Step |
Hyp |
Ref |
Expression |
1 |
|
pser.g |
|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) |
2 |
|
radcnv.a |
|- ( ph -> A : NN0 --> CC ) |
3 |
|
fveq2 |
|- ( r = 0 -> ( G ` r ) = ( G ` 0 ) ) |
4 |
3
|
seqeq3d |
|- ( r = 0 -> seq 0 ( + , ( G ` r ) ) = seq 0 ( + , ( G ` 0 ) ) ) |
5 |
4
|
eleq1d |
|- ( r = 0 -> ( seq 0 ( + , ( G ` r ) ) e. dom ~~> <-> seq 0 ( + , ( G ` 0 ) ) e. dom ~~> ) ) |
6 |
|
0red |
|- ( ph -> 0 e. RR ) |
7 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
8 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
9 |
|
snfi |
|- { 0 } e. Fin |
10 |
9
|
a1i |
|- ( ph -> { 0 } e. Fin ) |
11 |
|
0nn0 |
|- 0 e. NN0 |
12 |
11
|
a1i |
|- ( ph -> 0 e. NN0 ) |
13 |
12
|
snssd |
|- ( ph -> { 0 } C_ NN0 ) |
14 |
|
ifid |
|- if ( k e. { 0 } , ( ( G ` 0 ) ` k ) , ( ( G ` 0 ) ` k ) ) = ( ( G ` 0 ) ` k ) |
15 |
|
0cnd |
|- ( ph -> 0 e. CC ) |
16 |
1
|
pserval2 |
|- ( ( 0 e. CC /\ k e. NN0 ) -> ( ( G ` 0 ) ` k ) = ( ( A ` k ) x. ( 0 ^ k ) ) ) |
17 |
15 16
|
sylan |
|- ( ( ph /\ k e. NN0 ) -> ( ( G ` 0 ) ` k ) = ( ( A ` k ) x. ( 0 ^ k ) ) ) |
18 |
17
|
adantr |
|- ( ( ( ph /\ k e. NN0 ) /\ -. k e. { 0 } ) -> ( ( G ` 0 ) ` k ) = ( ( A ` k ) x. ( 0 ^ k ) ) ) |
19 |
|
simpr |
|- ( ( ph /\ k e. NN0 ) -> k e. NN0 ) |
20 |
|
elnn0 |
|- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) ) |
21 |
19 20
|
sylib |
|- ( ( ph /\ k e. NN0 ) -> ( k e. NN \/ k = 0 ) ) |
22 |
21
|
ord |
|- ( ( ph /\ k e. NN0 ) -> ( -. k e. NN -> k = 0 ) ) |
23 |
|
velsn |
|- ( k e. { 0 } <-> k = 0 ) |
24 |
22 23
|
syl6ibr |
|- ( ( ph /\ k e. NN0 ) -> ( -. k e. NN -> k e. { 0 } ) ) |
25 |
24
|
con1d |
|- ( ( ph /\ k e. NN0 ) -> ( -. k e. { 0 } -> k e. NN ) ) |
26 |
25
|
imp |
|- ( ( ( ph /\ k e. NN0 ) /\ -. k e. { 0 } ) -> k e. NN ) |
27 |
26
|
0expd |
|- ( ( ( ph /\ k e. NN0 ) /\ -. k e. { 0 } ) -> ( 0 ^ k ) = 0 ) |
28 |
27
|
oveq2d |
|- ( ( ( ph /\ k e. NN0 ) /\ -. k e. { 0 } ) -> ( ( A ` k ) x. ( 0 ^ k ) ) = ( ( A ` k ) x. 0 ) ) |
29 |
2
|
ffvelrnda |
|- ( ( ph /\ k e. NN0 ) -> ( A ` k ) e. CC ) |
30 |
29
|
adantr |
|- ( ( ( ph /\ k e. NN0 ) /\ -. k e. { 0 } ) -> ( A ` k ) e. CC ) |
31 |
30
|
mul01d |
|- ( ( ( ph /\ k e. NN0 ) /\ -. k e. { 0 } ) -> ( ( A ` k ) x. 0 ) = 0 ) |
32 |
18 28 31
|
3eqtrd |
|- ( ( ( ph /\ k e. NN0 ) /\ -. k e. { 0 } ) -> ( ( G ` 0 ) ` k ) = 0 ) |
33 |
32
|
ifeq2da |
|- ( ( ph /\ k e. NN0 ) -> if ( k e. { 0 } , ( ( G ` 0 ) ` k ) , ( ( G ` 0 ) ` k ) ) = if ( k e. { 0 } , ( ( G ` 0 ) ` k ) , 0 ) ) |
34 |
14 33
|
eqtr3id |
|- ( ( ph /\ k e. NN0 ) -> ( ( G ` 0 ) ` k ) = if ( k e. { 0 } , ( ( G ` 0 ) ` k ) , 0 ) ) |
35 |
13
|
sselda |
|- ( ( ph /\ k e. { 0 } ) -> k e. NN0 ) |
36 |
1 2 15
|
psergf |
|- ( ph -> ( G ` 0 ) : NN0 --> CC ) |
37 |
36
|
ffvelrnda |
|- ( ( ph /\ k e. NN0 ) -> ( ( G ` 0 ) ` k ) e. CC ) |
38 |
35 37
|
syldan |
|- ( ( ph /\ k e. { 0 } ) -> ( ( G ` 0 ) ` k ) e. CC ) |
39 |
7 8 10 13 34 38
|
fsumcvg3 |
|- ( ph -> seq 0 ( + , ( G ` 0 ) ) e. dom ~~> ) |
40 |
5 6 39
|
elrabd |
|- ( ph -> 0 e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |