Theorem drngmul0or

Description: A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014) (Proof shortened by SN, 25-Jun-2025)

	Ref	Expression
Hypotheses	drngmuleq0.b	$$⊢B={\mathrm{Base}}_R$$
	drngmuleq0.o	$$⊢\underset{\dot{}}{0}=0_R$$
	drngmuleq0.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	drngmuleq0.r	$$⊢\phi \to R\in \mathrm{DivRing}$$
	drngmuleq0.x	$$⊢\phi \to X\in B$$
	drngmuleq0.y	$$⊢\phi \to Y\in B$$
Assertion	drngmul0or	$$⊢\phi \to \left(X\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow \left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		drngmuleq0.b	$$⊢B={\mathrm{Base}}_R$$
2		drngmuleq0.o	$$⊢\underset{\dot{}}{0}=0_R$$
3		drngmuleq0.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
4		drngmuleq0.r	$$⊢\phi \to R\in \mathrm{DivRing}$$
5		drngmuleq0.x	$$⊢\phi \to X\in B$$
6		drngmuleq0.y	$$⊢\phi \to Y\in B$$
7		drngdomn	$$⊢R\in \mathrm{DivRing}\to R\in \mathrm{Domn}$$
8	4 7	syl	$$⊢\phi \to R\in \mathrm{Domn}$$
9	1 3 2	domneq0	$$⊢\left(R\in \mathrm{Domn}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow \left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\right)$$
10	8 5 6 9	syl3anc	$$⊢\phi \to \left(X\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow \left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\right)$$