Metamath Proof Explorer

Theorem dvhfset

Description: The constructed full vector space H for a lattice K . (Contributed by NM, 17-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypothesis dvhset.h 
|- H = ( LHyp ` K )
Assertion dvhfset 
|- ( K e. V -> ( DVecH ` K ) = ( w e. H |-> ( { <. ( 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 ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )

Proof

Step Hyp Ref Expression
1 dvhset.h 
 |-  H = ( LHyp ` K )
2 elex 
 |-  ( K e. V -> K e. _V )
3 fveq2 
 |-  ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
4 3 1 eqtr4di 
 |-  ( k = K -> ( LHyp ` k ) = H )
5 fveq2 
 |-  ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6 5 fveq1d 
 |-  ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7 fveq2 
 |-  ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
8 7 fveq1d 
 |-  ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
9 6 8 xpeq12d 
 |-  ( k = K -> ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) = ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) )
10 9 opeq2d 
 |-  ( k = K -> <. ( Base ` ndx ) , ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) >. = <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. )
11 6 mpteq1d 
 |-  ( k = K -> ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) = ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
12 11 opeq2d 
 |-  ( k = K -> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. = <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. )
13 9 9 12 mpoeq123dv 
 |-  ( k = K -> ( 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 ) ) ) >. ) = ( 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 ) ) ) >. ) )
14 13 opeq2d 
 |-  ( k = K -> <. ( +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 ) ) ) >. ) >. = <. ( +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 ) ) ) >. ) >. )
15 fveq2 
 |-  ( k = K -> ( EDRing ` k ) = ( EDRing ` K ) )
16 15 fveq1d 
 |-  ( k = K -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` w ) )
17 16 opeq2d 
 |-  ( k = K -> <. ( Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. = <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. )
18 10 14 17 tpeq123d 
 |-  ( k = K -> { <. ( 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 ) , ( ( EDRing ` k ) ` w ) >. } = { <. ( 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 ) , ( ( EDRing ` K ) ` w ) >. } )
19 eqidd 
 |-  ( k = K -> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. = <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. )
20 8 9 19 mpoeq123dv 
 |-  ( k = K -> ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) = ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) )
21 20 opeq2d 
 |-  ( k = K -> <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. = <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. )
22 21 sneqd 
 |-  ( k = K -> { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } = { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } )
23 18 22 uneq12d 
 |-  ( k = K -> ( { <. ( 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 ) , ( ( EDRing ` k ) ` w ) >. } 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 ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
24 4 23 mpteq12dv 
 |-  ( k = K -> ( w e. ( LHyp ` k ) |-> ( { <. ( 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 ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) = ( w e. H |-> ( { <. ( 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 ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
25 df-dvech 
 |-  DVecH = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( { <. ( 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 ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
26 24 25 1 mptfvmpt 
 |-  ( K e. _V -> ( DVecH ` K ) = ( w e. H |-> ( { <. ( 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 ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
27 2 26 syl 
 |-  ( K e. V -> ( DVecH ` K ) = ( w e. H |-> ( { <. ( 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 ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )