Metamath Proof Explorer


Theorem dvhlmod

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 dvhlmod
|- ( ph -> U e. LMod )

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 1 2 3 dvhlvec
 |-  ( ph -> U e. LVec )
5 lveclmod
 |-  ( U e. LVec -> U e. LMod )
6 4 5 syl
 |-  ( ph -> U e. LMod )