Theorem ring1nzdiv

Description: In a unitary ring, the ring unity is not a zero divisor. (Contributed by AV, 7-Mar-2025)

	Ref	Expression
Hypotheses	ringunitnzdiv.b	$$⊢B={\mathrm{Base}}_R$$
	ringunitnzdiv.z	$$⊢\underset{\dot{}}{0}=0_R$$
	ringunitnzdiv.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	ringunitnzdiv.r	$$⊢\phi \to R\in \mathrm{Ring}$$
	ringunitnzdiv.y	$$⊢\phi \to Y\in B$$
	ring1nzdiv.x	$$⊢\underset{\dot{}}{1}=1_R$$
Assertion	ring1nzdiv	$$⊢\phi \to \left(\underset{\dot{}}{1}\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow Y=\underset{\dot{}}{0}\right)$$

Proof

Step	Hyp	Ref	Expression
1		ringunitnzdiv.b	$$⊢B={\mathrm{Base}}_R$$
2		ringunitnzdiv.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		ringunitnzdiv.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
4		ringunitnzdiv.r	$$⊢\phi \to R\in \mathrm{Ring}$$
5		ringunitnzdiv.y	$$⊢\phi \to Y\in B$$
6		ring1nzdiv.x	$$⊢\underset{\dot{}}{1}=1_R$$
7		eqid	$$⊢\mathrm{Unit}\left(R\right)=\mathrm{Unit}\left(R\right)$$
8	7 6	1unit	$$⊢R\in \mathrm{Ring}\to \underset{\dot{}}{1}\in \mathrm{Unit}\left(R\right)$$
9	4 8	syl	$$⊢\phi \to \underset{\dot{}}{1}\in \mathrm{Unit}\left(R\right)$$
10	1 2 3 4 5 9	ringunitnzdiv	$$⊢\phi \to \left(\underset{\dot{}}{1}\underset{\dot{}}{·}Y=\underset{\dot{}}{0}\leftrightarrow Y=\underset{\dot{}}{0}\right)$$