Theorem dvasca

Description: The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom W ). (Contributed by NM, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvasca.h	|- H = ( LHyp ` K )
	dvasca.d	|- D = ( ( EDRing ` K ) ` W )
	dvasca.u	|- U = ( ( DVecA ` K ) ` W )
	dvasca.f	|- F = ( Scalar ` U )
Assertion	dvasca	|- ( ( K e. X /\ W e. H ) -> F = D )

Proof

Step	Hyp	Ref	Expression
1		dvasca.h	|- H = ( LHyp ` K )
2		dvasca.d	|- D = ( ( EDRing ` K ) ` W )
3		dvasca.u	|- U = ( ( DVecA ` K ) ` W )
4		dvasca.f	|- F = ( Scalar ` U )
5		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7	1 5 6 2 3	dvaset	|- ( ( K e. X /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) )
8	7	fveq2d	|- ( ( K e. X /\ W e. H ) -> ( Scalar ` U ) = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) ) )
9	2	fvexi	|- D e. _V
10		eqid	|- ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) = ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } )
11	10	lmodsca	|- ( D e. _V -> D = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) ) )
12	9 11	ax-mp	|- D = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) )
13	8 4 12	3eqtr4g	|- ( ( K e. X /\ W e. H ) -> F = D )