Theorem dvhvsca

Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013)

	Ref	Expression
Hypotheses	dvhfvsca.h	|- H = ( LHyp ` K )
	dvhfvsca.t	|- T = ( ( LTrn ` K ) ` W )
	dvhfvsca.e	|- E = ( ( TEndo ` K ) ` W )
	dvhfvsca.u	|- U = ( ( DVecH ` K ) ` W )
	dvhfvsca.s	|- .x. = ( .s ` U )
Assertion	dvhvsca	|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) = <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. )

Proof

Step	Hyp	Ref	Expression
1		dvhfvsca.h	|- H = ( LHyp ` K )
2		dvhfvsca.t	|- T = ( ( LTrn ` K ) ` W )
3		dvhfvsca.e	|- E = ( ( TEndo ` K ) ` W )
4		dvhfvsca.u	|- U = ( ( DVecH ` K ) ` W )
5		dvhfvsca.s	|- .x. = ( .s ` U )
6	1 2 3 4 5	dvhfvsca	|- ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) )
7	6	oveqd	|- ( ( K e. V /\ W e. H ) -> ( R .x. F ) = ( R ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) F ) )
8		eqid	|- ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. )
9	8	dvhvscaval	|- ( ( R e. E /\ F e. ( T X. E ) ) -> ( R ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) F ) = <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. )
10	7 9	sylan9eq	|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) = <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. )