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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 ⊢ ( +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 )