Ring.multWellDefined R reflexive x=y)
where
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
homIntoFieldOfFractionsIsInj : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → SetoidInjection S (fieldOfFractionsSetoid I) (embedIntoFieldOfFractions I)
SetoidInjection.wellDefined (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x=y = transitive (Ring.multCommutative R) (Ring.multWellDefined R reflexive x=y)
where
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
SetoidInjection.injective (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x~y = transitive (symmetric multIdentIsIdent) (transitive multCommutative (transitive x~y multIdentIsIdent))
where
open Ring R
open Setoid S
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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