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20 lines
911 B
Agda
20 lines
911 B
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Setoids.Setoids
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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module Rings.Ideals.Principal.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
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open Setoid S
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open Ring R
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open Equivalence eq
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open import Rings.Ideals.Principal.Definition R
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open import Rings.Ideals.Definition R
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open import Rings.Ideals.Lemmas R
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open import Rings.Divisible.Definition R
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generatorZeroImpliesAllZero : {c : _} {pred : A → Set c} → {i : Ideal pred} → (princ : PrincipalIdeal i) → PrincipalIdeal.generator princ ∼ 0R → {x : A} → pred x → x ∼ 0R
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generatorZeroImpliesAllZero record { generator = gen ; genIsInIdeal = genIsInIdeal ; genGenerates = genGenerates } gen=0 {x} predX = generatorZeroImpliesMembersZero {x} (divisibleWellDefined gen=0 reflexive (genGenerates predX))
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