Theorem opprnzr

Description: The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015)

	Ref	Expression
Hypothesis	opprnzr.1	|- O = ( oppR ` R )
Assertion	opprnzr	|- ( R e. NzRing -> O e. NzRing )

Proof

Step	Hyp	Ref	Expression
1		opprnzr.1	|- O = ( oppR ` R )
2		nzrring	|- ( R e. NzRing -> R e. Ring )
3	1	opprring	|- ( R e. Ring -> O e. Ring )
4	2 3	syl	|- ( R e. NzRing -> O e. Ring )
5		eqid	|- ( Base ` R ) = ( Base ` R )
6	5	isnzr2	|- ( R e. NzRing <-> ( R e. Ring /\ 2o ~<_ ( Base ` R ) ) )
7	6	simprbi	|- ( R e. NzRing -> 2o ~<_ ( Base ` R ) )
8	1 5	opprbas	|- ( Base ` R ) = ( Base ` O )
9	8	isnzr2	|- ( O e. NzRing <-> ( O e. Ring /\ 2o ~<_ ( Base ` R ) ) )
10	4 7 9	sylanbrc	|- ( R e. NzRing -> O e. NzRing )