Theorem opprnzrb

Description: The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr . (Contributed by SN, 20-Jun-2025)

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

Proof

Step	Hyp	Ref	Expression
1		opprnzr.1	|- O = ( oppR ` R )
2	1	opprringb	|- ( R e. Ring <-> O e. Ring )
3	2	anbi1i	|- ( ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) <-> ( O e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) )
4		eqid	|- ( 1r ` R ) = ( 1r ` R )
5		eqid	|- ( 0g ` R ) = ( 0g ` R )
6	4 5	isnzr	|- ( R e. NzRing <-> ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) )
7	1 4	oppr1	|- ( 1r ` R ) = ( 1r ` O )
8	1 5	oppr0	|- ( 0g ` R ) = ( 0g ` O )
9	7 8	isnzr	|- ( O e. NzRing <-> ( O e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) )
10	3 6 9	3bitr4i	|- ( R e. NzRing <-> O e. NzRing )