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 ) |