Theorem crngcom

Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015)

	Ref	Expression
Hypotheses	ringcl.b	|- B = ( Base ` R )
	ringcl.t	|- .x. = ( .r ` R )
Assertion	crngcom	|- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( Y .x. X ) )

Proof

Step	Hyp	Ref	Expression
1		ringcl.b	|- B = ( Base ` R )
2		ringcl.t	|- .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	cmncom	|- ( ( ( mulGrp ` R ) e. CMnd /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( Y .x. X ) )
8	4 7	syl3an1	|- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( Y .x. X ) )