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 -> 0 e. ZZ ) |
10 |
|
1rp |
|- 1 e. RR+ |
11 |
10
|
a1i |
|- ( ph -> 1 e. RR+ ) |
12 |
|
eqidd |
|- ( ( ph /\ m e. NN0 ) -> ( seq 0 ( + , A ) ` m ) = ( seq 0 ( + , A ) ` m ) ) |
13 |
8 9 11 12 7
|
climi0 |
|- ( ph -> E. j e. NN0 A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) |
14 |
13
|
adantr |
|- ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> E. j e. NN0 A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) |
15 |
|
simprl |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> j e. NN0 ) |
16 |
|
oveq2 |
|- ( n = i -> ( ( abs ` X ) ^ n ) = ( ( abs ` X ) ^ i ) ) |
17 |
|
eqid |
|- ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) = ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) |
18 |
|
ovex |
|- ( ( abs ` X ) ^ i ) e. _V |
19 |
16 17 18
|
fvmpt |
|- ( i e. NN0 -> ( ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ` i ) = ( ( abs ` X ) ^ i ) ) |
20 |
19
|
adantl |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ` i ) = ( ( abs ` X ) ^ i ) ) |
21 |
|
cnxmet |
|- ( abs o. - ) e. ( *Met ` CC ) |
22 |
|
0cn |
|- 0 e. CC |
23 |
|
1xr |
|- 1 e. RR* |
24 |
|
blssm |
|- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ 1 e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) 1 ) C_ CC ) |
25 |
21 22 23 24
|
mp3an |
|- ( 0 ( ball ` ( abs o. - ) ) 1 ) C_ CC |
26 |
|
simplr |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) |
27 |
25 26
|
sselid |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> X e. CC ) |
28 |
27
|
abscld |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( abs ` X ) e. RR ) |
29 |
|
reexpcl |
|- ( ( ( abs ` X ) e. RR /\ i e. NN0 ) -> ( ( abs ` X ) ^ i ) e. RR ) |
30 |
28 29
|
sylan |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( ( abs ` X ) ^ i ) e. RR ) |
31 |
20 30
|
eqeltrd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ` i ) e. RR ) |
32 |
|
fveq2 |
|- ( k = i -> ( seq 0 ( + , A ) ` k ) = ( seq 0 ( + , A ) ` i ) ) |
33 |
|
oveq2 |
|- ( k = i -> ( X ^ k ) = ( X ^ i ) ) |
34 |
32 33
|
oveq12d |
|- ( k = i -> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) = ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) |
35 |
|
eqid |
|- ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) = ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) |
36 |
|
ovex |
|- ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) e. _V |
37 |
34 35 36
|
fvmpt |
|- ( i e. NN0 -> ( ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ` i ) = ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) |
38 |
37
|
adantl |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ` i ) = ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) |
39 |
1
|
ffvelrnda |
|- ( ( ph /\ x e. NN0 ) -> ( A ` x ) e. CC ) |
40 |
8 9 39
|
serf |
|- ( ph -> seq 0 ( + , A ) : NN0 --> CC ) |
41 |
40
|
ad2antrr |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> seq 0 ( + , A ) : NN0 --> CC ) |
42 |
41
|
ffvelrnda |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( seq 0 ( + , A ) ` i ) e. CC ) |
43 |
|
expcl |
|- ( ( X e. CC /\ i e. NN0 ) -> ( X ^ i ) e. CC ) |
44 |
27 43
|
sylan |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( X ^ i ) e. CC ) |
45 |
42 44
|
mulcld |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) e. CC ) |
46 |
38 45
|
eqeltrd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ` i ) e. CC ) |
47 |
28
|
recnd |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( abs ` X ) e. CC ) |
48 |
|
absidm |
|- ( X e. CC -> ( abs ` ( abs ` X ) ) = ( abs ` X ) ) |
49 |
27 48
|
syl |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( abs ` ( abs ` X ) ) = ( abs ` X ) ) |
50 |
|
eqid |
|- ( abs o. - ) = ( abs o. - ) |
51 |
50
|
cnmetdval |
|- ( ( X e. CC /\ 0 e. CC ) -> ( X ( abs o. - ) 0 ) = ( abs ` ( X - 0 ) ) ) |
52 |
27 22 51
|
sylancl |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( X ( abs o. - ) 0 ) = ( abs ` ( X - 0 ) ) ) |
53 |
27
|
subid1d |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( X - 0 ) = X ) |
54 |
53
|
fveq2d |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( abs ` ( X - 0 ) ) = ( abs ` X ) ) |
55 |
52 54
|
eqtrd |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( X ( abs o. - ) 0 ) = ( abs ` X ) ) |
56 |
|
elbl3 |
|- ( ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 1 e. RR* ) /\ ( 0 e. CC /\ X e. CC ) ) -> ( X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( X ( abs o. - ) 0 ) < 1 ) ) |
57 |
21 23 56
|
mpanl12 |
|- ( ( 0 e. CC /\ X e. CC ) -> ( X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( X ( abs o. - ) 0 ) < 1 ) ) |
58 |
22 27 57
|
sylancr |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( X ( abs o. - ) 0 ) < 1 ) ) |
59 |
26 58
|
mpbid |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( X ( abs o. - ) 0 ) < 1 ) |
60 |
55 59
|
eqbrtrrd |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( abs ` X ) < 1 ) |
61 |
49 60
|
eqbrtrd |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> ( abs ` ( abs ` X ) ) < 1 ) |
62 |
47 61 20
|
geolim |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> seq 0 ( + , ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ) ~~> ( 1 / ( 1 - ( abs ` X ) ) ) ) |
63 |
|
climrel |
|- Rel ~~> |
64 |
63
|
releldmi |
|- ( seq 0 ( + , ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ) ~~> ( 1 / ( 1 - ( abs ` X ) ) ) -> seq 0 ( + , ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ) e. dom ~~> ) |
65 |
62 64
|
syl |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> seq 0 ( + , ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ) e. dom ~~> ) |
66 |
|
1red |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> 1 e. RR ) |
67 |
41
|
adantr |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> seq 0 ( + , A ) : NN0 --> CC ) |
68 |
|
eluznn0 |
|- ( ( j e. NN0 /\ i e. ( ZZ>= ` j ) ) -> i e. NN0 ) |
69 |
15 68
|
sylan |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> i e. NN0 ) |
70 |
67 69
|
ffvelrnd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( seq 0 ( + , A ) ` i ) e. CC ) |
71 |
69 44
|
syldan |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( X ^ i ) e. CC ) |
72 |
70 71
|
absmuld |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) = ( ( abs ` ( seq 0 ( + , A ) ` i ) ) x. ( abs ` ( X ^ i ) ) ) ) |
73 |
27
|
adantr |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> X e. CC ) |
74 |
73 69
|
absexpd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( X ^ i ) ) = ( ( abs ` X ) ^ i ) ) |
75 |
74
|
oveq2d |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( ( abs ` ( seq 0 ( + , A ) ` i ) ) x. ( abs ` ( X ^ i ) ) ) = ( ( abs ` ( seq 0 ( + , A ) ` i ) ) x. ( ( abs ` X ) ^ i ) ) ) |
76 |
72 75
|
eqtrd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) = ( ( abs ` ( seq 0 ( + , A ) ` i ) ) x. ( ( abs ` X ) ^ i ) ) ) |
77 |
70
|
abscld |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( seq 0 ( + , A ) ` i ) ) e. RR ) |
78 |
|
1red |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> 1 e. RR ) |
79 |
69 30
|
syldan |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( ( abs ` X ) ^ i ) e. RR ) |
80 |
71
|
absge0d |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> 0 <_ ( abs ` ( X ^ i ) ) ) |
81 |
80 74
|
breqtrd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> 0 <_ ( ( abs ` X ) ^ i ) ) |
82 |
|
simprr |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) |
83 |
|
2fveq3 |
|- ( m = i -> ( abs ` ( seq 0 ( + , A ) ` m ) ) = ( abs ` ( seq 0 ( + , A ) ` i ) ) ) |
84 |
83
|
breq1d |
|- ( m = i -> ( ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 <-> ( abs ` ( seq 0 ( + , A ) ` i ) ) < 1 ) ) |
85 |
84
|
rspccva |
|- ( ( A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( seq 0 ( + , A ) ` i ) ) < 1 ) |
86 |
82 85
|
sylan |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( seq 0 ( + , A ) ` i ) ) < 1 ) |
87 |
|
1re |
|- 1 e. RR |
88 |
|
ltle |
|- ( ( ( abs ` ( seq 0 ( + , A ) ` i ) ) e. RR /\ 1 e. RR ) -> ( ( abs ` ( seq 0 ( + , A ) ` i ) ) < 1 -> ( abs ` ( seq 0 ( + , A ) ` i ) ) <_ 1 ) ) |
89 |
77 87 88
|
sylancl |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( ( abs ` ( seq 0 ( + , A ) ` i ) ) < 1 -> ( abs ` ( seq 0 ( + , A ) ` i ) ) <_ 1 ) ) |
90 |
86 89
|
mpd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( seq 0 ( + , A ) ` i ) ) <_ 1 ) |
91 |
77 78 79 81 90
|
lemul1ad |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( ( abs ` ( seq 0 ( + , A ) ` i ) ) x. ( ( abs ` X ) ^ i ) ) <_ ( 1 x. ( ( abs ` X ) ^ i ) ) ) |
92 |
76 91
|
eqbrtrd |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) <_ ( 1 x. ( ( abs ` X ) ^ i ) ) ) |
93 |
69 37
|
syl |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ` i ) = ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) |
94 |
93
|
fveq2d |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ` i ) ) = ( abs ` ( ( seq 0 ( + , A ) ` i ) x. ( X ^ i ) ) ) ) |
95 |
69 19
|
syl |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ` i ) = ( ( abs ` X ) ^ i ) ) |
96 |
95
|
oveq2d |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( 1 x. ( ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ` i ) ) = ( 1 x. ( ( abs ` X ) ^ i ) ) ) |
97 |
92 94 96
|
3brtr4d |
|- ( ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) /\ i e. ( ZZ>= ` j ) ) -> ( abs ` ( ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ` i ) ) <_ ( 1 x. ( ( n e. NN0 |-> ( ( abs ` X ) ^ n ) ) ` i ) ) ) |
98 |
8 15 31 46 65 66 97
|
cvgcmpce |
|- ( ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) /\ ( j e. NN0 /\ A. m e. ( ZZ>= ` j ) ( abs ` ( seq 0 ( + , A ) ` m ) ) < 1 ) ) -> seq 0 ( + , ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ) e. dom ~~> ) |
99 |
14 98
|
rexlimddv |
|- ( ( ph /\ X e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> seq 0 ( + , ( k e. NN0 |-> ( ( seq 0 ( + , A ) ` k ) x. ( X ^ k ) ) ) ) e. dom ~~> ) |