Theorem dvhmulr

Description: Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvhfmul.h	|- H = ( LHyp ` K )
	dvhfmul.t	|- T = ( ( LTrn ` K ) ` W )
	dvhfmul.e	|- E = ( ( TEndo ` K ) ` W )
	dvhfmul.u	|- U = ( ( DVecH ` K ) ` W )
	dvhfmul.f	|- F = ( Scalar ` U )
	dvhfmul.m	|- .x. = ( .r ` F )
Assertion	dvhmulr	|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( R .x. S ) = ( R o. S ) )

Proof

Step	Hyp	Ref	Expression
1		dvhfmul.h	|- H = ( LHyp ` K )
2		dvhfmul.t	|- T = ( ( LTrn ` K ) ` W )
3		dvhfmul.e	|- E = ( ( TEndo ` K ) ` W )
4		dvhfmul.u	|- U = ( ( DVecH ` K ) ` W )
5		dvhfmul.f	|- F = ( Scalar ` U )
6		dvhfmul.m	|- .x. = ( .r ` F )
7	1 2 3 4 5 6	dvhfmulr	|- ( ( K e. V /\ W e. H ) -> .x. = ( r e. E , s e. E |-> ( r o. s ) ) )
8	7	oveqd	|- ( ( K e. V /\ W e. H ) -> ( R .x. S ) = ( R ( r e. E , s e. E |-> ( r o. s ) ) S ) )
9		coexg	|- ( ( R e. E /\ S e. E ) -> ( R o. S ) e. _V )
10		coeq1	|- ( r = R -> ( r o. s ) = ( R o. s ) )
11		coeq2	|- ( s = S -> ( R o. s ) = ( R o. S ) )
12		eqid	|- ( r e. E , s e. E |-> ( r o. s ) ) = ( r e. E , s e. E |-> ( r o. s ) )
13	10 11 12	ovmpog	|- ( ( R e. E /\ S e. E /\ ( R o. S ) e. _V ) -> ( R ( r e. E , s e. E |-> ( r o. s ) ) S ) = ( R o. S ) )
14	9 13	mpd3an3	|- ( ( R e. E /\ S e. E ) -> ( R ( r e. E , s e. E |-> ( r o. s ) ) S ) = ( R o. S ) )
15	8 14	sylan9eq	|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( R .x. S ) = ( R o. S ) )