Step |
Hyp |
Ref |
Expression |
1 |
|
abelth.1 |
|- ( ph -> A : NN0 --> CC ) |
2 |
|
abelth.2 |
|- ( ph -> seq 0 ( + , A ) e. dom ~~> ) |
3 |
|
abelth.3 |
|- ( ph -> M e. RR ) |
4 |
|
abelth.4 |
|- ( ph -> 0 <_ M ) |
5 |
|
abelth.5 |
|- S = { z e. CC | ( abs ` ( 1 - z ) ) <_ ( M x. ( 1 - ( abs ` z ) ) ) } |
6 |
|
abelth.6 |
|- F = ( x e. S |-> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) ) |
7 |
|
0nn0 |
|- 0 e. NN0 |
8 |
7
|
a1i |
|- ( k e. NN0 -> 0 e. NN0 ) |
9 |
|
ffvelrn |
|- ( ( A : NN0 --> CC /\ 0 e. NN0 ) -> ( A ` 0 ) e. CC ) |
10 |
1 8 9
|
syl2an |
|- ( ( ph /\ k e. NN0 ) -> ( A ` 0 ) e. CC ) |
11 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
12 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
13 |
|
eqidd |
|- ( ( ph /\ m e. NN0 ) -> ( A ` m ) = ( A ` m ) ) |
14 |
1
|
ffvelrnda |
|- ( ( ph /\ m e. NN0 ) -> ( A ` m ) e. CC ) |
15 |
11 12 13 14 2
|
isumcl |
|- ( ph -> sum_ m e. NN0 ( A ` m ) e. CC ) |
16 |
15
|
adantr |
|- ( ( ph /\ k e. NN0 ) -> sum_ m e. NN0 ( A ` m ) e. CC ) |
17 |
10 16
|
subcld |
|- ( ( ph /\ k e. NN0 ) -> ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) e. CC ) |
18 |
1
|
ffvelrnda |
|- ( ( ph /\ k e. NN0 ) -> ( A ` k ) e. CC ) |
19 |
17 18
|
ifcld |
|- ( ( ph /\ k e. NN0 ) -> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) e. CC ) |
20 |
19
|
fmpttd |
|- ( ph -> ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) : NN0 --> CC ) |
21 |
7
|
a1i |
|- ( ph -> 0 e. NN0 ) |
22 |
20
|
ffvelrnda |
|- ( ( ph /\ i e. NN0 ) -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) e. CC ) |
23 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
24 |
|
1z |
|- 1 e. ZZ |
25 |
23 24
|
eqeltrri |
|- ( 0 + 1 ) e. ZZ |
26 |
25
|
a1i |
|- ( ph -> ( 0 + 1 ) e. ZZ ) |
27 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
28 |
23
|
fveq2i |
|- ( ZZ>= ` 1 ) = ( ZZ>= ` ( 0 + 1 ) ) |
29 |
27 28
|
eqtri |
|- NN = ( ZZ>= ` ( 0 + 1 ) ) |
30 |
29
|
eleq2i |
|- ( i e. NN <-> i e. ( ZZ>= ` ( 0 + 1 ) ) ) |
31 |
|
nnnn0 |
|- ( i e. NN -> i e. NN0 ) |
32 |
31
|
adantl |
|- ( ( ph /\ i e. NN ) -> i e. NN0 ) |
33 |
|
eqeq1 |
|- ( k = i -> ( k = 0 <-> i = 0 ) ) |
34 |
|
fveq2 |
|- ( k = i -> ( A ` k ) = ( A ` i ) ) |
35 |
33 34
|
ifbieq2d |
|- ( k = i -> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) = if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) ) |
36 |
|
eqid |
|- ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) = ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) |
37 |
|
ovex |
|- ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) e. _V |
38 |
|
fvex |
|- ( A ` i ) e. _V |
39 |
37 38
|
ifex |
|- if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) e. _V |
40 |
35 36 39
|
fvmpt |
|- ( i e. NN0 -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) = if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) ) |
41 |
32 40
|
syl |
|- ( ( ph /\ i e. NN ) -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) = if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) ) |
42 |
|
nnne0 |
|- ( i e. NN -> i =/= 0 ) |
43 |
42
|
adantl |
|- ( ( ph /\ i e. NN ) -> i =/= 0 ) |
44 |
43
|
neneqd |
|- ( ( ph /\ i e. NN ) -> -. i = 0 ) |
45 |
44
|
iffalsed |
|- ( ( ph /\ i e. NN ) -> if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) = ( A ` i ) ) |
46 |
41 45
|
eqtrd |
|- ( ( ph /\ i e. NN ) -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) = ( A ` i ) ) |
47 |
30 46
|
sylan2br |
|- ( ( ph /\ i e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) = ( A ` i ) ) |
48 |
26 47
|
seqfeq |
|- ( ph -> seq ( 0 + 1 ) ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) = seq ( 0 + 1 ) ( + , A ) ) |
49 |
11 12 13 14 2
|
isumclim2 |
|- ( ph -> seq 0 ( + , A ) ~~> sum_ m e. NN0 ( A ` m ) ) |
50 |
11 21 18 49
|
clim2ser |
|- ( ph -> seq ( 0 + 1 ) ( + , A ) ~~> ( sum_ m e. NN0 ( A ` m ) - ( seq 0 ( + , A ) ` 0 ) ) ) |
51 |
|
0z |
|- 0 e. ZZ |
52 |
|
seq1 |
|- ( 0 e. ZZ -> ( seq 0 ( + , A ) ` 0 ) = ( A ` 0 ) ) |
53 |
51 52
|
ax-mp |
|- ( seq 0 ( + , A ) ` 0 ) = ( A ` 0 ) |
54 |
53
|
oveq2i |
|- ( sum_ m e. NN0 ( A ` m ) - ( seq 0 ( + , A ) ` 0 ) ) = ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) |
55 |
50 54
|
breqtrdi |
|- ( ph -> seq ( 0 + 1 ) ( + , A ) ~~> ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) ) |
56 |
48 55
|
eqbrtrd |
|- ( ph -> seq ( 0 + 1 ) ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ~~> ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) ) |
57 |
11 21 22 56
|
clim2ser2 |
|- ( ph -> seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ~~> ( ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) + ( seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ` 0 ) ) ) |
58 |
|
seq1 |
|- ( 0 e. ZZ -> ( seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ` 0 ) = ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` 0 ) ) |
59 |
51 58
|
ax-mp |
|- ( seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ` 0 ) = ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` 0 ) |
60 |
|
iftrue |
|- ( k = 0 -> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) = ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) |
61 |
60 36 37
|
fvmpt |
|- ( 0 e. NN0 -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` 0 ) = ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) |
62 |
7 61
|
ax-mp |
|- ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` 0 ) = ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) |
63 |
59 62
|
eqtri |
|- ( seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ` 0 ) = ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) |
64 |
63
|
oveq2i |
|- ( ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) + ( seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ` 0 ) ) = ( ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) + ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) |
65 |
1 7 9
|
sylancl |
|- ( ph -> ( A ` 0 ) e. CC ) |
66 |
|
npncan2 |
|- ( ( sum_ m e. NN0 ( A ` m ) e. CC /\ ( A ` 0 ) e. CC ) -> ( ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) + ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) = 0 ) |
67 |
15 65 66
|
syl2anc |
|- ( ph -> ( ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) + ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) = 0 ) |
68 |
64 67
|
syl5eq |
|- ( ph -> ( ( sum_ m e. NN0 ( A ` m ) - ( A ` 0 ) ) + ( seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ` 0 ) ) = 0 ) |
69 |
57 68
|
breqtrd |
|- ( ph -> seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ~~> 0 ) |
70 |
|
seqex |
|- seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) e. _V |
71 |
|
c0ex |
|- 0 e. _V |
72 |
70 71
|
breldm |
|- ( seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ~~> 0 -> seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) e. dom ~~> ) |
73 |
69 72
|
syl |
|- ( ph -> seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) e. dom ~~> ) |
74 |
|
eqid |
|- ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) = ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) |
75 |
20 73 3 4 5 74 69
|
abelthlem8 |
|- ( ( ph /\ R e. RR+ ) -> E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) ) < R ) ) |
76 |
1 2 3 4 5
|
abelthlem2 |
|- ( ph -> ( 1 e. S /\ ( S \ { 1 } ) C_ ( 0 ( ball ` ( abs o. - ) ) 1 ) ) ) |
77 |
76
|
simpld |
|- ( ph -> 1 e. S ) |
78 |
77
|
adantr |
|- ( ( ph /\ y e. S ) -> 1 e. S ) |
79 |
40
|
adantl |
|- ( ( x = 1 /\ i e. NN0 ) -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) = if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) ) |
80 |
|
oveq1 |
|- ( x = 1 -> ( x ^ i ) = ( 1 ^ i ) ) |
81 |
|
nn0z |
|- ( i e. NN0 -> i e. ZZ ) |
82 |
|
1exp |
|- ( i e. ZZ -> ( 1 ^ i ) = 1 ) |
83 |
81 82
|
syl |
|- ( i e. NN0 -> ( 1 ^ i ) = 1 ) |
84 |
80 83
|
sylan9eq |
|- ( ( x = 1 /\ i e. NN0 ) -> ( x ^ i ) = 1 ) |
85 |
79 84
|
oveq12d |
|- ( ( x = 1 /\ i e. NN0 ) -> ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) = ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) ) |
86 |
85
|
sumeq2dv |
|- ( x = 1 -> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) = sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) ) |
87 |
|
sumex |
|- sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) e. _V |
88 |
86 74 87
|
fvmpt |
|- ( 1 e. S -> ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) = sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) ) |
89 |
78 88
|
syl |
|- ( ( ph /\ y e. S ) -> ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) = sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) ) |
90 |
|
0zd |
|- ( ( ph /\ y e. S ) -> 0 e. ZZ ) |
91 |
40
|
adantl |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) = if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) ) |
92 |
65 15
|
subcld |
|- ( ph -> ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) e. CC ) |
93 |
92
|
ad2antrr |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) e. CC ) |
94 |
1
|
ffvelrnda |
|- ( ( ph /\ i e. NN0 ) -> ( A ` i ) e. CC ) |
95 |
94
|
adantlr |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( A ` i ) e. CC ) |
96 |
93 95
|
ifcld |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) e. CC ) |
97 |
96
|
mulid1d |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) = if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) ) |
98 |
91 97
|
eqtr4d |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) = ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) ) |
99 |
97 96
|
eqeltrd |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) e. CC ) |
100 |
|
oveq1 |
|- ( x = 1 -> ( x ^ n ) = ( 1 ^ n ) ) |
101 |
|
nn0z |
|- ( n e. NN0 -> n e. ZZ ) |
102 |
|
1exp |
|- ( n e. ZZ -> ( 1 ^ n ) = 1 ) |
103 |
101 102
|
syl |
|- ( n e. NN0 -> ( 1 ^ n ) = 1 ) |
104 |
100 103
|
sylan9eq |
|- ( ( x = 1 /\ n e. NN0 ) -> ( x ^ n ) = 1 ) |
105 |
104
|
oveq2d |
|- ( ( x = 1 /\ n e. NN0 ) -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` n ) x. 1 ) ) |
106 |
105
|
sumeq2dv |
|- ( x = 1 -> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) = sum_ n e. NN0 ( ( A ` n ) x. 1 ) ) |
107 |
|
fveq2 |
|- ( n = m -> ( A ` n ) = ( A ` m ) ) |
108 |
107
|
oveq1d |
|- ( n = m -> ( ( A ` n ) x. 1 ) = ( ( A ` m ) x. 1 ) ) |
109 |
108
|
cbvsumv |
|- sum_ n e. NN0 ( ( A ` n ) x. 1 ) = sum_ m e. NN0 ( ( A ` m ) x. 1 ) |
110 |
106 109
|
eqtrdi |
|- ( x = 1 -> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) = sum_ m e. NN0 ( ( A ` m ) x. 1 ) ) |
111 |
|
sumex |
|- sum_ m e. NN0 ( ( A ` m ) x. 1 ) e. _V |
112 |
110 6 111
|
fvmpt |
|- ( 1 e. S -> ( F ` 1 ) = sum_ m e. NN0 ( ( A ` m ) x. 1 ) ) |
113 |
77 112
|
syl |
|- ( ph -> ( F ` 1 ) = sum_ m e. NN0 ( ( A ` m ) x. 1 ) ) |
114 |
14
|
mulid1d |
|- ( ( ph /\ m e. NN0 ) -> ( ( A ` m ) x. 1 ) = ( A ` m ) ) |
115 |
114
|
sumeq2dv |
|- ( ph -> sum_ m e. NN0 ( ( A ` m ) x. 1 ) = sum_ m e. NN0 ( A ` m ) ) |
116 |
113 115
|
eqtrd |
|- ( ph -> ( F ` 1 ) = sum_ m e. NN0 ( A ` m ) ) |
117 |
116
|
oveq1d |
|- ( ph -> ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) = ( sum_ m e. NN0 ( A ` m ) - sum_ m e. NN0 ( A ` m ) ) ) |
118 |
15
|
subidd |
|- ( ph -> ( sum_ m e. NN0 ( A ` m ) - sum_ m e. NN0 ( A ` m ) ) = 0 ) |
119 |
117 118
|
eqtrd |
|- ( ph -> ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) = 0 ) |
120 |
69 119
|
breqtrrd |
|- ( ph -> seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ~~> ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) ) |
121 |
120
|
adantr |
|- ( ( ph /\ y e. S ) -> seq 0 ( + , ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ) ~~> ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) ) |
122 |
11 90 98 99 121
|
isumclim |
|- ( ( ph /\ y e. S ) -> sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. 1 ) = ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) ) |
123 |
89 122
|
eqtrd |
|- ( ( ph /\ y e. S ) -> ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) = ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) ) |
124 |
|
oveq1 |
|- ( x = y -> ( x ^ i ) = ( y ^ i ) ) |
125 |
40 124
|
oveqan12rd |
|- ( ( x = y /\ i e. NN0 ) -> ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) = ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
126 |
125
|
sumeq2dv |
|- ( x = y -> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) = sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
127 |
|
sumex |
|- sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) e. _V |
128 |
126 74 127
|
fvmpt |
|- ( y e. S -> ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) = sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
129 |
128
|
adantl |
|- ( ( ph /\ y e. S ) -> ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) = sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
130 |
|
oveq2 |
|- ( k = i -> ( y ^ k ) = ( y ^ i ) ) |
131 |
35 130
|
oveq12d |
|- ( k = i -> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) = ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
132 |
|
eqid |
|- ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) = ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) |
133 |
|
ovex |
|- ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) e. _V |
134 |
131 132 133
|
fvmpt |
|- ( i e. NN0 -> ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` i ) = ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
135 |
134
|
adantl |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` i ) = ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
136 |
5
|
ssrab3 |
|- S C_ CC |
137 |
136
|
a1i |
|- ( ph -> S C_ CC ) |
138 |
137
|
sselda |
|- ( ( ph /\ y e. S ) -> y e. CC ) |
139 |
|
expcl |
|- ( ( y e. CC /\ i e. NN0 ) -> ( y ^ i ) e. CC ) |
140 |
138 139
|
sylan |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( y ^ i ) e. CC ) |
141 |
96 140
|
mulcld |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) e. CC ) |
142 |
7
|
a1i |
|- ( ( ph /\ y e. S ) -> 0 e. NN0 ) |
143 |
19
|
adantlr |
|- ( ( ( ph /\ y e. S ) /\ k e. NN0 ) -> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) e. CC ) |
144 |
|
expcl |
|- ( ( y e. CC /\ k e. NN0 ) -> ( y ^ k ) e. CC ) |
145 |
138 144
|
sylan |
|- ( ( ( ph /\ y e. S ) /\ k e. NN0 ) -> ( y ^ k ) e. CC ) |
146 |
143 145
|
mulcld |
|- ( ( ( ph /\ y e. S ) /\ k e. NN0 ) -> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) e. CC ) |
147 |
146
|
fmpttd |
|- ( ( ph /\ y e. S ) -> ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) : NN0 --> CC ) |
148 |
147
|
ffvelrnda |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` i ) e. CC ) |
149 |
45
|
oveq1d |
|- ( ( ph /\ i e. NN ) -> ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) = ( ( A ` i ) x. ( y ^ i ) ) ) |
150 |
32 134
|
syl |
|- ( ( ph /\ i e. NN ) -> ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` i ) = ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) ) |
151 |
34 130
|
oveq12d |
|- ( k = i -> ( ( A ` k ) x. ( y ^ k ) ) = ( ( A ` i ) x. ( y ^ i ) ) ) |
152 |
|
eqid |
|- ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) = ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) |
153 |
|
ovex |
|- ( ( A ` i ) x. ( y ^ i ) ) e. _V |
154 |
151 152 153
|
fvmpt |
|- ( i e. NN0 -> ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` i ) = ( ( A ` i ) x. ( y ^ i ) ) ) |
155 |
32 154
|
syl |
|- ( ( ph /\ i e. NN ) -> ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` i ) = ( ( A ` i ) x. ( y ^ i ) ) ) |
156 |
149 150 155
|
3eqtr4d |
|- ( ( ph /\ i e. NN ) -> ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` i ) = ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` i ) ) |
157 |
30 156
|
sylan2br |
|- ( ( ph /\ i e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` i ) = ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` i ) ) |
158 |
26 157
|
seqfeq |
|- ( ph -> seq ( 0 + 1 ) ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) = seq ( 0 + 1 ) ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ) |
159 |
158
|
adantr |
|- ( ( ph /\ y e. S ) -> seq ( 0 + 1 ) ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) = seq ( 0 + 1 ) ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ) |
160 |
18
|
adantlr |
|- ( ( ( ph /\ y e. S ) /\ k e. NN0 ) -> ( A ` k ) e. CC ) |
161 |
160 145
|
mulcld |
|- ( ( ( ph /\ y e. S ) /\ k e. NN0 ) -> ( ( A ` k ) x. ( y ^ k ) ) e. CC ) |
162 |
161
|
fmpttd |
|- ( ( ph /\ y e. S ) -> ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) : NN0 --> CC ) |
163 |
162
|
ffvelrnda |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` i ) e. CC ) |
164 |
154
|
adantl |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` i ) = ( ( A ` i ) x. ( y ^ i ) ) ) |
165 |
95 140
|
mulcld |
|- ( ( ( ph /\ y e. S ) /\ i e. NN0 ) -> ( ( A ` i ) x. ( y ^ i ) ) e. CC ) |
166 |
1 2 3 4 5
|
abelthlem3 |
|- ( ( ph /\ y e. S ) -> seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) e. dom ~~> ) |
167 |
11 90 164 165 166
|
isumclim2 |
|- ( ( ph /\ y e. S ) -> seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ~~> sum_ i e. NN0 ( ( A ` i ) x. ( y ^ i ) ) ) |
168 |
|
fveq2 |
|- ( n = i -> ( A ` n ) = ( A ` i ) ) |
169 |
|
oveq2 |
|- ( n = i -> ( x ^ n ) = ( x ^ i ) ) |
170 |
168 169
|
oveq12d |
|- ( n = i -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` i ) x. ( x ^ i ) ) ) |
171 |
170
|
cbvsumv |
|- sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) = sum_ i e. NN0 ( ( A ` i ) x. ( x ^ i ) ) |
172 |
124
|
oveq2d |
|- ( x = y -> ( ( A ` i ) x. ( x ^ i ) ) = ( ( A ` i ) x. ( y ^ i ) ) ) |
173 |
172
|
sumeq2sdv |
|- ( x = y -> sum_ i e. NN0 ( ( A ` i ) x. ( x ^ i ) ) = sum_ i e. NN0 ( ( A ` i ) x. ( y ^ i ) ) ) |
174 |
171 173
|
syl5eq |
|- ( x = y -> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) = sum_ i e. NN0 ( ( A ` i ) x. ( y ^ i ) ) ) |
175 |
|
sumex |
|- sum_ i e. NN0 ( ( A ` i ) x. ( y ^ i ) ) e. _V |
176 |
174 6 175
|
fvmpt |
|- ( y e. S -> ( F ` y ) = sum_ i e. NN0 ( ( A ` i ) x. ( y ^ i ) ) ) |
177 |
176
|
adantl |
|- ( ( ph /\ y e. S ) -> ( F ` y ) = sum_ i e. NN0 ( ( A ` i ) x. ( y ^ i ) ) ) |
178 |
167 177
|
breqtrrd |
|- ( ( ph /\ y e. S ) -> seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ~~> ( F ` y ) ) |
179 |
11 142 163 178
|
clim2ser |
|- ( ( ph /\ y e. S ) -> seq ( 0 + 1 ) ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ~~> ( ( F ` y ) - ( seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ` 0 ) ) ) |
180 |
|
seq1 |
|- ( 0 e. ZZ -> ( seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ` 0 ) = ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` 0 ) ) |
181 |
51 180
|
ax-mp |
|- ( seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ` 0 ) = ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` 0 ) |
182 |
|
fveq2 |
|- ( k = 0 -> ( A ` k ) = ( A ` 0 ) ) |
183 |
|
oveq2 |
|- ( k = 0 -> ( y ^ k ) = ( y ^ 0 ) ) |
184 |
182 183
|
oveq12d |
|- ( k = 0 -> ( ( A ` k ) x. ( y ^ k ) ) = ( ( A ` 0 ) x. ( y ^ 0 ) ) ) |
185 |
|
ovex |
|- ( ( A ` 0 ) x. ( y ^ 0 ) ) e. _V |
186 |
184 152 185
|
fvmpt |
|- ( 0 e. NN0 -> ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` 0 ) = ( ( A ` 0 ) x. ( y ^ 0 ) ) ) |
187 |
7 186
|
ax-mp |
|- ( ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ` 0 ) = ( ( A ` 0 ) x. ( y ^ 0 ) ) |
188 |
181 187
|
eqtri |
|- ( seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ` 0 ) = ( ( A ` 0 ) x. ( y ^ 0 ) ) |
189 |
138
|
exp0d |
|- ( ( ph /\ y e. S ) -> ( y ^ 0 ) = 1 ) |
190 |
189
|
oveq2d |
|- ( ( ph /\ y e. S ) -> ( ( A ` 0 ) x. ( y ^ 0 ) ) = ( ( A ` 0 ) x. 1 ) ) |
191 |
65
|
adantr |
|- ( ( ph /\ y e. S ) -> ( A ` 0 ) e. CC ) |
192 |
191
|
mulid1d |
|- ( ( ph /\ y e. S ) -> ( ( A ` 0 ) x. 1 ) = ( A ` 0 ) ) |
193 |
190 192
|
eqtrd |
|- ( ( ph /\ y e. S ) -> ( ( A ` 0 ) x. ( y ^ 0 ) ) = ( A ` 0 ) ) |
194 |
188 193
|
syl5eq |
|- ( ( ph /\ y e. S ) -> ( seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ` 0 ) = ( A ` 0 ) ) |
195 |
194
|
oveq2d |
|- ( ( ph /\ y e. S ) -> ( ( F ` y ) - ( seq 0 ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ` 0 ) ) = ( ( F ` y ) - ( A ` 0 ) ) ) |
196 |
179 195
|
breqtrd |
|- ( ( ph /\ y e. S ) -> seq ( 0 + 1 ) ( + , ( k e. NN0 |-> ( ( A ` k ) x. ( y ^ k ) ) ) ) ~~> ( ( F ` y ) - ( A ` 0 ) ) ) |
197 |
159 196
|
eqbrtrd |
|- ( ( ph /\ y e. S ) -> seq ( 0 + 1 ) ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ~~> ( ( F ` y ) - ( A ` 0 ) ) ) |
198 |
11 142 148 197
|
clim2ser2 |
|- ( ( ph /\ y e. S ) -> seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ~~> ( ( ( F ` y ) - ( A ` 0 ) ) + ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ` 0 ) ) ) |
199 |
|
seq1 |
|- ( 0 e. ZZ -> ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ` 0 ) = ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` 0 ) ) |
200 |
51 199
|
ax-mp |
|- ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ` 0 ) = ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` 0 ) |
201 |
60 183
|
oveq12d |
|- ( k = 0 -> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) = ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. ( y ^ 0 ) ) ) |
202 |
|
ovex |
|- ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. ( y ^ 0 ) ) e. _V |
203 |
201 132 202
|
fvmpt |
|- ( 0 e. NN0 -> ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` 0 ) = ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. ( y ^ 0 ) ) ) |
204 |
7 203
|
ax-mp |
|- ( ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ` 0 ) = ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. ( y ^ 0 ) ) |
205 |
200 204
|
eqtri |
|- ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ` 0 ) = ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. ( y ^ 0 ) ) |
206 |
189
|
oveq2d |
|- ( ( ph /\ y e. S ) -> ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. ( y ^ 0 ) ) = ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. 1 ) ) |
207 |
15
|
adantr |
|- ( ( ph /\ y e. S ) -> sum_ m e. NN0 ( A ` m ) e. CC ) |
208 |
191 207
|
subcld |
|- ( ( ph /\ y e. S ) -> ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) e. CC ) |
209 |
208
|
mulid1d |
|- ( ( ph /\ y e. S ) -> ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. 1 ) = ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) |
210 |
206 209
|
eqtrd |
|- ( ( ph /\ y e. S ) -> ( ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) x. ( y ^ 0 ) ) = ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) |
211 |
205 210
|
syl5eq |
|- ( ( ph /\ y e. S ) -> ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ` 0 ) = ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) |
212 |
211
|
oveq2d |
|- ( ( ph /\ y e. S ) -> ( ( ( F ` y ) - ( A ` 0 ) ) + ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ` 0 ) ) = ( ( ( F ` y ) - ( A ` 0 ) ) + ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) ) |
213 |
1 2 3 4 5 6
|
abelthlem4 |
|- ( ph -> F : S --> CC ) |
214 |
213
|
ffvelrnda |
|- ( ( ph /\ y e. S ) -> ( F ` y ) e. CC ) |
215 |
214 191 207
|
npncand |
|- ( ( ph /\ y e. S ) -> ( ( ( F ` y ) - ( A ` 0 ) ) + ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) ) = ( ( F ` y ) - sum_ m e. NN0 ( A ` m ) ) ) |
216 |
212 215
|
eqtrd |
|- ( ( ph /\ y e. S ) -> ( ( ( F ` y ) - ( A ` 0 ) ) + ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ` 0 ) ) = ( ( F ` y ) - sum_ m e. NN0 ( A ` m ) ) ) |
217 |
198 216
|
breqtrd |
|- ( ( ph /\ y e. S ) -> seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) x. ( y ^ k ) ) ) ) ~~> ( ( F ` y ) - sum_ m e. NN0 ( A ` m ) ) ) |
218 |
11 90 135 141 217
|
isumclim |
|- ( ( ph /\ y e. S ) -> sum_ i e. NN0 ( if ( i = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` i ) ) x. ( y ^ i ) ) = ( ( F ` y ) - sum_ m e. NN0 ( A ` m ) ) ) |
219 |
129 218
|
eqtrd |
|- ( ( ph /\ y e. S ) -> ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) = ( ( F ` y ) - sum_ m e. NN0 ( A ` m ) ) ) |
220 |
123 219
|
oveq12d |
|- ( ( ph /\ y e. S ) -> ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) = ( ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) - ( ( F ` y ) - sum_ m e. NN0 ( A ` m ) ) ) ) |
221 |
213
|
adantr |
|- ( ( ph /\ y e. S ) -> F : S --> CC ) |
222 |
221 78
|
ffvelrnd |
|- ( ( ph /\ y e. S ) -> ( F ` 1 ) e. CC ) |
223 |
222 214 207
|
nnncan2d |
|- ( ( ph /\ y e. S ) -> ( ( ( F ` 1 ) - sum_ m e. NN0 ( A ` m ) ) - ( ( F ` y ) - sum_ m e. NN0 ( A ` m ) ) ) = ( ( F ` 1 ) - ( F ` y ) ) ) |
224 |
220 223
|
eqtrd |
|- ( ( ph /\ y e. S ) -> ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) = ( ( F ` 1 ) - ( F ` y ) ) ) |
225 |
224
|
fveq2d |
|- ( ( ph /\ y e. S ) -> ( abs ` ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) ) = ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) ) |
226 |
225
|
breq1d |
|- ( ( ph /\ y e. S ) -> ( ( abs ` ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) ) < R <-> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) |
227 |
226
|
imbi2d |
|- ( ( ph /\ y e. S ) -> ( ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) ) < R ) <-> ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) ) |
228 |
227
|
ralbidva |
|- ( ph -> ( A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) ) < R ) <-> A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) ) |
229 |
228
|
rexbidv |
|- ( ph -> ( E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) ) < R ) <-> E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) ) |
230 |
229
|
adantr |
|- ( ( ph /\ R e. RR+ ) -> ( E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` 1 ) - ( ( x e. S |-> sum_ i e. NN0 ( ( ( k e. NN0 |-> if ( k = 0 , ( ( A ` 0 ) - sum_ m e. NN0 ( A ` m ) ) , ( A ` k ) ) ) ` i ) x. ( x ^ i ) ) ) ` y ) ) ) < R ) <-> E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) ) |
231 |
75 230
|
mpbid |
|- ( ( ph /\ R e. RR+ ) -> E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) |