Step |
Hyp |
Ref |
Expression |
1 |
|
plycj.2 |
|- G = ( ( * o. F ) o. * ) |
2 |
|
coecj.3 |
|- A = ( coeff ` F ) |
3 |
|
cjcl |
|- ( x e. CC -> ( * ` x ) e. CC ) |
4 |
3
|
adantl |
|- ( ( F e. ( Poly ` S ) /\ x e. CC ) -> ( * ` x ) e. CC ) |
5 |
|
plyssc |
|- ( Poly ` S ) C_ ( Poly ` CC ) |
6 |
5
|
sseli |
|- ( F e. ( Poly ` S ) -> F e. ( Poly ` CC ) ) |
7 |
1 4 6
|
plycj |
|- ( F e. ( Poly ` S ) -> G e. ( Poly ` CC ) ) |
8 |
|
dgrcl |
|- ( F e. ( Poly ` S ) -> ( deg ` F ) e. NN0 ) |
9 |
|
cjf |
|- * : CC --> CC |
10 |
2
|
coef3 |
|- ( F e. ( Poly ` S ) -> A : NN0 --> CC ) |
11 |
|
fco |
|- ( ( * : CC --> CC /\ A : NN0 --> CC ) -> ( * o. A ) : NN0 --> CC ) |
12 |
9 10 11
|
sylancr |
|- ( F e. ( Poly ` S ) -> ( * o. A ) : NN0 --> CC ) |
13 |
|
fvco3 |
|- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( ( * o. A ) ` k ) = ( * ` ( A ` k ) ) ) |
14 |
10 13
|
sylan |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( ( * o. A ) ` k ) = ( * ` ( A ` k ) ) ) |
15 |
|
cj0 |
|- ( * ` 0 ) = 0 |
16 |
15
|
eqcomi |
|- 0 = ( * ` 0 ) |
17 |
16
|
a1i |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> 0 = ( * ` 0 ) ) |
18 |
14 17
|
eqeq12d |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( ( ( * o. A ) ` k ) = 0 <-> ( * ` ( A ` k ) ) = ( * ` 0 ) ) ) |
19 |
10
|
ffvelcdmda |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( A ` k ) e. CC ) |
20 |
|
0cnd |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> 0 e. CC ) |
21 |
|
cj11 |
|- ( ( ( A ` k ) e. CC /\ 0 e. CC ) -> ( ( * ` ( A ` k ) ) = ( * ` 0 ) <-> ( A ` k ) = 0 ) ) |
22 |
19 20 21
|
syl2anc |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( ( * ` ( A ` k ) ) = ( * ` 0 ) <-> ( A ` k ) = 0 ) ) |
23 |
18 22
|
bitrd |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( ( ( * o. A ) ` k ) = 0 <-> ( A ` k ) = 0 ) ) |
24 |
23
|
necon3bid |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( ( ( * o. A ) ` k ) =/= 0 <-> ( A ` k ) =/= 0 ) ) |
25 |
|
eqid |
|- ( deg ` F ) = ( deg ` F ) |
26 |
2 25
|
dgrub2 |
|- ( F e. ( Poly ` S ) -> ( A " ( ZZ>= ` ( ( deg ` F ) + 1 ) ) ) = { 0 } ) |
27 |
|
plyco0 |
|- ( ( ( deg ` F ) e. NN0 /\ A : NN0 --> CC ) -> ( ( A " ( ZZ>= ` ( ( deg ` F ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) ) |
28 |
8 10 27
|
syl2anc |
|- ( F e. ( Poly ` S ) -> ( ( A " ( ZZ>= ` ( ( deg ` F ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) ) |
29 |
26 28
|
mpbid |
|- ( F e. ( Poly ` S ) -> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) |
30 |
29
|
r19.21bi |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( ( A ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) |
31 |
24 30
|
sylbid |
|- ( ( F e. ( Poly ` S ) /\ k e. NN0 ) -> ( ( ( * o. A ) ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) |
32 |
31
|
ralrimiva |
|- ( F e. ( Poly ` S ) -> A. k e. NN0 ( ( ( * o. A ) ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) |
33 |
|
plyco0 |
|- ( ( ( deg ` F ) e. NN0 /\ ( * o. A ) : NN0 --> CC ) -> ( ( ( * o. A ) " ( ZZ>= ` ( ( deg ` F ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( ( * o. A ) ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) ) |
34 |
8 12 33
|
syl2anc |
|- ( F e. ( Poly ` S ) -> ( ( ( * o. A ) " ( ZZ>= ` ( ( deg ` F ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( ( * o. A ) ` k ) =/= 0 -> k <_ ( deg ` F ) ) ) ) |
35 |
32 34
|
mpbird |
|- ( F e. ( Poly ` S ) -> ( ( * o. A ) " ( ZZ>= ` ( ( deg ` F ) + 1 ) ) ) = { 0 } ) |
36 |
25 1 2
|
plycjlem |
|- ( F e. ( Poly ` S ) -> G = ( y e. CC |-> sum_ z e. ( 0 ... ( deg ` F ) ) ( ( ( * o. A ) ` z ) x. ( y ^ z ) ) ) ) |
37 |
7 8 12 35 36
|
coeeq |
|- ( F e. ( Poly ` S ) -> ( coeff ` G ) = ( * o. A ) ) |