Theorem dvalvec

Description: The constructed partial vector space A for a lattice K is a left vector space. (Contributed by NM, 11-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvalvec.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvalvec.v	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dvalvec	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑈 ∈ LVec )

Proof

Step	Hyp	Ref	Expression
1		dvalvec.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvalvec.v	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
3		eqid	⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		eqid	⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
5		eqid	⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
7		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8		eqid	⊢ ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) )
9		eqid	⊢ ( .r ‘ ( Scalar ‘ 𝑈 ) ) = ( .r ‘ ( Scalar ‘ 𝑈 ) )
10		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
11	1 2 3 4 5 6 7 8 9 10	dvalveclem	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑈 ∈ LVec )