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https://github.com/Smaug123/agdaproofs
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43 lines
2.3 KiB
Agda
43 lines
2.3 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Rings.Definition
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open import Rings.Homomorphisms.Definition
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open import Rings.IntegralDomains.Definition
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open import Setoids.Setoids
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open import Sets.EquivalenceRelations
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module Fields.FieldOfFractions.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
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open import Fields.FieldOfFractions.Setoid I
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open import Fields.FieldOfFractions.Ring I
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embedIntoFieldOfFractions : A → fieldOfFractionsSet
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embedIntoFieldOfFractions a = a ,, (Ring.1R R , IntegralDomain.nontrivial I)
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homIntoFieldOfFractions : RingHom R fieldOfFractionsRing embedIntoFieldOfFractions
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RingHom.preserves1 homIntoFieldOfFractions = Equivalence.reflexive (Setoid.eq S)
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RingHom.ringHom homIntoFieldOfFractions {a} {b} = Equivalence.transitive (Setoid.eq S) (Ring.*WellDefined R (Equivalence.reflexive (Setoid.eq S)) (Ring.identIsIdent R)) (Ring.*Commutative R)
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GroupHom.groupHom (RingHom.groupHom homIntoFieldOfFractions) {x} {y} = need
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where
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open Setoid S
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open Equivalence eq
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need : ((x + y) * (Ring.1R R * Ring.1R R)) ∼ (Ring.1R R * ((x * Ring.1R R) + (Ring.1R R * y)))
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need = transitive (transitive (Ring.*WellDefined R reflexive (Ring.identIsIdent R)) (transitive (Ring.*Commutative R) (transitive (Ring.identIsIdent R) (Group.+WellDefined (Ring.additiveGroup R) (symmetric (transitive (Ring.*Commutative R) (Ring.identIsIdent R))) (symmetric (Ring.identIsIdent R)))))) (symmetric (Ring.identIsIdent R))
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GroupHom.wellDefined (RingHom.groupHom homIntoFieldOfFractions) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
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where
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open Equivalence (Setoid.eq S)
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homIntoFieldOfFractionsIsInj : SetoidInjection S fieldOfFractionsSetoid embedIntoFieldOfFractions
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SetoidInjection.wellDefined homIntoFieldOfFractionsIsInj x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
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where
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open Equivalence (Setoid.eq S)
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SetoidInjection.injective homIntoFieldOfFractionsIsInj x~y = transitive (symmetric identIsIdent) (transitive *Commutative (transitive x~y identIsIdent))
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where
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open Ring R
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open Setoid S
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open Equivalence eq
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