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Theorems · Theorem · ring theory

List.prod_eq_zero_iff

∀ {M₀ : Type u_4} [inst : MonoidWithZero M₀] [Nontrivial M₀] [NoZeroDivisors M₀] {l : List M₀}, l.prod = 0 ↔ 0 ∈ l

Product of elements of a list l equals zero if and only if 0 ∈ l. See also List.prod_eq_zero for an implication that needs weaker typeclass assumptions.

Defined in
Mathlib.Algebra.BigOperators.Ring.List
Cited by
4 results in Mathlib
Foundations
Depth 14 from the axioms · uses propext
Assumes
MonoidWithZeroNontrivialNoZeroDivisors
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Multiset.prod_eq_zero_iff · cited by 7Multiset.prod_eq_zero_iffNat.primeFactorsList_unique · cited by 3Nat.primeFactorsList_uniq…jacobiSym.eq_zero_iff_not_coprime · cited by 3jacobiSym.eq_zero_iff_not…List.prod_ne_zero · cited by 3List.prod_ne_zeroNontrivial · cited by 2416NontrivialNoZeroDivisors · cited by 545NoZeroDivisorsMonoidWithZero · cited by 456MonoidWithZeroList.prod_eq_zero_iffCITED BYCITES

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Cited by4

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