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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	$⊢ φ \to (K \in HL \land W \in H)$
Assertion	dvhlmod	$⊢ φ \to U \in LMod$

Proof

Step	Hyp	Ref	Expression
1		dvhlvec.h	$⊢ H = LHyp (K)$
2		dvhlvec.u	$⊢ U = DVecH (K) (W)$
3		dvhlvec.k	$⊢ φ \to (K \in HL \land W \in H)$
4	1 2 3	dvhlvec	$⊢ φ \to U \in LVec$
5		lveclmod	$⊢ U \in LVec \to U \in LMod$
6	4 5	syl	$⊢ φ \to U \in LMod$