Theorem lmodvscl

Description: Closure of scalar product for a left module. ( hvmulcl analog.) (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvscl.v	|- V = ( Base ` W )
	lmodvscl.f	|- F = ( Scalar ` W )
	lmodvscl.s	|- .x. = ( .s ` W )
	lmodvscl.k	|- K = ( Base ` F )
Assertion	lmodvscl	|- ( ( W e. LMod /\ R e. K /\ X e. V ) -> ( R .x. X ) e. V )

Proof

Step	Hyp	Ref	Expression
1		lmodvscl.v	|- V = ( Base ` W )
2		lmodvscl.f	|- F = ( Scalar ` W )
3		lmodvscl.s	|- .x. = ( .s ` W )
4		lmodvscl.k	|- K = ( Base ` F )
5		biid	|- ( W e. LMod <-> W e. LMod )
6		pm4.24	|- ( R e. K <-> ( R e. K /\ R e. K ) )
7		pm4.24	|- ( X e. V <-> ( X e. V /\ X e. V ) )
8		eqid	|- ( +g ` W ) = ( +g ` W )
9		eqid	|- ( +g ` F ) = ( +g ` F )
10		eqid	|- ( .r ` F ) = ( .r ` F )
11		eqid	|- ( 1r ` F ) = ( 1r ` F )
12	1 8 3 2 4 9 10 11	lmodlema	|- ( ( W e. LMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( ( R .x. X ) e. V /\ ( R .x. ( X ( +g ` W ) X ) ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) /\ ( ( R ( +g ` F ) R ) .x. X ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) ) /\ ( ( ( R ( .r ` F ) R ) .x. X ) = ( R .x. ( R .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X ) ) )
13	12	simpld	|- ( ( W e. LMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( R .x. X ) e. V /\ ( R .x. ( X ( +g ` W ) X ) ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) /\ ( ( R ( +g ` F ) R ) .x. X ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) ) )
14	13	simp1d	|- ( ( W e. LMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( R .x. X ) e. V )
15	5 6 7 14	syl3anb	|- ( ( W e. LMod /\ R e. K /\ X e. V ) -> ( R .x. X ) e. V )