Metamath Proof Explorer

Theorem dvhsca

Description: The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014)

Ref Expression
Hypotheses dvhsca.h 
|- H = ( LHyp ` K )
dvhsca.d 
|- D = ( ( EDRing ` K ) ` W )
dvhsca.u 
|- U = ( ( DVecH ` K ) ` W )
dvhsca.f 
|- F = ( Scalar ` U )
Assertion dvhsca 
|- ( ( K e. X /\ W e. H ) -> F = D )

Proof

Step Hyp Ref Expression
1 dvhsca.h 
 |-  H = ( LHyp ` K )
2 dvhsca.d 
 |-  D = ( ( EDRing ` K ) ` W )
3 dvhsca.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dvhsca.f 
 |-  F = ( Scalar ` U )
5 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7 1 5 6 2 3 dvhset 
 |-  ( ( K e. X /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) , g e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
8 7 fveq2d 
 |-  ( ( K e. X /\ W e. H ) -> ( Scalar ` U ) = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) , g e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
9 2 fvexi 
 |-  D e. _V
10 eqid 
 |-  ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) , g e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) = ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) , g e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } )
11 10 lmodsca 
 |-  ( D e. _V -> D = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) , g e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
12 9 11 ax-mp 
 |-  D = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) , g e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
13 8 4 12 3eqtr4g 
 |-  ( ( K e. X /\ W e. H ) -> F = D )