# 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 ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
Assertion dvhlvec ( 𝜑𝑈 ∈ LVec )

### 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 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 ( +g𝑈 ) = ( +g𝑈 )
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 ( ·𝑠𝑈 ) = ( ·𝑠𝑈 )
14 4 1 5 6 2 7 8 9 10 11 12 13 dvhlveclem ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝑈 ∈ LVec )
15 3 14 syl ( 𝜑𝑈 ∈ LVec )