Theorem rngass

Description: Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011) (Revised by AV, 13-Feb-2025)

	Ref	Expression
Hypotheses	rngass.b	|- B = ( Base ` R )
	rngass.t	|- .x. = ( .r ` R )
Assertion	rngass	|- ( ( R e. Rng /\ ( 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		rngass.b	|- B = ( Base ` R )
2		rngass.t	|- .x. = ( .r ` R )
3		eqid	|- ( mulGrp ` R ) = ( mulGrp ` R )
4	3	rngmgp	|- ( R e. Rng -> ( mulGrp ` R ) e. Smgrp )
5	3 1	mgpbas	|- B = ( Base ` ( mulGrp ` R ) )
6	3 2	mgpplusg	|- .x. = ( +g ` ( mulGrp ` R ) )
7	5 6	sgrpass	|- ( ( ( mulGrp ` R ) e. Smgrp /\ ( 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. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )