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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 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ 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 ∧ 𝑊 ∈ 𝐻 ) )
4	1 2 3	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
5		lveclmod	⊢ ( 𝑈 ∈ LVec → 𝑈 ∈ LMod )
6	4 5	syl	⊢ ( 𝜑 → 𝑈 ∈ LMod )