Theorem dvabase

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, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvabase.h	|- H = ( LHyp ` K )
	dvabase.e	|- E = ( ( TEndo ` K ) ` W )
	dvabase.u	|- U = ( ( DVecA ` K ) ` W )
	dvabase.f	|- F = ( Scalar ` U )
	dvabase.c	|- C = ( Base ` F )
Assertion	dvabase	|- ( ( K e. X /\ W e. H ) -> C = E )

Proof

Step	Hyp	Ref	Expression
1		dvabase.h	|- H = ( LHyp ` K )
2		dvabase.e	|- E = ( ( TEndo ` K ) ` W )
3		dvabase.u	|- U = ( ( DVecA ` K ) ` W )
4		dvabase.f	|- F = ( Scalar ` U )
5		dvabase.c	|- C = ( Base ` F )
6		eqid	|- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
7	1 6 3 4	dvasca	|- ( ( K e. X /\ W e. H ) -> F = ( ( EDRing ` K ) ` W ) )
8	7	fveq2d	|- ( ( K e. X /\ W e. H ) -> ( Base ` F ) = ( Base ` ( ( EDRing ` K ) ` W ) ) )
9	5 8	syl5eq	|- ( ( K e. X /\ W e. H ) -> C = ( Base ` ( ( EDRing ` K ) ` W ) ) )
10		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11		eqid	|- ( Base ` ( ( EDRing ` K ) ` W ) ) = ( Base ` ( ( EDRing ` K ) ` W ) )
12	1 10 2 6 11	erngbase	|- ( ( K e. X /\ W e. H ) -> ( Base ` ( ( EDRing ` K ) ` W ) ) = E )
13	9 12	eqtrd	|- ( ( K e. X /\ W e. H ) -> C = E )