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 |
|
abelth.7 |
|- ( ph -> seq 0 ( + , A ) ~~> 0 ) |
8 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
9 |
|
0zd |
|- ( ( ph /\ R e. RR+ ) -> 0 e. ZZ ) |
10 |
|
id |
|- ( R e. RR+ -> R e. RR+ ) |
11 |
3 4
|
ge0p1rpd |
|- ( ph -> ( M + 1 ) e. RR+ ) |
12 |
|
rpdivcl |
|- ( ( R e. RR+ /\ ( M + 1 ) e. RR+ ) -> ( R / ( M + 1 ) ) e. RR+ ) |
13 |
10 11 12
|
syl2anr |
|- ( ( ph /\ R e. RR+ ) -> ( R / ( M + 1 ) ) e. RR+ ) |
14 |
|
eqidd |
|- ( ( ( ph /\ R e. RR+ ) /\ k e. NN0 ) -> ( seq 0 ( + , A ) ` k ) = ( seq 0 ( + , A ) ` k ) ) |
15 |
7
|
adantr |
|- ( ( ph /\ R e. RR+ ) -> seq 0 ( + , A ) ~~> 0 ) |
16 |
8 9 13 14 15
|
climi0 |
|- ( ( ph /\ R e. RR+ ) -> E. j e. NN0 A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) |
17 |
13
|
adantr |
|- ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) -> ( R / ( M + 1 ) ) e. RR+ ) |
18 |
|
fzfid |
|- ( ph -> ( 0 ... ( j - 1 ) ) e. Fin ) |
19 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
20 |
1
|
ffvelrnda |
|- ( ( ph /\ w e. NN0 ) -> ( A ` w ) e. CC ) |
21 |
8 19 20
|
serf |
|- ( ph -> seq 0 ( + , A ) : NN0 --> CC ) |
22 |
|
elfznn0 |
|- ( i e. ( 0 ... ( j - 1 ) ) -> i e. NN0 ) |
23 |
|
ffvelrn |
|- ( ( seq 0 ( + , A ) : NN0 --> CC /\ i e. NN0 ) -> ( seq 0 ( + , A ) ` i ) e. CC ) |
24 |
21 22 23
|
syl2an |
|- ( ( ph /\ i e. ( 0 ... ( j - 1 ) ) ) -> ( seq 0 ( + , A ) ` i ) e. CC ) |
25 |
24
|
abscld |
|- ( ( ph /\ i e. ( 0 ... ( j - 1 ) ) ) -> ( abs ` ( seq 0 ( + , A ) ` i ) ) e. RR ) |
26 |
18 25
|
fsumrecl |
|- ( ph -> sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) e. RR ) |
27 |
26
|
ad2antrr |
|- ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) -> sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) e. RR ) |
28 |
24
|
absge0d |
|- ( ( ph /\ i e. ( 0 ... ( j - 1 ) ) ) -> 0 <_ ( abs ` ( seq 0 ( + , A ) ` i ) ) ) |
29 |
18 25 28
|
fsumge0 |
|- ( ph -> 0 <_ sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) ) |
30 |
29
|
ad2antrr |
|- ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) -> 0 <_ sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) ) |
31 |
27 30
|
ge0p1rpd |
|- ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) -> ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) e. RR+ ) |
32 |
17 31
|
rpdivcld |
|- ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) -> ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) e. RR+ ) |
33 |
1 2 3 4 5
|
abelthlem2 |
|- ( ph -> ( 1 e. S /\ ( S \ { 1 } ) C_ ( 0 ( ball ` ( abs o. - ) ) 1 ) ) ) |
34 |
33
|
simpld |
|- ( ph -> 1 e. S ) |
35 |
|
oveq1 |
|- ( x = 1 -> ( x ^ n ) = ( 1 ^ n ) ) |
36 |
|
nn0z |
|- ( n e. NN0 -> n e. ZZ ) |
37 |
|
1exp |
|- ( n e. ZZ -> ( 1 ^ n ) = 1 ) |
38 |
36 37
|
syl |
|- ( n e. NN0 -> ( 1 ^ n ) = 1 ) |
39 |
35 38
|
sylan9eq |
|- ( ( x = 1 /\ n e. NN0 ) -> ( x ^ n ) = 1 ) |
40 |
39
|
oveq2d |
|- ( ( x = 1 /\ n e. NN0 ) -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` n ) x. 1 ) ) |
41 |
40
|
sumeq2dv |
|- ( x = 1 -> sum_ n e. NN0 ( ( A ` n ) x. ( x ^ n ) ) = sum_ n e. NN0 ( ( A ` n ) x. 1 ) ) |
42 |
|
sumex |
|- sum_ n e. NN0 ( ( A ` n ) x. 1 ) e. _V |
43 |
41 6 42
|
fvmpt |
|- ( 1 e. S -> ( F ` 1 ) = sum_ n e. NN0 ( ( A ` n ) x. 1 ) ) |
44 |
34 43
|
syl |
|- ( ph -> ( F ` 1 ) = sum_ n e. NN0 ( ( A ` n ) x. 1 ) ) |
45 |
1
|
ffvelrnda |
|- ( ( ph /\ n e. NN0 ) -> ( A ` n ) e. CC ) |
46 |
45
|
mulid1d |
|- ( ( ph /\ n e. NN0 ) -> ( ( A ` n ) x. 1 ) = ( A ` n ) ) |
47 |
46
|
eqcomd |
|- ( ( ph /\ n e. NN0 ) -> ( A ` n ) = ( ( A ` n ) x. 1 ) ) |
48 |
46 45
|
eqeltrd |
|- ( ( ph /\ n e. NN0 ) -> ( ( A ` n ) x. 1 ) e. CC ) |
49 |
8 19 47 48 7
|
isumclim |
|- ( ph -> sum_ n e. NN0 ( ( A ` n ) x. 1 ) = 0 ) |
50 |
44 49
|
eqtrd |
|- ( ph -> ( F ` 1 ) = 0 ) |
51 |
50
|
adantr |
|- ( ( ph /\ y e. S ) -> ( F ` 1 ) = 0 ) |
52 |
51
|
oveq1d |
|- ( ( ph /\ y e. S ) -> ( ( F ` 1 ) - ( F ` y ) ) = ( 0 - ( F ` y ) ) ) |
53 |
|
df-neg |
|- -u ( F ` y ) = ( 0 - ( F ` y ) ) |
54 |
52 53
|
eqtr4di |
|- ( ( ph /\ y e. S ) -> ( ( F ` 1 ) - ( F ` y ) ) = -u ( F ` y ) ) |
55 |
54
|
fveq2d |
|- ( ( ph /\ y e. S ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) = ( abs ` -u ( F ` y ) ) ) |
56 |
1 2 3 4 5 6
|
abelthlem4 |
|- ( ph -> F : S --> CC ) |
57 |
56
|
ffvelrnda |
|- ( ( ph /\ y e. S ) -> ( F ` y ) e. CC ) |
58 |
57
|
absnegd |
|- ( ( ph /\ y e. S ) -> ( abs ` -u ( F ` y ) ) = ( abs ` ( F ` y ) ) ) |
59 |
55 58
|
eqtrd |
|- ( ( ph /\ y e. S ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) = ( abs ` ( F ` y ) ) ) |
60 |
59
|
adantlr |
|- ( ( ( ph /\ R e. RR+ ) /\ y e. S ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) = ( abs ` ( F ` y ) ) ) |
61 |
60
|
ad2ant2r |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) = ( abs ` ( F ` y ) ) ) |
62 |
|
fveq2 |
|- ( y = 1 -> ( F ` y ) = ( F ` 1 ) ) |
63 |
62 50
|
sylan9eqr |
|- ( ( ph /\ y = 1 ) -> ( F ` y ) = 0 ) |
64 |
63
|
abs00bd |
|- ( ( ph /\ y = 1 ) -> ( abs ` ( F ` y ) ) = 0 ) |
65 |
64
|
ad5ant15 |
|- ( ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) /\ y = 1 ) -> ( abs ` ( F ` y ) ) = 0 ) |
66 |
|
simpllr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) -> R e. RR+ ) |
67 |
66
|
rpgt0d |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) -> 0 < R ) |
68 |
67
|
adantr |
|- ( ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) /\ y = 1 ) -> 0 < R ) |
69 |
65 68
|
eqbrtrd |
|- ( ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) /\ y = 1 ) -> ( abs ` ( F ` y ) ) < R ) |
70 |
1
|
ad3antrrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> A : NN0 --> CC ) |
71 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> seq 0 ( + , A ) e. dom ~~> ) |
72 |
3
|
ad3antrrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> M e. RR ) |
73 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> 0 <_ M ) |
74 |
7
|
ad3antrrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> seq 0 ( + , A ) ~~> 0 ) |
75 |
|
simprll |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> y e. S ) |
76 |
|
simprr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> y =/= 1 ) |
77 |
|
eldifsn |
|- ( y e. ( S \ { 1 } ) <-> ( y e. S /\ y =/= 1 ) ) |
78 |
75 76 77
|
sylanbrc |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> y e. ( S \ { 1 } ) ) |
79 |
13
|
ad2antrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> ( R / ( M + 1 ) ) e. RR+ ) |
80 |
|
simplrl |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> j e. NN0 ) |
81 |
|
simplrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) |
82 |
|
2fveq3 |
|- ( k = m -> ( abs ` ( seq 0 ( + , A ) ` k ) ) = ( abs ` ( seq 0 ( + , A ) ` m ) ) ) |
83 |
82
|
breq1d |
|- ( k = m -> ( ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) <-> ( abs ` ( seq 0 ( + , A ) ` m ) ) < ( R / ( M + 1 ) ) ) ) |
84 |
83
|
cbvralvw |
|- ( A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) <-> A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < ( R / ( M + 1 ) ) ) |
85 |
81 84
|
sylib |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < ( R / ( M + 1 ) ) ) |
86 |
|
simprlr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) |
87 |
|
2fveq3 |
|- ( i = n -> ( abs ` ( seq 0 ( + , A ) ` i ) ) = ( abs ` ( seq 0 ( + , A ) ` n ) ) ) |
88 |
87
|
cbvsumv |
|- sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) = sum_ n e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` n ) ) |
89 |
88
|
oveq1i |
|- ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) = ( sum_ n e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` n ) ) + 1 ) |
90 |
89
|
oveq2i |
|- ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) = ( ( R / ( M + 1 ) ) / ( sum_ n e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` n ) ) + 1 ) ) |
91 |
86 90
|
breqtrdi |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ n e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` n ) ) + 1 ) ) ) |
92 |
70 71 72 73 5 6 74 78 79 80 85 91
|
abelthlem7 |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> ( abs ` ( F ` y ) ) < ( ( M + 1 ) x. ( R / ( M + 1 ) ) ) ) |
93 |
|
rpcn |
|- ( R e. RR+ -> R e. CC ) |
94 |
93
|
adantl |
|- ( ( ph /\ R e. RR+ ) -> R e. CC ) |
95 |
11
|
adantr |
|- ( ( ph /\ R e. RR+ ) -> ( M + 1 ) e. RR+ ) |
96 |
95
|
rpcnd |
|- ( ( ph /\ R e. RR+ ) -> ( M + 1 ) e. CC ) |
97 |
95
|
rpne0d |
|- ( ( ph /\ R e. RR+ ) -> ( M + 1 ) =/= 0 ) |
98 |
94 96 97
|
divcan2d |
|- ( ( ph /\ R e. RR+ ) -> ( ( M + 1 ) x. ( R / ( M + 1 ) ) ) = R ) |
99 |
98
|
ad2antrr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> ( ( M + 1 ) x. ( R / ( M + 1 ) ) ) = R ) |
100 |
92 99
|
breqtrd |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) /\ y =/= 1 ) ) -> ( abs ` ( F ` y ) ) < R ) |
101 |
100
|
anassrs |
|- ( ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) /\ y =/= 1 ) -> ( abs ` ( F ` y ) ) < R ) |
102 |
69 101
|
pm2.61dane |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) -> ( abs ` ( F ` y ) ) < R ) |
103 |
61 102
|
eqbrtrd |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ ( y e. S /\ ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) |
104 |
103
|
expr |
|- ( ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) /\ y e. S ) -> ( ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) |
105 |
104
|
ralrimiva |
|- ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) -> A. y e. S ( ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) |
106 |
|
breq2 |
|- ( w = ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) -> ( ( abs ` ( 1 - y ) ) < w <-> ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) ) ) |
107 |
106
|
rspceaimv |
|- ( ( ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) e. RR+ /\ A. y e. S ( ( abs ` ( 1 - y ) ) < ( ( R / ( M + 1 ) ) / ( sum_ i e. ( 0 ... ( j - 1 ) ) ( abs ` ( seq 0 ( + , A ) ` i ) ) + 1 ) ) -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) -> E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) |
108 |
32 105 107
|
syl2anc |
|- ( ( ( ph /\ R e. RR+ ) /\ ( j e. NN0 /\ A. k e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` k ) ) < ( R / ( M + 1 ) ) ) ) -> E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) |
109 |
16 108
|
rexlimddv |
|- ( ( ph /\ R e. RR+ ) -> E. w e. RR+ A. y e. S ( ( abs ` ( 1 - y ) ) < w -> ( abs ` ( ( F ` 1 ) - ( F ` y ) ) ) < R ) ) |