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 e. HL /\ W e. H ) -> U e. 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 e. HL /\ W e. H ) -> U e. LVec )