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Metamath Proof Explorer

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 )