Step |
Hyp |
Ref |
Expression |
1 |
|
dvaset.h |
|- H = ( LHyp ` K ) |
2 |
|
dvaset.t |
|- T = ( ( LTrn ` K ) ` W ) |
3 |
|
dvaset.e |
|- E = ( ( TEndo ` K ) ` W ) |
4 |
|
dvaset.d |
|- D = ( ( EDRing ` K ) ` W ) |
5 |
|
dvaset.u |
|- U = ( ( DVecA ` K ) ` W ) |
6 |
1
|
dvafset |
|- ( K e. X -> ( DVecA ` K ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) ) |
7 |
6
|
fveq1d |
|- ( K e. X -> ( ( DVecA ` K ) ` W ) = ( ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) ` W ) ) |
8 |
|
fveq2 |
|- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) ) |
9 |
8 2
|
eqtr4di |
|- ( w = W -> ( ( LTrn ` K ) ` w ) = T ) |
10 |
9
|
opeq2d |
|- ( w = W -> <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. = <. ( Base ` ndx ) , T >. ) |
11 |
|
eqidd |
|- ( w = W -> ( f o. g ) = ( f o. g ) ) |
12 |
9 9 11
|
mpoeq123dv |
|- ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) = ( f e. T , g e. T |-> ( f o. g ) ) ) |
13 |
12
|
opeq2d |
|- ( w = W -> <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. = <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. ) |
14 |
|
fveq2 |
|- ( w = W -> ( ( EDRing ` K ) ` w ) = ( ( EDRing ` K ) ` W ) ) |
15 |
14 4
|
eqtr4di |
|- ( w = W -> ( ( EDRing ` K ) ` w ) = D ) |
16 |
15
|
opeq2d |
|- ( w = W -> <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. = <. ( Scalar ` ndx ) , D >. ) |
17 |
10 13 16
|
tpeq123d |
|- ( w = W -> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } ) |
18 |
|
fveq2 |
|- ( w = W -> ( ( TEndo ` K ) ` w ) = ( ( TEndo ` K ) ` W ) ) |
19 |
18 3
|
eqtr4di |
|- ( w = W -> ( ( TEndo ` K ) ` w ) = E ) |
20 |
|
eqidd |
|- ( w = W -> ( s ` f ) = ( s ` f ) ) |
21 |
19 9 20
|
mpoeq123dv |
|- ( w = W -> ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) = ( s e. E , f e. T |-> ( s ` f ) ) ) |
22 |
21
|
opeq2d |
|- ( w = W -> <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. = <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. ) |
23 |
22
|
sneqd |
|- ( w = W -> { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } = { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) |
24 |
17 23
|
uneq12d |
|- ( w = W -> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) ) |
25 |
|
eqid |
|- ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) |
26 |
|
tpex |
|- { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } e. _V |
27 |
|
snex |
|- { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } e. _V |
28 |
26 27
|
unex |
|- ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) e. _V |
29 |
24 25 28
|
fvmpt |
|- ( W e. H -> ( ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) ` W ) = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) ) |
30 |
7 29
|
sylan9eq |
|- ( ( K e. X /\ W e. H ) -> ( ( DVecA ` K ) ` W ) = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) ) |
31 |
5 30
|
syl5eq |
|- ( ( K e. X /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) ) |