Theorem srgass

Description: Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	srgcl.b	|- B = ( Base ` R )
	srgcl.t	|- .x. = ( .r ` R )
Assertion	srgass	|- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )

Proof

Step	Hyp	Ref	Expression
1		srgcl.b	|- B = ( Base ` R )
2		srgcl.t	|- .x. = ( .r ` R )
3		eqid	|- ( mulGrp ` R ) = ( mulGrp ` R )
4	3	srgmgp	|- ( R e. SRing -> ( mulGrp ` R ) e. Mnd )
5	3 1	mgpbas	|- B = ( Base ` ( mulGrp ` R ) )
6	3 2	mgpplusg	|- .x. = ( +g ` ( mulGrp ` R ) )
7	5 6	mndass	|- ( ( ( mulGrp ` R ) e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
8	4 7	sylan	|- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )