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Metamath Proof Explorer

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 ⊢ + U = + U
5 eqid ⊢ TEndo ⁡ K ⁡ W = TEndo ⁡ K ⁡ W
6 eqid ⊢ Scalar ⁡ U = Scalar ⁡ U
7 eqid ⊢ Base K = Base K
8 eqid ⊢ + Scalar ⁡ U = + Scalar ⁡ U
9 eqid ⊢ ⋅ Scalar ⁡ U = ⋅ Scalar ⁡ U
10 eqid ⊢ ⋅ U = ⋅ U
11 1 2 3 4 5 6 7 8 9 10 dvalveclem ⊢ K ∈ HL ∧ W ∈ H → U ∈ LVec