Theorem ringcl

Description: Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

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

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	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	mndcl	|- ( ( ( mulGrp ` R ) e. Mnd /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
8	4 7	syl3an1	|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )