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 ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
Assertion dvhlmod ( 𝜑𝑈 ∈ LMod )

Proof

Step Hyp Ref Expression
1 dvhlvec.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dvhlvec.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dvhlvec.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
4 1 2 3 dvhlvec ( 𝜑𝑈 ∈ LVec )
5 lveclmod ( 𝑈 ∈ LVec → 𝑈 ∈ LMod )
6 4 5 syl ( 𝜑𝑈 ∈ LMod )