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Metamath Proof Explorer

Theorem dvhfvsca

Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013) (Revised by Mario Carneiro, 23-Jun-2014)

Ref Expression
Hypotheses dvhfvsca.h 
|- H = ( LHyp ` K )
dvhfvsca.t 
|- T = ( ( LTrn ` K ) ` W )
dvhfvsca.e 
|- E = ( ( TEndo ` K ) ` W )
dvhfvsca.u 
|- U = ( ( DVecH ` K ) ` W )
dvhfvsca.s 
|- .x. = ( .s ` U )
Assertion dvhfvsca 
|- ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) )

Proof

Step Hyp Ref Expression
1 dvhfvsca.h 
 |-  H = ( LHyp ` K )
2 dvhfvsca.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvhfvsca.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 dvhfvsca.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dvhfvsca.s 
 |-  .x. = ( .s ` U )
6 eqid 
 |-  ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
7 1 2 3 6 4 dvhset 
 |-  ( ( K e. V /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
8 7 fveq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( .s ` U ) = ( .s ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
9 3 fvexi 
 |-  E e. _V
10 2 fvexi 
 |-  T e. _V
11 10 9 xpex 
 |-  ( T X. E ) e. _V
12 9 11 mpoex 
 |-  ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) e. _V
13 eqid 
 |-  ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) = ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } )
14 13 lmodvsca 
 |-  ( ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) e. _V -> ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) = ( .s ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
15 12 14 ax-mp 
 |-  ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) = ( .s ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
16 8 5 15 3eqtr4g 
 |-  ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) )