Theorem ringvcl

Description: Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015)

	Ref	Expression
Hypotheses	ringvcl.b	|- B = ( Base ` R )
	ringvcl.t	|- .x. = ( .r ` R )
Assertion	ringvcl	|- ( ( R e. Ring /\ X e. ( B ^m I ) /\ Y e. ( B ^m I ) ) -> ( X oF .x. Y ) e. ( B ^m I ) )

Proof

Step	Hyp	Ref	Expression
1		ringvcl.b	|- B = ( Base ` R )
2		ringvcl.t	|- .x. = ( .r ` R )
3		eqid	|- ( mulGrp ` R ) = ( mulGrp ` R )
4	3	ringmgp	|- ( R e. Ring -> ( mulGrp ` R ) e. Mnd )
5	3 1	mgpbas	|- B = ( Base ` ( mulGrp ` R ) )
6	3 2	mgpplusg	|- .x. = ( +g ` ( mulGrp ` R ) )
7	5 6	mndvcl	|- ( ( ( mulGrp ` R ) e. Mnd /\ X e. ( B ^m I ) /\ Y e. ( B ^m I ) ) -> ( X oF .x. Y ) e. ( B ^m I ) )
8	4 7	syl3an1	|- ( ( R e. Ring /\ X e. ( B ^m I ) /\ Y e. ( B ^m I ) ) -> ( X oF .x. Y ) e. ( B ^m I ) )