| Step |
Hyp |
Ref |
Expression |
| 1 |
|
plycj.2 |
|- G = ( ( * o. F ) o. * ) |
| 2 |
|
plycj.3 |
|- ( ( ph /\ x e. S ) -> ( * ` x ) e. S ) |
| 3 |
|
plycj.4 |
|- ( ph -> F e. ( Poly ` S ) ) |
| 4 |
|
eqid |
|- ( deg ` F ) = ( deg ` F ) |
| 5 |
|
eqid |
|- ( coeff ` F ) = ( coeff ` F ) |
| 6 |
4 1 5
|
plycjlem |
|- ( F e. ( Poly ` S ) -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( * o. ( coeff ` F ) ) ` k ) x. ( z ^ k ) ) ) ) |
| 7 |
3 6
|
syl |
|- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( * o. ( coeff ` F ) ) ` k ) x. ( z ^ k ) ) ) ) |
| 8 |
|
plybss |
|- ( F e. ( Poly ` S ) -> S C_ CC ) |
| 9 |
3 8
|
syl |
|- ( ph -> S C_ CC ) |
| 10 |
|
0cnd |
|- ( ph -> 0 e. CC ) |
| 11 |
10
|
snssd |
|- ( ph -> { 0 } C_ CC ) |
| 12 |
9 11
|
unssd |
|- ( ph -> ( S u. { 0 } ) C_ CC ) |
| 13 |
|
dgrcl |
|- ( F e. ( Poly ` S ) -> ( deg ` F ) e. NN0 ) |
| 14 |
3 13
|
syl |
|- ( ph -> ( deg ` F ) e. NN0 ) |
| 15 |
5
|
coef |
|- ( F e. ( Poly ` S ) -> ( coeff ` F ) : NN0 --> ( S u. { 0 } ) ) |
| 16 |
3 15
|
syl |
|- ( ph -> ( coeff ` F ) : NN0 --> ( S u. { 0 } ) ) |
| 17 |
|
elfznn0 |
|- ( k e. ( 0 ... ( deg ` F ) ) -> k e. NN0 ) |
| 18 |
|
fvco3 |
|- ( ( ( coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ k e. NN0 ) -> ( ( * o. ( coeff ` F ) ) ` k ) = ( * ` ( ( coeff ` F ) ` k ) ) ) |
| 19 |
16 17 18
|
syl2an |
|- ( ( ph /\ k e. ( 0 ... ( deg ` F ) ) ) -> ( ( * o. ( coeff ` F ) ) ` k ) = ( * ` ( ( coeff ` F ) ` k ) ) ) |
| 20 |
|
ffvelcdm |
|- ( ( ( coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ k e. NN0 ) -> ( ( coeff ` F ) ` k ) e. ( S u. { 0 } ) ) |
| 21 |
16 17 20
|
syl2an |
|- ( ( ph /\ k e. ( 0 ... ( deg ` F ) ) ) -> ( ( coeff ` F ) ` k ) e. ( S u. { 0 } ) ) |
| 22 |
2
|
ralrimiva |
|- ( ph -> A. x e. S ( * ` x ) e. S ) |
| 23 |
|
fveq2 |
|- ( x = ( ( coeff ` F ) ` k ) -> ( * ` x ) = ( * ` ( ( coeff ` F ) ` k ) ) ) |
| 24 |
23
|
eleq1d |
|- ( x = ( ( coeff ` F ) ` k ) -> ( ( * ` x ) e. S <-> ( * ` ( ( coeff ` F ) ` k ) ) e. S ) ) |
| 25 |
24
|
rspccv |
|- ( A. x e. S ( * ` x ) e. S -> ( ( ( coeff ` F ) ` k ) e. S -> ( * ` ( ( coeff ` F ) ` k ) ) e. S ) ) |
| 26 |
22 25
|
syl |
|- ( ph -> ( ( ( coeff ` F ) ` k ) e. S -> ( * ` ( ( coeff ` F ) ` k ) ) e. S ) ) |
| 27 |
|
elsni |
|- ( ( ( coeff ` F ) ` k ) e. { 0 } -> ( ( coeff ` F ) ` k ) = 0 ) |
| 28 |
27
|
fveq2d |
|- ( ( ( coeff ` F ) ` k ) e. { 0 } -> ( * ` ( ( coeff ` F ) ` k ) ) = ( * ` 0 ) ) |
| 29 |
|
cj0 |
|- ( * ` 0 ) = 0 |
| 30 |
28 29
|
eqtrdi |
|- ( ( ( coeff ` F ) ` k ) e. { 0 } -> ( * ` ( ( coeff ` F ) ` k ) ) = 0 ) |
| 31 |
|
fvex |
|- ( * ` ( ( coeff ` F ) ` k ) ) e. _V |
| 32 |
31
|
elsn |
|- ( ( * ` ( ( coeff ` F ) ` k ) ) e. { 0 } <-> ( * ` ( ( coeff ` F ) ` k ) ) = 0 ) |
| 33 |
30 32
|
sylibr |
|- ( ( ( coeff ` F ) ` k ) e. { 0 } -> ( * ` ( ( coeff ` F ) ` k ) ) e. { 0 } ) |
| 34 |
33
|
a1i |
|- ( ph -> ( ( ( coeff ` F ) ` k ) e. { 0 } -> ( * ` ( ( coeff ` F ) ` k ) ) e. { 0 } ) ) |
| 35 |
26 34
|
orim12d |
|- ( ph -> ( ( ( ( coeff ` F ) ` k ) e. S \/ ( ( coeff ` F ) ` k ) e. { 0 } ) -> ( ( * ` ( ( coeff ` F ) ` k ) ) e. S \/ ( * ` ( ( coeff ` F ) ` k ) ) e. { 0 } ) ) ) |
| 36 |
|
elun |
|- ( ( ( coeff ` F ) ` k ) e. ( S u. { 0 } ) <-> ( ( ( coeff ` F ) ` k ) e. S \/ ( ( coeff ` F ) ` k ) e. { 0 } ) ) |
| 37 |
|
elun |
|- ( ( * ` ( ( coeff ` F ) ` k ) ) e. ( S u. { 0 } ) <-> ( ( * ` ( ( coeff ` F ) ` k ) ) e. S \/ ( * ` ( ( coeff ` F ) ` k ) ) e. { 0 } ) ) |
| 38 |
35 36 37
|
3imtr4g |
|- ( ph -> ( ( ( coeff ` F ) ` k ) e. ( S u. { 0 } ) -> ( * ` ( ( coeff ` F ) ` k ) ) e. ( S u. { 0 } ) ) ) |
| 39 |
38
|
adantr |
|- ( ( ph /\ k e. ( 0 ... ( deg ` F ) ) ) -> ( ( ( coeff ` F ) ` k ) e. ( S u. { 0 } ) -> ( * ` ( ( coeff ` F ) ` k ) ) e. ( S u. { 0 } ) ) ) |
| 40 |
21 39
|
mpd |
|- ( ( ph /\ k e. ( 0 ... ( deg ` F ) ) ) -> ( * ` ( ( coeff ` F ) ` k ) ) e. ( S u. { 0 } ) ) |
| 41 |
19 40
|
eqeltrd |
|- ( ( ph /\ k e. ( 0 ... ( deg ` F ) ) ) -> ( ( * o. ( coeff ` F ) ) ` k ) e. ( S u. { 0 } ) ) |
| 42 |
12 14 41
|
elplyd |
|- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( * o. ( coeff ` F ) ) ` k ) x. ( z ^ k ) ) ) e. ( Poly ` ( S u. { 0 } ) ) ) |
| 43 |
7 42
|
eqeltrd |
|- ( ph -> G e. ( Poly ` ( S u. { 0 } ) ) ) |
| 44 |
|
plyun0 |
|- ( Poly ` ( S u. { 0 } ) ) = ( Poly ` S ) |
| 45 |
43 44
|
eleqtrdi |
|- ( ph -> G e. ( Poly ` S ) ) |