Theorem ringassd

Description: Associative law for multiplication in a ring. (Contributed by SN, 14-Aug-2024)

	Ref	Expression
Hypotheses	ringassd.b	|- B = ( Base ` R )
	ringassd.t	|- .x. = ( .r ` R )
	ringassd.r	|- ( ph -> R e. Ring )
	ringassd.x	|- ( ph -> X e. B )
	ringassd.y	|- ( ph -> Y e. B )
	ringassd.z	|- ( ph -> Z e. B )
Assertion	ringassd	|- ( ph -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringassd.b	|- B = ( Base ` R )
2		ringassd.t	|- .x. = ( .r ` R )
3		ringassd.r	|- ( ph -> R e. Ring )
4		ringassd.x	|- ( ph -> X e. B )
5		ringassd.y	|- ( ph -> Y e. B )
6		ringassd.z	|- ( ph -> Z e. B )
7	1 2	ringass	|- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
8	3 4 5 6 7	syl13anc	|- ( ph -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )