Theorem drngmuleq0

Description: An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014)

	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	$$⊢\phi \to Y\in B$$
	drngmuleq0.e	$$⊢\phi \to Y\ne \underset{\dot{}}{0}$$
Assertion	drngmuleq0	$$⊢\phi \to \left(X\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow X=\underset{\dot{}}{0}\right)$$

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		drngmuleq0.e	$$⊢\phi \to Y\ne \underset{\dot{}}{0}$$
8	1 2 3 4 5 6	drngmul0or	$$⊢\phi \to \left(X\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow \left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\right)$$
9		df-ne	$$⊢Y\ne \underset{\dot{}}{0}\leftrightarrow \neg Y=\underset{\dot{}}{0}$$
10		orel2	$$⊢\neg Y=\underset{\dot{}}{0}\to \left(\left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\to X=\underset{\dot{}}{0}\right)$$
11		orc	$$⊢X=\underset{\dot{}}{0}\to \left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)$$
12	10 11	impbid1	$$⊢\neg Y=\underset{\dot{}}{0}\to \left(\left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\leftrightarrow X=\underset{\dot{}}{0}\right)$$
13	9 12	sylbi	$$⊢Y\ne \underset{\dot{}}{0}\to \left(\left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\leftrightarrow X=\underset{\dot{}}{0}\right)$$
14	7 13	syl	$$⊢\phi \to \left(\left(X=\underset{\dot{}}{0}\vee Y=\underset{\dot{}}{0}\right)\leftrightarrow X=\underset{\dot{}}{0}\right)$$
15	8 14	bitrd	$$⊢\phi \to \left(X\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow X=\underset{\dot{}}{0}\right)$$