Theorem isnzr

Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015)

	Ref	Expression
Hypotheses	isnzr.o	$$⊢\underset{\dot{}}{1}=1_R$$
	isnzr.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	isnzr	$$⊢R\in \mathrm{NzRing}\leftrightarrow \left(R\in \mathrm{Ring}\wedge \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)$$

Proof

Step	Hyp	Ref	Expression
1		isnzr.o	$$⊢\underset{\dot{}}{1}=1_R$$
2		isnzr.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		fveq2	$$⊢r=R\to 1_r=1_R$$
4	3 1	eqtr4di	$$⊢r=R\to 1_r=\underset{\dot{}}{1}$$
5		fveq2	$$⊢r=R\to 0_r=0_R$$
6	5 2	eqtr4di	$$⊢r=R\to 0_r=\underset{\dot{}}{0}$$
7	4 6	neeq12d	$$⊢r=R\to \left(1_r\ne 0_r\leftrightarrow \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)$$
8		df-nzr	$$⊢\mathrm{NzRing}=\left\{r\in \mathrm{Ring}|1_r\ne 0_r\right\}$$
9	7 8	elrab2	$$⊢R\in \mathrm{NzRing}\leftrightarrow \left(R\in \mathrm{Ring}\wedge \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)$$