Theorem dvhlvec

Description: The full vector space U constructed from a Hilbert lattice K (given a fiducial hyperplane W ) is a left module. (Contributed by NM, 23-May-2015)

	Ref	Expression
Hypotheses	dvhlvec.h	|- H = ( LHyp ` K )
	dvhlvec.u	|- U = ( ( DVecH ` K ) ` W )
	dvhlvec.k	|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion	dvhlvec	|- ( ph -> U e. LVec )

Proof

Step	Hyp	Ref	Expression
1		dvhlvec.h	|- H = ( LHyp ` K )
2		dvhlvec.u	|- U = ( ( DVecH ` K ) ` W )
3		dvhlvec.k	|- ( ph -> ( K e. HL /\ W e. H ) )
4		eqid	|- ( Base ` K ) = ( Base ` K )
5		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7		eqid	|- ( Scalar ` U ) = ( Scalar ` U )
8		eqid	|- ( +g ` ( Scalar ` U ) ) = ( +g ` ( Scalar ` U ) )
9		eqid	|- ( +g ` U ) = ( +g ` U )
10		eqid	|- ( 0g ` ( Scalar ` U ) ) = ( 0g ` ( Scalar ` U ) )
11		eqid	|- ( invg ` ( Scalar ` U ) ) = ( invg ` ( Scalar ` U ) )
12		eqid	|- ( .r ` ( Scalar ` U ) ) = ( .r ` ( Scalar ` U ) )
13		eqid	|- ( .s ` U ) = ( .s ` U )
14	4 1 5 6 2 7 8 9 10 11 12 13	dvhlveclem	|- ( ( K e. HL /\ W e. H ) -> U e. LVec )
15	3 14	syl	|- ( ph -> U e. LVec )