Step |
Hyp |
Ref |
Expression |
1 |
|
abelth2.1 |
|- ( ph -> A : NN0 --> CC ) |
2 |
|
abelth2.2 |
|- ( ph -> seq 0 ( + , A ) e. dom ~~> ) |
3 |
|
abelth2.3 |
|- F = ( x e. ( 0 [,] 1 ) |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) |
4 |
|
unitssre |
|- ( 0 [,] 1 ) C_ RR |
5 |
|
ax-resscn |
|- RR C_ CC |
6 |
4 5
|
sstri |
|- ( 0 [,] 1 ) C_ CC |
7 |
6
|
a1i |
|- ( ph -> ( 0 [,] 1 ) C_ CC ) |
8 |
|
1re |
|- 1 e. RR |
9 |
|
simpr |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> z e. ( 0 [,] 1 ) ) |
10 |
|
elicc01 |
|- ( z e. ( 0 [,] 1 ) <-> ( z e. RR /\ 0 <_ z /\ z <_ 1 ) ) |
11 |
9 10
|
sylib |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( z e. RR /\ 0 <_ z /\ z <_ 1 ) ) |
12 |
11
|
simp1d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> z e. RR ) |
13 |
|
resubcl |
|- ( ( 1 e. RR /\ z e. RR ) -> ( 1 - z ) e. RR ) |
14 |
8 12 13
|
sylancr |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( 1 - z ) e. RR ) |
15 |
14
|
leidd |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( 1 - z ) <_ ( 1 - z ) ) |
16 |
|
1red |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> 1 e. RR ) |
17 |
11
|
simp3d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> z <_ 1 ) |
18 |
12 16 17
|
abssubge0d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( abs ` ( 1 - z ) ) = ( 1 - z ) ) |
19 |
11
|
simp2d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> 0 <_ z ) |
20 |
12 19
|
absidd |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( abs ` z ) = z ) |
21 |
20
|
oveq2d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( 1 - ( abs ` z ) ) = ( 1 - z ) ) |
22 |
21
|
oveq2d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( 1 x. ( 1 - ( abs ` z ) ) ) = ( 1 x. ( 1 - z ) ) ) |
23 |
14
|
recnd |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( 1 - z ) e. CC ) |
24 |
23
|
mulid2d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( 1 x. ( 1 - z ) ) = ( 1 - z ) ) |
25 |
22 24
|
eqtrd |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( 1 x. ( 1 - ( abs ` z ) ) ) = ( 1 - z ) ) |
26 |
15 18 25
|
3brtr4d |
|- ( ( ph /\ z e. ( 0 [,] 1 ) ) -> ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) ) |
27 |
7 26
|
ssrabdv |
|- ( ph -> ( 0 [,] 1 ) C_ { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } ) |
28 |
27
|
resmptd |
|- ( ph -> ( ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) |` ( 0 [,] 1 ) ) = ( x e. ( 0 [,] 1 ) |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) ) |
29 |
28 3
|
eqtr4di |
|- ( ph -> ( ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) |` ( 0 [,] 1 ) ) = F ) |
30 |
|
1red |
|- ( ph -> 1 e. RR ) |
31 |
|
0le1 |
|- 0 <_ 1 |
32 |
31
|
a1i |
|- ( ph -> 0 <_ 1 ) |
33 |
|
eqid |
|- { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } = { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |
34 |
|
eqid |
|- ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) = ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) |
35 |
1 2 30 32 33 34
|
abelth |
|- ( ph -> ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) e. ( { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } -cn-> CC ) ) |
36 |
|
rescncf |
|- ( ( 0 [,] 1 ) C_ { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } -> ( ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) e. ( { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } -cn-> CC ) -> ( ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) |` ( 0 [,] 1 ) ) e. ( ( 0 [,] 1 ) -cn-> CC ) ) ) |
37 |
27 35 36
|
sylc |
|- ( ph -> ( ( x e. { z e. CC | ( abs ` ( 1 - z ) ) <_ ( 1 x. ( 1 - ( abs ` z ) ) ) } |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) |` ( 0 [,] 1 ) ) e. ( ( 0 [,] 1 ) -cn-> CC ) ) |
38 |
29 37
|
eqeltrrd |
|- ( ph -> F e. ( ( 0 [,] 1 ) -cn-> CC ) ) |