Theorem cringm4

Description: Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024)

	Ref	Expression
Hypotheses	cringm4.1	|- B = ( Base ` R )
	cringm4.2	|- .x. = ( .r ` R )
Assertion	cringm4	|- ( ( R e. CRing /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .x. Y ) .x. ( Z .x. W ) ) = ( ( X .x. Z ) .x. ( Y .x. W ) ) )

Proof

Step	Hyp	Ref	Expression
1		cringm4.1	|- B = ( Base ` R )
2		cringm4.2	|- .x. = ( .r ` R )
3		eqid	|- ( mulGrp ` R ) = ( mulGrp ` R )
4	3	crngmgp	|- ( R e. CRing -> ( mulGrp ` R ) e. CMnd )
5	3 1	mgpbas	|- B = ( Base ` ( mulGrp ` R ) )
6	3 2	mgpplusg	|- .x. = ( +g ` ( mulGrp ` R ) )
7	5 6	cmn4	|- ( ( ( mulGrp ` R ) e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .x. Y ) .x. ( Z .x. W ) ) = ( ( X .x. Z ) .x. ( Y .x. W ) ) )
8	4 7	syl3an1	|- ( ( R e. CRing /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .x. Y ) .x. ( Z .x. W ) ) = ( ( X .x. Z ) .x. ( Y .x. W ) ) )