Theorem dvafmulr

Description: Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvafmul.h	|- H = ( LHyp ` K )
	dvafmul.t	|- T = ( ( LTrn ` K ) ` W )
	dvafmul.e	|- E = ( ( TEndo ` K ) ` W )
	dvafmul.u	|- U = ( ( DVecA ` K ) ` W )
	dvafmul.f	|- F = ( Scalar ` U )
	dvafmul.p	|- .x. = ( .r ` F )
Assertion	dvafmulr	|- ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , t e. E |-> ( s o. t ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvafmul.h	|- H = ( LHyp ` K )
2		dvafmul.t	|- T = ( ( LTrn ` K ) ` W )
3		dvafmul.e	|- E = ( ( TEndo ` K ) ` W )
4		dvafmul.u	|- U = ( ( DVecA ` K ) ` W )
5		dvafmul.f	|- F = ( Scalar ` U )
6		dvafmul.p	|- .x. = ( .r ` F )
7		eqid	|- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
8	1 7 4 5	dvasca	|- ( ( K e. V /\ W e. H ) -> F = ( ( EDRing ` K ) ` W ) )
9	8	fveq2d	|- ( ( K e. V /\ W e. H ) -> ( .r ` F ) = ( .r ` ( ( EDRing ` K ) ` W ) ) )
10	6 9	eqtrid	|- ( ( K e. V /\ W e. H ) -> .x. = ( .r ` ( ( EDRing ` K ) ` W ) ) )
11		eqid	|- ( .r ` ( ( EDRing ` K ) ` W ) ) = ( .r ` ( ( EDRing ` K ) ` W ) )
12	1 2 3 7 11	erngfmul	|- ( ( K e. V /\ W e. H ) -> ( .r ` ( ( EDRing ` K ) ` W ) ) = ( s e. E , t e. E |-> ( s o. t ) ) )
13	10 12	eqtrd	|- ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , t e. E |-> ( s o. t ) ) )