Step |
Hyp |
Ref |
Expression |
1 |
|
ssid |
|- CC C_ CC |
2 |
|
plyconst |
|- ( ( CC C_ CC /\ A e. CC ) -> ( CC X. { A } ) e. ( Poly ` CC ) ) |
3 |
1 2
|
mpan |
|- ( A e. CC -> ( CC X. { A } ) e. ( Poly ` CC ) ) |
4 |
|
0nn0 |
|- 0 e. NN0 |
5 |
4
|
a1i |
|- ( A e. CC -> 0 e. NN0 ) |
6 |
|
simpl |
|- ( ( A e. CC /\ k e. ( 0 ... 0 ) ) -> A e. CC ) |
7 |
|
fconstmpt |
|- ( CC X. { A } ) = ( z e. CC |-> A ) |
8 |
|
0z |
|- 0 e. ZZ |
9 |
|
exp0 |
|- ( z e. CC -> ( z ^ 0 ) = 1 ) |
10 |
9
|
oveq2d |
|- ( z e. CC -> ( A x. ( z ^ 0 ) ) = ( A x. 1 ) ) |
11 |
|
mulid1 |
|- ( A e. CC -> ( A x. 1 ) = A ) |
12 |
10 11
|
sylan9eqr |
|- ( ( A e. CC /\ z e. CC ) -> ( A x. ( z ^ 0 ) ) = A ) |
13 |
|
simpl |
|- ( ( A e. CC /\ z e. CC ) -> A e. CC ) |
14 |
12 13
|
eqeltrd |
|- ( ( A e. CC /\ z e. CC ) -> ( A x. ( z ^ 0 ) ) e. CC ) |
15 |
|
oveq2 |
|- ( k = 0 -> ( z ^ k ) = ( z ^ 0 ) ) |
16 |
15
|
oveq2d |
|- ( k = 0 -> ( A x. ( z ^ k ) ) = ( A x. ( z ^ 0 ) ) ) |
17 |
16
|
fsum1 |
|- ( ( 0 e. ZZ /\ ( A x. ( z ^ 0 ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) = ( A x. ( z ^ 0 ) ) ) |
18 |
8 14 17
|
sylancr |
|- ( ( A e. CC /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) = ( A x. ( z ^ 0 ) ) ) |
19 |
18 12
|
eqtrd |
|- ( ( A e. CC /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) = A ) |
20 |
19
|
mpteq2dva |
|- ( A e. CC -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) ) = ( z e. CC |-> A ) ) |
21 |
7 20
|
eqtr4id |
|- ( A e. CC -> ( CC X. { A } ) = ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) ) ) |
22 |
3 5 6 21
|
dgrle |
|- ( A e. CC -> ( deg ` ( CC X. { A } ) ) <_ 0 ) |
23 |
|
dgrcl |
|- ( ( CC X. { A } ) e. ( Poly ` CC ) -> ( deg ` ( CC X. { A } ) ) e. NN0 ) |
24 |
|
nn0le0eq0 |
|- ( ( deg ` ( CC X. { A } ) ) e. NN0 -> ( ( deg ` ( CC X. { A } ) ) <_ 0 <-> ( deg ` ( CC X. { A } ) ) = 0 ) ) |
25 |
3 23 24
|
3syl |
|- ( A e. CC -> ( ( deg ` ( CC X. { A } ) ) <_ 0 <-> ( deg ` ( CC X. { A } ) ) = 0 ) ) |
26 |
22 25
|
mpbid |
|- ( A e. CC -> ( deg ` ( CC X. { A } ) ) = 0 ) |