Theorem oppreqg

Description: Group coset equivalence relation for the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025)

	Ref	Expression
Hypotheses	oppreqg.o	|- O = ( oppR ` R )
	oppreqg.b	|- B = ( Base ` R )
Assertion	oppreqg	|- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) )

Proof

Step	Hyp	Ref	Expression
1		oppreqg.o	|- O = ( oppR ` R )
2		oppreqg.b	|- B = ( Base ` R )
3		eqid	|- ( invg ` R ) = ( invg ` R )
4		eqid	|- ( +g ` R ) = ( +g ` R )
5		eqid	|- ( R ~QG I ) = ( R ~QG I )
6	2 3 4 5	eqgfval	|- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = { <. x , y >. | ( { x , y } C_ B /\ ( ( ( invg ` R ) ` x ) ( +g ` R ) y ) e. I ) } )
7	1	fvexi	|- O e. _V
8	1 2	opprbas	|- B = ( Base ` O )
9	1 3	opprneg	|- ( invg ` R ) = ( invg ` O )
10	1 4	oppradd	|- ( +g ` R ) = ( +g ` O )
11		eqid	|- ( O ~QG I ) = ( O ~QG I )
12	8 9 10 11	eqgfval	|- ( ( O e. _V /\ I C_ B ) -> ( O ~QG I ) = { <. x , y >. | ( { x , y } C_ B /\ ( ( ( invg ` R ) ` x ) ( +g ` R ) y ) e. I ) } )
13	7 12	mpan	|- ( I C_ B -> ( O ~QG I ) = { <. x , y >. | ( { x , y } C_ B /\ ( ( ( invg ` R ) ` x ) ( +g ` R ) y ) e. I ) } )
14	13	adantl	|- ( ( R e. V /\ I C_ B ) -> ( O ~QG I ) = { <. x , y >. | ( { x , y } C_ B /\ ( ( ( invg ` R ) ` x ) ( +g ` R ) y ) e. I ) } )
15	6 14	eqtr4d	|- ( ( R e. V /\ I C_ B ) -> ( R ~QG I ) = ( O ~QG I ) )