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	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dvhlvec.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	dvhlvec.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
Assertion	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$

Proof

Step	Hyp	Ref	Expression
1		dvhlvec.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		dvhlvec.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		dvhlvec.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
4	1 2 3	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
5		lveclmod	$$⊢U\in \mathrm{LVec}\to U\in \mathrm{LMod}$$
6	4 5	syl	$$⊢\phi \to U\in \mathrm{LMod}$$