Theorem opprrngb

Description: A class is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng . (Contributed by AV, 15-Feb-2025)

	Ref	Expression
Hypothesis	opprbas.1	|- O = ( oppR ` R )
Assertion	opprrngb	|- ( R e. Rng <-> O e. Rng )

Proof

Step	Hyp	Ref	Expression
1		opprbas.1	|- O = ( oppR ` R )
2	1	opprrng	|- ( R e. Rng -> O e. Rng )
3		eqid	|- ( oppR ` O ) = ( oppR ` O )
4	3	opprrng	|- ( O e. Rng -> ( oppR ` O ) e. Rng )
5		eqidd	|- ( T. -> ( Base ` R ) = ( Base ` R ) )
6		eqid	|- ( Base ` R ) = ( Base ` R )
7	1 6	opprbas	|- ( Base ` R ) = ( Base ` O )
8	3 7	opprbas	|- ( Base ` R ) = ( Base ` ( oppR ` O ) )
9	8	a1i	|- ( T. -> ( Base ` R ) = ( Base ` ( oppR ` O ) ) )
10		eqid	|- ( +g ` R ) = ( +g ` R )
11	1 10	oppradd	|- ( +g ` R ) = ( +g ` O )
12	3 11	oppradd	|- ( +g ` R ) = ( +g ` ( oppR ` O ) )
13	12	oveqi	|- ( x ( +g ` R ) y ) = ( x ( +g ` ( oppR ` O ) ) y )
14	13	a1i	|- ( ( T. /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` ( oppR ` O ) ) y ) )
15		eqid	|- ( .r ` O ) = ( .r ` O )
16		eqid	|- ( .r ` ( oppR ` O ) ) = ( .r ` ( oppR ` O ) )
17	7 15 3 16	opprmul	|- ( x ( .r ` ( oppR ` O ) ) y ) = ( y ( .r ` O ) x )
18		eqid	|- ( .r ` R ) = ( .r ` R )
19	6 18 1 15	opprmul	|- ( y ( .r ` O ) x ) = ( x ( .r ` R ) y )
20	17 19	eqtr2i	|- ( x ( .r ` R ) y ) = ( x ( .r ` ( oppR ` O ) ) y )
21	20	a1i	|- ( ( T. /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` ( oppR ` O ) ) y ) )
22	5 9 14 21	rngpropd	|- ( T. -> ( R e. Rng <-> ( oppR ` O ) e. Rng ) )
23	22	mptru	|- ( R e. Rng <-> ( oppR ` O ) e. Rng )
24	4 23	sylibr	|- ( O e. Rng -> R e. Rng )
25	2 24	impbii	|- ( R e. Rng <-> O e. Rng )