Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( coeff ` F ) = ( coeff ` F ) |
2 |
|
eqid |
|- ( deg ` F ) = ( deg ` F ) |
3 |
1 2
|
coeid |
|- ( F e. ( Poly ` S ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) |
4 |
3
|
adantr |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) |
5 |
|
simplr |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( deg ` F ) = 0 ) |
6 |
5
|
oveq2d |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( 0 ... ( deg ` F ) ) = ( 0 ... 0 ) ) |
7 |
6
|
sumeq1d |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... 0 ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) |
8 |
|
0z |
|- 0 e. ZZ |
9 |
|
exp0 |
|- ( z e. CC -> ( z ^ 0 ) = 1 ) |
10 |
9
|
adantl |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( z ^ 0 ) = 1 ) |
11 |
10
|
oveq2d |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( ( ( coeff ` F ) ` 0 ) x. ( z ^ 0 ) ) = ( ( ( coeff ` F ) ` 0 ) x. 1 ) ) |
12 |
1
|
coef3 |
|- ( F e. ( Poly ` S ) -> ( coeff ` F ) : NN0 --> CC ) |
13 |
|
0nn0 |
|- 0 e. NN0 |
14 |
|
ffvelrn |
|- ( ( ( coeff ` F ) : NN0 --> CC /\ 0 e. NN0 ) -> ( ( coeff ` F ) ` 0 ) e. CC ) |
15 |
12 13 14
|
sylancl |
|- ( F e. ( Poly ` S ) -> ( ( coeff ` F ) ` 0 ) e. CC ) |
16 |
15
|
ad2antrr |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( ( coeff ` F ) ` 0 ) e. CC ) |
17 |
16
|
mulid1d |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( ( ( coeff ` F ) ` 0 ) x. 1 ) = ( ( coeff ` F ) ` 0 ) ) |
18 |
11 17
|
eqtrd |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( ( ( coeff ` F ) ` 0 ) x. ( z ^ 0 ) ) = ( ( coeff ` F ) ` 0 ) ) |
19 |
18 16
|
eqeltrd |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> ( ( ( coeff ` F ) ` 0 ) x. ( z ^ 0 ) ) e. CC ) |
20 |
|
fveq2 |
|- ( k = 0 -> ( ( coeff ` F ) ` k ) = ( ( coeff ` F ) ` 0 ) ) |
21 |
|
oveq2 |
|- ( k = 0 -> ( z ^ k ) = ( z ^ 0 ) ) |
22 |
20 21
|
oveq12d |
|- ( k = 0 -> ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) = ( ( ( coeff ` F ) ` 0 ) x. ( z ^ 0 ) ) ) |
23 |
22
|
fsum1 |
|- ( ( 0 e. ZZ /\ ( ( ( coeff ` F ) ` 0 ) x. ( z ^ 0 ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) = ( ( ( coeff ` F ) ` 0 ) x. ( z ^ 0 ) ) ) |
24 |
8 19 23
|
sylancr |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) = ( ( ( coeff ` F ) ` 0 ) x. ( z ^ 0 ) ) ) |
25 |
24 18
|
eqtrd |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) = ( ( coeff ` F ) ` 0 ) ) |
26 |
7 25
|
eqtrd |
|- ( ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) = ( ( coeff ` F ) ` 0 ) ) |
27 |
26
|
mpteq2dva |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> ( z e. CC |-> sum_ k e. ( 0 ... ( deg ` F ) ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> ( ( coeff ` F ) ` 0 ) ) ) |
28 |
4 27
|
eqtrd |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> F = ( z e. CC |-> ( ( coeff ` F ) ` 0 ) ) ) |
29 |
|
fconstmpt |
|- ( CC X. { ( ( coeff ` F ) ` 0 ) } ) = ( z e. CC |-> ( ( coeff ` F ) ` 0 ) ) |
30 |
28 29
|
eqtr4di |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> F = ( CC X. { ( ( coeff ` F ) ` 0 ) } ) ) |
31 |
30
|
fveq1d |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> ( F ` 0 ) = ( ( CC X. { ( ( coeff ` F ) ` 0 ) } ) ` 0 ) ) |
32 |
|
0cn |
|- 0 e. CC |
33 |
|
fvex |
|- ( ( coeff ` F ) ` 0 ) e. _V |
34 |
33
|
fvconst2 |
|- ( 0 e. CC -> ( ( CC X. { ( ( coeff ` F ) ` 0 ) } ) ` 0 ) = ( ( coeff ` F ) ` 0 ) ) |
35 |
32 34
|
ax-mp |
|- ( ( CC X. { ( ( coeff ` F ) ` 0 ) } ) ` 0 ) = ( ( coeff ` F ) ` 0 ) |
36 |
31 35
|
eqtrdi |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> ( F ` 0 ) = ( ( coeff ` F ) ` 0 ) ) |
37 |
36
|
sneqd |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> { ( F ` 0 ) } = { ( ( coeff ` F ) ` 0 ) } ) |
38 |
37
|
xpeq2d |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> ( CC X. { ( F ` 0 ) } ) = ( CC X. { ( ( coeff ` F ) ` 0 ) } ) ) |
39 |
30 38
|
eqtr4d |
|- ( ( F e. ( Poly ` S ) /\ ( deg ` F ) = 0 ) -> F = ( CC X. { ( F ` 0 ) } ) ) |
40 |
39
|
ex |
|- ( F e. ( Poly ` S ) -> ( ( deg ` F ) = 0 -> F = ( CC X. { ( F ` 0 ) } ) ) ) |
41 |
|
plyf |
|- ( F e. ( Poly ` S ) -> F : CC --> CC ) |
42 |
|
ffvelrn |
|- ( ( F : CC --> CC /\ 0 e. CC ) -> ( F ` 0 ) e. CC ) |
43 |
41 32 42
|
sylancl |
|- ( F e. ( Poly ` S ) -> ( F ` 0 ) e. CC ) |
44 |
|
0dgr |
|- ( ( F ` 0 ) e. CC -> ( deg ` ( CC X. { ( F ` 0 ) } ) ) = 0 ) |
45 |
43 44
|
syl |
|- ( F e. ( Poly ` S ) -> ( deg ` ( CC X. { ( F ` 0 ) } ) ) = 0 ) |
46 |
|
fveqeq2 |
|- ( F = ( CC X. { ( F ` 0 ) } ) -> ( ( deg ` F ) = 0 <-> ( deg ` ( CC X. { ( F ` 0 ) } ) ) = 0 ) ) |
47 |
45 46
|
syl5ibrcom |
|- ( F e. ( Poly ` S ) -> ( F = ( CC X. { ( F ` 0 ) } ) -> ( deg ` F ) = 0 ) ) |
48 |
40 47
|
impbid |
|- ( F e. ( Poly ` S ) -> ( ( deg ` F ) = 0 <-> F = ( CC X. { ( F ` 0 ) } ) ) ) |