Step |
Hyp |
Ref |
Expression |
1 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
2 |
|
0zd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> 0 e. ZZ ) |
3 |
|
eqeq1 |
|- ( k = n -> ( k = 0 <-> n = 0 ) ) |
4 |
|
oveq2 |
|- ( k = n -> ( 1 / k ) = ( 1 / n ) ) |
5 |
3 4
|
ifbieq2d |
|- ( k = n -> if ( k = 0 , 0 , ( 1 / k ) ) = if ( n = 0 , 0 , ( 1 / n ) ) ) |
6 |
|
oveq2 |
|- ( k = n -> ( A ^ k ) = ( A ^ n ) ) |
7 |
5 6
|
oveq12d |
|- ( k = n -> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
8 |
|
eqid |
|- ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) = ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) |
9 |
|
ovex |
|- ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) e. _V |
10 |
7 8 9
|
fvmpt |
|- ( n e. NN0 -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` n ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
11 |
10
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` n ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
12 |
|
0cnd |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) /\ n = 0 ) -> 0 e. CC ) |
13 |
|
simpr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> n e. NN0 ) |
14 |
|
elnn0 |
|- ( n e. NN0 <-> ( n e. NN \/ n = 0 ) ) |
15 |
13 14
|
sylib |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> ( n e. NN \/ n = 0 ) ) |
16 |
15
|
ord |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> ( -. n e. NN -> n = 0 ) ) |
17 |
16
|
con1d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> ( -. n = 0 -> n e. NN ) ) |
18 |
17
|
imp |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN ) |
19 |
18
|
nnrecred |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) /\ -. n = 0 ) -> ( 1 / n ) e. RR ) |
20 |
19
|
recnd |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) /\ -. n = 0 ) -> ( 1 / n ) e. CC ) |
21 |
12 20
|
ifclda |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> if ( n = 0 , 0 , ( 1 / n ) ) e. CC ) |
22 |
|
expcl |
|- ( ( A e. CC /\ n e. NN0 ) -> ( A ^ n ) e. CC ) |
23 |
22
|
adantlr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> ( A ^ n ) e. CC ) |
24 |
21 23
|
mulcld |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN0 ) -> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) e. CC ) |
25 |
|
logtayllem |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) e. dom ~~> ) |
26 |
1 2 11 24 25
|
isumclim2 |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ~~> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
27 |
|
simpl |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> A e. CC ) |
28 |
|
0cn |
|- 0 e. CC |
29 |
|
eqid |
|- ( abs o. - ) = ( abs o. - ) |
30 |
29
|
cnmetdval |
|- ( ( A e. CC /\ 0 e. CC ) -> ( A ( abs o. - ) 0 ) = ( abs ` ( A - 0 ) ) ) |
31 |
27 28 30
|
sylancl |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( A ( abs o. - ) 0 ) = ( abs ` ( A - 0 ) ) ) |
32 |
|
subid1 |
|- ( A e. CC -> ( A - 0 ) = A ) |
33 |
32
|
adantr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( A - 0 ) = A ) |
34 |
33
|
fveq2d |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` ( A - 0 ) ) = ( abs ` A ) ) |
35 |
31 34
|
eqtrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( A ( abs o. - ) 0 ) = ( abs ` A ) ) |
36 |
|
simpr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` A ) < 1 ) |
37 |
35 36
|
eqbrtrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( A ( abs o. - ) 0 ) < 1 ) |
38 |
|
cnxmet |
|- ( abs o. - ) e. ( *Met ` CC ) |
39 |
|
1xr |
|- 1 e. RR* |
40 |
|
elbl3 |
|- ( ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 1 e. RR* ) /\ ( 0 e. CC /\ A e. CC ) ) -> ( A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( A ( abs o. - ) 0 ) < 1 ) ) |
41 |
38 39 40
|
mpanl12 |
|- ( ( 0 e. CC /\ A e. CC ) -> ( A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( A ( abs o. - ) 0 ) < 1 ) ) |
42 |
28 27 41
|
sylancr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( A ( abs o. - ) 0 ) < 1 ) ) |
43 |
37 42
|
mpbird |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) |
44 |
|
tru |
|- T. |
45 |
|
eqid |
|- ( 0 ( ball ` ( abs o. - ) ) 1 ) = ( 0 ( ball ` ( abs o. - ) ) 1 ) |
46 |
|
0cnd |
|- ( T. -> 0 e. CC ) |
47 |
39
|
a1i |
|- ( T. -> 1 e. RR* ) |
48 |
|
ax-1cn |
|- 1 e. CC |
49 |
|
blssm |
|- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ 1 e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) 1 ) C_ CC ) |
50 |
38 28 39 49
|
mp3an |
|- ( 0 ( ball ` ( abs o. - ) ) 1 ) C_ CC |
51 |
50
|
sseli |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> y e. CC ) |
52 |
|
subcl |
|- ( ( 1 e. CC /\ y e. CC ) -> ( 1 - y ) e. CC ) |
53 |
48 51 52
|
sylancr |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 - y ) e. CC ) |
54 |
51
|
abscld |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` y ) e. RR ) |
55 |
29
|
cnmetdval |
|- ( ( y e. CC /\ 0 e. CC ) -> ( y ( abs o. - ) 0 ) = ( abs ` ( y - 0 ) ) ) |
56 |
51 28 55
|
sylancl |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( y ( abs o. - ) 0 ) = ( abs ` ( y - 0 ) ) ) |
57 |
51
|
subid1d |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( y - 0 ) = y ) |
58 |
57
|
fveq2d |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` ( y - 0 ) ) = ( abs ` y ) ) |
59 |
56 58
|
eqtrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( y ( abs o. - ) 0 ) = ( abs ` y ) ) |
60 |
|
elbl3 |
|- ( ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 1 e. RR* ) /\ ( 0 e. CC /\ y e. CC ) ) -> ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( y ( abs o. - ) 0 ) < 1 ) ) |
61 |
38 39 60
|
mpanl12 |
|- ( ( 0 e. CC /\ y e. CC ) -> ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( y ( abs o. - ) 0 ) < 1 ) ) |
62 |
28 51 61
|
sylancr |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) <-> ( y ( abs o. - ) 0 ) < 1 ) ) |
63 |
62
|
ibi |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( y ( abs o. - ) 0 ) < 1 ) |
64 |
59 63
|
eqbrtrrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` y ) < 1 ) |
65 |
54 64
|
gtned |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 1 =/= ( abs ` y ) ) |
66 |
|
abs1 |
|- ( abs ` 1 ) = 1 |
67 |
|
fveq2 |
|- ( 1 = y -> ( abs ` 1 ) = ( abs ` y ) ) |
68 |
66 67
|
eqtr3id |
|- ( 1 = y -> 1 = ( abs ` y ) ) |
69 |
68
|
necon3i |
|- ( 1 =/= ( abs ` y ) -> 1 =/= y ) |
70 |
65 69
|
syl |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 1 =/= y ) |
71 |
|
subeq0 |
|- ( ( 1 e. CC /\ y e. CC ) -> ( ( 1 - y ) = 0 <-> 1 = y ) ) |
72 |
71
|
necon3bid |
|- ( ( 1 e. CC /\ y e. CC ) -> ( ( 1 - y ) =/= 0 <-> 1 =/= y ) ) |
73 |
48 51 72
|
sylancr |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( 1 - y ) =/= 0 <-> 1 =/= y ) ) |
74 |
70 73
|
mpbird |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 - y ) =/= 0 ) |
75 |
53 74
|
logcld |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( log ` ( 1 - y ) ) e. CC ) |
76 |
75
|
negcld |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> -u ( log ` ( 1 - y ) ) e. CC ) |
77 |
76
|
adantl |
|- ( ( T. /\ y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> -u ( log ` ( 1 - y ) ) e. CC ) |
78 |
77
|
fmpttd |
|- ( T. -> ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) : ( 0 ( ball ` ( abs o. - ) ) 1 ) --> CC ) |
79 |
51
|
absge0d |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 0 <_ ( abs ` y ) ) |
80 |
54
|
rexrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` y ) e. RR* ) |
81 |
|
peano2re |
|- ( ( abs ` y ) e. RR -> ( ( abs ` y ) + 1 ) e. RR ) |
82 |
54 81
|
syl |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( abs ` y ) + 1 ) e. RR ) |
83 |
82
|
rehalfcld |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( ( abs ` y ) + 1 ) / 2 ) e. RR ) |
84 |
83
|
rexrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( ( abs ` y ) + 1 ) / 2 ) e. RR* ) |
85 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
86 |
|
eqeq1 |
|- ( m = j -> ( m = 0 <-> j = 0 ) ) |
87 |
|
oveq2 |
|- ( m = j -> ( 1 / m ) = ( 1 / j ) ) |
88 |
86 87
|
ifbieq2d |
|- ( m = j -> if ( m = 0 , 0 , ( 1 / m ) ) = if ( j = 0 , 0 , ( 1 / j ) ) ) |
89 |
|
eqid |
|- ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) = ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) |
90 |
|
c0ex |
|- 0 e. _V |
91 |
|
ovex |
|- ( 1 / j ) e. _V |
92 |
90 91
|
ifex |
|- if ( j = 0 , 0 , ( 1 / j ) ) e. _V |
93 |
88 89 92
|
fvmpt |
|- ( j e. NN0 -> ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` j ) = if ( j = 0 , 0 , ( 1 / j ) ) ) |
94 |
93
|
eqcomd |
|- ( j e. NN0 -> if ( j = 0 , 0 , ( 1 / j ) ) = ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` j ) ) |
95 |
94
|
oveq1d |
|- ( j e. NN0 -> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) = ( ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` j ) x. ( x ^ j ) ) ) |
96 |
95
|
mpteq2ia |
|- ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) = ( j e. NN0 |-> ( ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` j ) x. ( x ^ j ) ) ) |
97 |
96
|
mpteq2i |
|- ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) = ( x e. CC |-> ( j e. NN0 |-> ( ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` j ) x. ( x ^ j ) ) ) ) |
98 |
|
0cnd |
|- ( ( ( T. /\ m e. NN0 ) /\ m = 0 ) -> 0 e. CC ) |
99 |
|
nn0cn |
|- ( m e. NN0 -> m e. CC ) |
100 |
99
|
adantl |
|- ( ( T. /\ m e. NN0 ) -> m e. CC ) |
101 |
|
neqne |
|- ( -. m = 0 -> m =/= 0 ) |
102 |
|
reccl |
|- ( ( m e. CC /\ m =/= 0 ) -> ( 1 / m ) e. CC ) |
103 |
100 101 102
|
syl2an |
|- ( ( ( T. /\ m e. NN0 ) /\ -. m = 0 ) -> ( 1 / m ) e. CC ) |
104 |
98 103
|
ifclda |
|- ( ( T. /\ m e. NN0 ) -> if ( m = 0 , 0 , ( 1 / m ) ) e. CC ) |
105 |
104
|
fmpttd |
|- ( T. -> ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) : NN0 --> CC ) |
106 |
|
recn |
|- ( r e. RR -> r e. CC ) |
107 |
|
oveq1 |
|- ( x = r -> ( x ^ j ) = ( r ^ j ) ) |
108 |
107
|
oveq2d |
|- ( x = r -> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) = ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) |
109 |
108
|
mpteq2dv |
|- ( x = r -> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) |
110 |
|
eqid |
|- ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) = ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) |
111 |
|
nn0ex |
|- NN0 e. _V |
112 |
111
|
mptex |
|- ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) e. _V |
113 |
109 110 112
|
fvmpt |
|- ( r e. CC -> ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` r ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) |
114 |
106 113
|
syl |
|- ( r e. RR -> ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` r ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) |
115 |
114
|
eqcomd |
|- ( r e. RR -> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) = ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` r ) ) |
116 |
115
|
seqeq3d |
|- ( r e. RR -> seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) = seq 0 ( + , ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` r ) ) ) |
117 |
116
|
eleq1d |
|- ( r e. RR -> ( seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> <-> seq 0 ( + , ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` r ) ) e. dom ~~> ) ) |
118 |
117
|
rabbiia |
|- { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } = { r e. RR | seq 0 ( + , ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` r ) ) e. dom ~~> } |
119 |
118
|
supeq1i |
|- sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) = sup ( { r e. RR | seq 0 ( + , ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` r ) ) e. dom ~~> } , RR* , < ) |
120 |
97 105 119
|
radcnvcl |
|- ( T. -> sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. ( 0 [,] +oo ) ) |
121 |
85 120
|
sselid |
|- ( T. -> sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. RR* ) |
122 |
44 121
|
mp1i |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. RR* ) |
123 |
|
1re |
|- 1 e. RR |
124 |
|
avglt1 |
|- ( ( ( abs ` y ) e. RR /\ 1 e. RR ) -> ( ( abs ` y ) < 1 <-> ( abs ` y ) < ( ( ( abs ` y ) + 1 ) / 2 ) ) ) |
125 |
54 123 124
|
sylancl |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( abs ` y ) < 1 <-> ( abs ` y ) < ( ( ( abs ` y ) + 1 ) / 2 ) ) ) |
126 |
64 125
|
mpbid |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` y ) < ( ( ( abs ` y ) + 1 ) / 2 ) ) |
127 |
|
0red |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 0 e. RR ) |
128 |
127 54 83 79 126
|
lelttrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 0 < ( ( ( abs ` y ) + 1 ) / 2 ) ) |
129 |
127 83 128
|
ltled |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 0 <_ ( ( ( abs ` y ) + 1 ) / 2 ) ) |
130 |
83 129
|
absidd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` ( ( ( abs ` y ) + 1 ) / 2 ) ) = ( ( ( abs ` y ) + 1 ) / 2 ) ) |
131 |
44 105
|
mp1i |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) : NN0 --> CC ) |
132 |
83
|
recnd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( ( abs ` y ) + 1 ) / 2 ) e. CC ) |
133 |
|
oveq1 |
|- ( x = ( ( ( abs ` y ) + 1 ) / 2 ) -> ( x ^ j ) = ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) |
134 |
133
|
oveq2d |
|- ( x = ( ( ( abs ` y ) + 1 ) / 2 ) -> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) = ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) |
135 |
134
|
mpteq2dv |
|- ( x = ( ( ( abs ` y ) + 1 ) / 2 ) -> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) ) |
136 |
111
|
mptex |
|- ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) e. _V |
137 |
135 110 136
|
fvmpt |
|- ( ( ( ( abs ` y ) + 1 ) / 2 ) e. CC -> ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` ( ( ( abs ` y ) + 1 ) / 2 ) ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) ) |
138 |
132 137
|
syl |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` ( ( ( abs ` y ) + 1 ) / 2 ) ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) ) |
139 |
138
|
seqeq3d |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> seq 0 ( + , ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` ( ( ( abs ` y ) + 1 ) / 2 ) ) ) = seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) ) ) |
140 |
|
avglt2 |
|- ( ( ( abs ` y ) e. RR /\ 1 e. RR ) -> ( ( abs ` y ) < 1 <-> ( ( ( abs ` y ) + 1 ) / 2 ) < 1 ) ) |
141 |
54 123 140
|
sylancl |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( abs ` y ) < 1 <-> ( ( ( abs ` y ) + 1 ) / 2 ) < 1 ) ) |
142 |
64 141
|
mpbid |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( ( abs ` y ) + 1 ) / 2 ) < 1 ) |
143 |
130 142
|
eqbrtrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` ( ( ( abs ` y ) + 1 ) / 2 ) ) < 1 ) |
144 |
|
logtayllem |
|- ( ( ( ( ( abs ` y ) + 1 ) / 2 ) e. CC /\ ( abs ` ( ( ( abs ` y ) + 1 ) / 2 ) ) < 1 ) -> seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) ) e. dom ~~> ) |
145 |
132 143 144
|
syl2anc |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( ( ( ( abs ` y ) + 1 ) / 2 ) ^ j ) ) ) ) e. dom ~~> ) |
146 |
139 145
|
eqeltrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> seq 0 ( + , ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` ( ( ( abs ` y ) + 1 ) / 2 ) ) ) e. dom ~~> ) |
147 |
97 131 119 132 146
|
radcnvle |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` ( ( ( abs ` y ) + 1 ) / 2 ) ) <_ sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) |
148 |
130 147
|
eqbrtrrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( ( abs ` y ) + 1 ) / 2 ) <_ sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) |
149 |
80 84 122 126 148
|
xrltletrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` y ) < sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) |
150 |
|
0re |
|- 0 e. RR |
151 |
|
elico2 |
|- ( ( 0 e. RR /\ sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. RR* ) -> ( ( abs ` y ) e. ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) |
152 |
150 122 151
|
sylancr |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( abs ` y ) e. ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) |
153 |
54 79 149 152
|
mpbir3and |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` y ) e. ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |
154 |
|
absf |
|- abs : CC --> RR |
155 |
|
ffn |
|- ( abs : CC --> RR -> abs Fn CC ) |
156 |
|
elpreima |
|- ( abs Fn CC -> ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) <-> ( y e. CC /\ ( abs ` y ) e. ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) ) |
157 |
154 155 156
|
mp2b |
|- ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) <-> ( y e. CC /\ ( abs ` y ) e. ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) |
158 |
51 153 157
|
sylanbrc |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) |
159 |
|
cnvimass |
|- ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) C_ dom abs |
160 |
154
|
fdmi |
|- dom abs = CC |
161 |
159 160
|
sseqtri |
|- ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) C_ CC |
162 |
161
|
sseli |
|- ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) -> y e. CC ) |
163 |
|
oveq1 |
|- ( x = y -> ( x ^ j ) = ( y ^ j ) ) |
164 |
163
|
oveq2d |
|- ( x = y -> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) = ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) |
165 |
164
|
mpteq2dv |
|- ( x = y -> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) ) |
166 |
111
|
mptex |
|- ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) e. _V |
167 |
165 110 166
|
fvmpt |
|- ( y e. CC -> ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) ) |
168 |
167
|
adantr |
|- ( ( y e. CC /\ n e. NN0 ) -> ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) ) |
169 |
168
|
fveq1d |
|- ( ( y e. CC /\ n e. NN0 ) -> ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` n ) = ( ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) ` n ) ) |
170 |
|
eqeq1 |
|- ( j = n -> ( j = 0 <-> n = 0 ) ) |
171 |
|
oveq2 |
|- ( j = n -> ( 1 / j ) = ( 1 / n ) ) |
172 |
170 171
|
ifbieq2d |
|- ( j = n -> if ( j = 0 , 0 , ( 1 / j ) ) = if ( n = 0 , 0 , ( 1 / n ) ) ) |
173 |
|
oveq2 |
|- ( j = n -> ( y ^ j ) = ( y ^ n ) ) |
174 |
172 173
|
oveq12d |
|- ( j = n -> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) |
175 |
|
eqid |
|- ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) = ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) |
176 |
|
ovex |
|- ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) e. _V |
177 |
174 175 176
|
fvmpt |
|- ( n e. NN0 -> ( ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) ` n ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) |
178 |
177
|
adantl |
|- ( ( y e. CC /\ n e. NN0 ) -> ( ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( y ^ j ) ) ) ` n ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) |
179 |
169 178
|
eqtr2d |
|- ( ( y e. CC /\ n e. NN0 ) -> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` n ) ) |
180 |
179
|
sumeq2dv |
|- ( y e. CC -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = sum_ n e. NN0 ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` n ) ) |
181 |
162 180
|
syl |
|- ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = sum_ n e. NN0 ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` n ) ) |
182 |
181
|
mpteq2ia |
|- ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) = ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` n ) ) |
183 |
|
eqid |
|- ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) = ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |
184 |
|
eqid |
|- if ( sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` z ) + sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` z ) + 1 ) ) = if ( sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` z ) + sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` z ) + 1 ) ) |
185 |
97 182 105 119 183 184
|
psercn |
|- ( T. -> ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) e. ( ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) -cn-> CC ) ) |
186 |
|
cncff |
|- ( ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) e. ( ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) -cn-> CC ) -> ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) : ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) --> CC ) |
187 |
185 186
|
syl |
|- ( T. -> ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) : ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) --> CC ) |
188 |
187
|
fvmptelrn |
|- ( ( T. /\ y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) e. CC ) |
189 |
158 188
|
sylan2 |
|- ( ( T. /\ y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) e. CC ) |
190 |
189
|
fmpttd |
|- ( T. -> ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) : ( 0 ( ball ` ( abs o. - ) ) 1 ) --> CC ) |
191 |
|
cnelprrecn |
|- CC e. { RR , CC } |
192 |
191
|
a1i |
|- ( T. -> CC e. { RR , CC } ) |
193 |
75
|
adantl |
|- ( ( T. /\ y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> ( log ` ( 1 - y ) ) e. CC ) |
194 |
|
ovexd |
|- ( ( T. /\ y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> ( ( 1 / ( 1 - y ) ) x. -u 1 ) e. _V ) |
195 |
29
|
cnmetdval |
|- ( ( 1 e. CC /\ ( 1 - y ) e. CC ) -> ( 1 ( abs o. - ) ( 1 - y ) ) = ( abs ` ( 1 - ( 1 - y ) ) ) ) |
196 |
48 53 195
|
sylancr |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 ( abs o. - ) ( 1 - y ) ) = ( abs ` ( 1 - ( 1 - y ) ) ) ) |
197 |
|
nncan |
|- ( ( 1 e. CC /\ y e. CC ) -> ( 1 - ( 1 - y ) ) = y ) |
198 |
48 51 197
|
sylancr |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 - ( 1 - y ) ) = y ) |
199 |
198
|
fveq2d |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( abs ` ( 1 - ( 1 - y ) ) ) = ( abs ` y ) ) |
200 |
196 199
|
eqtrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 ( abs o. - ) ( 1 - y ) ) = ( abs ` y ) ) |
201 |
200 64
|
eqbrtrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 ( abs o. - ) ( 1 - y ) ) < 1 ) |
202 |
|
elbl |
|- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 1 e. CC /\ 1 e. RR* ) -> ( ( 1 - y ) e. ( 1 ( ball ` ( abs o. - ) ) 1 ) <-> ( ( 1 - y ) e. CC /\ ( 1 ( abs o. - ) ( 1 - y ) ) < 1 ) ) ) |
203 |
38 48 39 202
|
mp3an |
|- ( ( 1 - y ) e. ( 1 ( ball ` ( abs o. - ) ) 1 ) <-> ( ( 1 - y ) e. CC /\ ( 1 ( abs o. - ) ( 1 - y ) ) < 1 ) ) |
204 |
53 201 203
|
sylanbrc |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 - y ) e. ( 1 ( ball ` ( abs o. - ) ) 1 ) ) |
205 |
204
|
adantl |
|- ( ( T. /\ y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> ( 1 - y ) e. ( 1 ( ball ` ( abs o. - ) ) 1 ) ) |
206 |
|
neg1cn |
|- -u 1 e. CC |
207 |
206
|
a1i |
|- ( ( T. /\ y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) -> -u 1 e. CC ) |
208 |
|
eqid |
|- ( 1 ( ball ` ( abs o. - ) ) 1 ) = ( 1 ( ball ` ( abs o. - ) ) 1 ) |
209 |
208
|
dvlog2lem |
|- ( 1 ( ball ` ( abs o. - ) ) 1 ) C_ ( CC \ ( -oo (,] 0 ) ) |
210 |
209
|
sseli |
|- ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) -> x e. ( CC \ ( -oo (,] 0 ) ) ) |
211 |
210
|
eldifad |
|- ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) -> x e. CC ) |
212 |
|
eqid |
|- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) ) |
213 |
212
|
logdmn0 |
|- ( x e. ( CC \ ( -oo (,] 0 ) ) -> x =/= 0 ) |
214 |
210 213
|
syl |
|- ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) -> x =/= 0 ) |
215 |
211 214
|
logcld |
|- ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) -> ( log ` x ) e. CC ) |
216 |
215
|
adantl |
|- ( ( T. /\ x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) ) -> ( log ` x ) e. CC ) |
217 |
|
ovexd |
|- ( ( T. /\ x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) ) -> ( 1 / x ) e. _V ) |
218 |
|
simpr |
|- ( ( T. /\ y e. CC ) -> y e. CC ) |
219 |
48 218 52
|
sylancr |
|- ( ( T. /\ y e. CC ) -> ( 1 - y ) e. CC ) |
220 |
206
|
a1i |
|- ( ( T. /\ y e. CC ) -> -u 1 e. CC ) |
221 |
|
1cnd |
|- ( ( T. /\ y e. CC ) -> 1 e. CC ) |
222 |
|
0cnd |
|- ( ( T. /\ y e. CC ) -> 0 e. CC ) |
223 |
|
1cnd |
|- ( T. -> 1 e. CC ) |
224 |
192 223
|
dvmptc |
|- ( T. -> ( CC _D ( y e. CC |-> 1 ) ) = ( y e. CC |-> 0 ) ) |
225 |
192
|
dvmptid |
|- ( T. -> ( CC _D ( y e. CC |-> y ) ) = ( y e. CC |-> 1 ) ) |
226 |
192 221 222 224 218 221 225
|
dvmptsub |
|- ( T. -> ( CC _D ( y e. CC |-> ( 1 - y ) ) ) = ( y e. CC |-> ( 0 - 1 ) ) ) |
227 |
|
df-neg |
|- -u 1 = ( 0 - 1 ) |
228 |
227
|
mpteq2i |
|- ( y e. CC |-> -u 1 ) = ( y e. CC |-> ( 0 - 1 ) ) |
229 |
226 228
|
eqtr4di |
|- ( T. -> ( CC _D ( y e. CC |-> ( 1 - y ) ) ) = ( y e. CC |-> -u 1 ) ) |
230 |
50
|
a1i |
|- ( T. -> ( 0 ( ball ` ( abs o. - ) ) 1 ) C_ CC ) |
231 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
232 |
231
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
233 |
232
|
toponrestid |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
234 |
231
|
cnfldtopn |
|- ( TopOpen ` CCfld ) = ( MetOpen ` ( abs o. - ) ) |
235 |
234
|
blopn |
|- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ 1 e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) 1 ) e. ( TopOpen ` CCfld ) ) |
236 |
38 28 39 235
|
mp3an |
|- ( 0 ( ball ` ( abs o. - ) ) 1 ) e. ( TopOpen ` CCfld ) |
237 |
236
|
a1i |
|- ( T. -> ( 0 ( ball ` ( abs o. - ) ) 1 ) e. ( TopOpen ` CCfld ) ) |
238 |
192 219 220 229 230 233 231 237
|
dvmptres |
|- ( T. -> ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 - y ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u 1 ) ) |
239 |
|
logf1o |
|- log : ( CC \ { 0 } ) -1-1-onto-> ran log |
240 |
|
f1of |
|- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) --> ran log ) |
241 |
239 240
|
ax-mp |
|- log : ( CC \ { 0 } ) --> ran log |
242 |
212
|
logdmss |
|- ( CC \ ( -oo (,] 0 ) ) C_ ( CC \ { 0 } ) |
243 |
209 242
|
sstri |
|- ( 1 ( ball ` ( abs o. - ) ) 1 ) C_ ( CC \ { 0 } ) |
244 |
|
fssres |
|- ( ( log : ( CC \ { 0 } ) --> ran log /\ ( 1 ( ball ` ( abs o. - ) ) 1 ) C_ ( CC \ { 0 } ) ) -> ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) : ( 1 ( ball ` ( abs o. - ) ) 1 ) --> ran log ) |
245 |
241 243 244
|
mp2an |
|- ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) : ( 1 ( ball ` ( abs o. - ) ) 1 ) --> ran log |
246 |
245
|
a1i |
|- ( T. -> ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) : ( 1 ( ball ` ( abs o. - ) ) 1 ) --> ran log ) |
247 |
246
|
feqmptd |
|- ( T. -> ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) = ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) ` x ) ) ) |
248 |
|
fvres |
|- ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) -> ( ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) ` x ) = ( log ` x ) ) |
249 |
248
|
mpteq2ia |
|- ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) ` x ) ) = ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( log ` x ) ) |
250 |
247 249
|
eqtrdi |
|- ( T. -> ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) = ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( log ` x ) ) ) |
251 |
250
|
oveq2d |
|- ( T. -> ( CC _D ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) ) = ( CC _D ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( log ` x ) ) ) ) |
252 |
208
|
dvlog2 |
|- ( CC _D ( log |` ( 1 ( ball ` ( abs o. - ) ) 1 ) ) ) = ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / x ) ) |
253 |
251 252
|
eqtr3di |
|- ( T. -> ( CC _D ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( log ` x ) ) ) = ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / x ) ) ) |
254 |
|
fveq2 |
|- ( x = ( 1 - y ) -> ( log ` x ) = ( log ` ( 1 - y ) ) ) |
255 |
|
oveq2 |
|- ( x = ( 1 - y ) -> ( 1 / x ) = ( 1 / ( 1 - y ) ) ) |
256 |
192 192 205 207 216 217 238 253 254 255
|
dvmptco |
|- ( T. -> ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( log ` ( 1 - y ) ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( ( 1 / ( 1 - y ) ) x. -u 1 ) ) ) |
257 |
192 193 194 256
|
dvmptneg |
|- ( T. -> ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( ( 1 / ( 1 - y ) ) x. -u 1 ) ) ) |
258 |
53 74
|
reccld |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 / ( 1 - y ) ) e. CC ) |
259 |
|
mulcom |
|- ( ( ( 1 / ( 1 - y ) ) e. CC /\ -u 1 e. CC ) -> ( ( 1 / ( 1 - y ) ) x. -u 1 ) = ( -u 1 x. ( 1 / ( 1 - y ) ) ) ) |
260 |
258 206 259
|
sylancl |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( 1 / ( 1 - y ) ) x. -u 1 ) = ( -u 1 x. ( 1 / ( 1 - y ) ) ) ) |
261 |
258
|
mulm1d |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( -u 1 x. ( 1 / ( 1 - y ) ) ) = -u ( 1 / ( 1 - y ) ) ) |
262 |
260 261
|
eqtrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( 1 / ( 1 - y ) ) x. -u 1 ) = -u ( 1 / ( 1 - y ) ) ) |
263 |
262
|
negeqd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> -u ( ( 1 / ( 1 - y ) ) x. -u 1 ) = -u -u ( 1 / ( 1 - y ) ) ) |
264 |
258
|
negnegd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> -u -u ( 1 / ( 1 - y ) ) = ( 1 / ( 1 - y ) ) ) |
265 |
263 264
|
eqtrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> -u ( ( 1 / ( 1 - y ) ) x. -u 1 ) = ( 1 / ( 1 - y ) ) ) |
266 |
265
|
mpteq2ia |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( ( 1 / ( 1 - y ) ) x. -u 1 ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / ( 1 - y ) ) ) |
267 |
257 266
|
eqtrdi |
|- ( T. -> ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / ( 1 - y ) ) ) ) |
268 |
267
|
dmeqd |
|- ( T. -> dom ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ) = dom ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / ( 1 - y ) ) ) ) |
269 |
|
dmmptg |
|- ( A. y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ( 1 / ( 1 - y ) ) e. _V -> dom ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / ( 1 - y ) ) ) = ( 0 ( ball ` ( abs o. - ) ) 1 ) ) |
270 |
|
ovexd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( 1 / ( 1 - y ) ) e. _V ) |
271 |
269 270
|
mprg |
|- dom ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / ( 1 - y ) ) ) = ( 0 ( ball ` ( abs o. - ) ) 1 ) |
272 |
268 271
|
eqtrdi |
|- ( T. -> dom ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ) = ( 0 ( ball ` ( abs o. - ) ) 1 ) ) |
273 |
|
sumex |
|- sum_ n e. NN ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) e. _V |
274 |
273
|
a1i |
|- ( ( T. /\ y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) -> sum_ n e. NN ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) e. _V ) |
275 |
|
fveq2 |
|- ( n = k -> ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` n ) = ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` k ) ) |
276 |
275
|
cbvsumv |
|- sum_ n e. NN0 ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` n ) = sum_ k e. NN0 ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` k ) |
277 |
181 276
|
eqtrdi |
|- ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = sum_ k e. NN0 ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` k ) ) |
278 |
277
|
mpteq2ia |
|- ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) = ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ k e. NN0 ( ( ( x e. CC |-> ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( x ^ j ) ) ) ) ` y ) ` k ) ) |
279 |
|
eqid |
|- ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` z ) + if ( sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` z ) + sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` z ) + 1 ) ) ) / 2 ) ) = ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` z ) + if ( sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` z ) + sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` z ) + 1 ) ) ) / 2 ) ) |
280 |
97 278 105 119 183 184 279
|
pserdv2 |
|- ( T. -> ( CC _D ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ) = ( y e. ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |-> sum_ n e. NN ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) ) |
281 |
158
|
ssriv |
|- ( 0 ( ball ` ( abs o. - ) ) 1 ) C_ ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) |
282 |
281
|
a1i |
|- ( T. -> ( 0 ( ball ` ( abs o. - ) ) 1 ) C_ ( `' abs " ( 0 [,) sup ( { r e. RR | seq 0 ( + , ( j e. NN0 |-> ( if ( j = 0 , 0 , ( 1 / j ) ) x. ( r ^ j ) ) ) ) e. dom ~~> } , RR* , < ) ) ) ) |
283 |
192 188 274 280 282 233 231 237
|
dvmptres |
|- ( T. -> ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) ) |
284 |
|
nnnn0 |
|- ( n e. NN -> n e. NN0 ) |
285 |
284
|
adantl |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> n e. NN0 ) |
286 |
|
eqeq1 |
|- ( m = n -> ( m = 0 <-> n = 0 ) ) |
287 |
|
oveq2 |
|- ( m = n -> ( 1 / m ) = ( 1 / n ) ) |
288 |
286 287
|
ifbieq2d |
|- ( m = n -> if ( m = 0 , 0 , ( 1 / m ) ) = if ( n = 0 , 0 , ( 1 / n ) ) ) |
289 |
|
ovex |
|- ( 1 / n ) e. _V |
290 |
90 289
|
ifex |
|- if ( n = 0 , 0 , ( 1 / n ) ) e. _V |
291 |
288 89 290
|
fvmpt |
|- ( n e. NN0 -> ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) = if ( n = 0 , 0 , ( 1 / n ) ) ) |
292 |
285 291
|
syl |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) = if ( n = 0 , 0 , ( 1 / n ) ) ) |
293 |
|
nnne0 |
|- ( n e. NN -> n =/= 0 ) |
294 |
293
|
adantl |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> n =/= 0 ) |
295 |
294
|
neneqd |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> -. n = 0 ) |
296 |
295
|
iffalsed |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> if ( n = 0 , 0 , ( 1 / n ) ) = ( 1 / n ) ) |
297 |
292 296
|
eqtrd |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) = ( 1 / n ) ) |
298 |
297
|
oveq2d |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) = ( n x. ( 1 / n ) ) ) |
299 |
|
nncn |
|- ( n e. NN -> n e. CC ) |
300 |
299
|
adantl |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> n e. CC ) |
301 |
300 294
|
recidd |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( n x. ( 1 / n ) ) = 1 ) |
302 |
298 301
|
eqtrd |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) = 1 ) |
303 |
302
|
oveq1d |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) = ( 1 x. ( y ^ ( n - 1 ) ) ) ) |
304 |
|
nnm1nn0 |
|- ( n e. NN -> ( n - 1 ) e. NN0 ) |
305 |
|
expcl |
|- ( ( y e. CC /\ ( n - 1 ) e. NN0 ) -> ( y ^ ( n - 1 ) ) e. CC ) |
306 |
51 304 305
|
syl2an |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( y ^ ( n - 1 ) ) e. CC ) |
307 |
306
|
mulid2d |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( 1 x. ( y ^ ( n - 1 ) ) ) = ( y ^ ( n - 1 ) ) ) |
308 |
303 307
|
eqtrd |
|- ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) /\ n e. NN ) -> ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) = ( y ^ ( n - 1 ) ) ) |
309 |
308
|
sumeq2dv |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> sum_ n e. NN ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) = sum_ n e. NN ( y ^ ( n - 1 ) ) ) |
310 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
311 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
312 |
311
|
fveq2i |
|- ( ZZ>= ` 1 ) = ( ZZ>= ` ( 0 + 1 ) ) |
313 |
310 312
|
eqtri |
|- NN = ( ZZ>= ` ( 0 + 1 ) ) |
314 |
|
oveq1 |
|- ( n = ( 1 + m ) -> ( n - 1 ) = ( ( 1 + m ) - 1 ) ) |
315 |
314
|
oveq2d |
|- ( n = ( 1 + m ) -> ( y ^ ( n - 1 ) ) = ( y ^ ( ( 1 + m ) - 1 ) ) ) |
316 |
|
1zzd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 1 e. ZZ ) |
317 |
|
0zd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> 0 e. ZZ ) |
318 |
1 313 315 316 317 306
|
isumshft |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> sum_ n e. NN ( y ^ ( n - 1 ) ) = sum_ m e. NN0 ( y ^ ( ( 1 + m ) - 1 ) ) ) |
319 |
|
pncan2 |
|- ( ( 1 e. CC /\ m e. CC ) -> ( ( 1 + m ) - 1 ) = m ) |
320 |
48 99 319
|
sylancr |
|- ( m e. NN0 -> ( ( 1 + m ) - 1 ) = m ) |
321 |
320
|
oveq2d |
|- ( m e. NN0 -> ( y ^ ( ( 1 + m ) - 1 ) ) = ( y ^ m ) ) |
322 |
321
|
sumeq2i |
|- sum_ m e. NN0 ( y ^ ( ( 1 + m ) - 1 ) ) = sum_ m e. NN0 ( y ^ m ) |
323 |
318 322
|
eqtrdi |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> sum_ n e. NN ( y ^ ( n - 1 ) ) = sum_ m e. NN0 ( y ^ m ) ) |
324 |
|
geoisum |
|- ( ( y e. CC /\ ( abs ` y ) < 1 ) -> sum_ m e. NN0 ( y ^ m ) = ( 1 / ( 1 - y ) ) ) |
325 |
51 64 324
|
syl2anc |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> sum_ m e. NN0 ( y ^ m ) = ( 1 / ( 1 - y ) ) ) |
326 |
309 323 325
|
3eqtrd |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> sum_ n e. NN ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) = ( 1 / ( 1 - y ) ) ) |
327 |
326
|
mpteq2ia |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN ( ( n x. ( ( m e. NN0 |-> if ( m = 0 , 0 , ( 1 / m ) ) ) ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / ( 1 - y ) ) ) |
328 |
283 327
|
eqtrdi |
|- ( T. -> ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> ( 1 / ( 1 - y ) ) ) ) |
329 |
267 328
|
eqtr4d |
|- ( T. -> ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ) = ( CC _D ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ) ) |
330 |
|
1rp |
|- 1 e. RR+ |
331 |
|
blcntr |
|- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ 1 e. RR+ ) -> 0 e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) |
332 |
38 28 330 331
|
mp3an |
|- 0 e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |
333 |
332
|
a1i |
|- ( T. -> 0 e. ( 0 ( ball ` ( abs o. - ) ) 1 ) ) |
334 |
|
oveq2 |
|- ( y = 0 -> ( 1 - y ) = ( 1 - 0 ) ) |
335 |
|
1m0e1 |
|- ( 1 - 0 ) = 1 |
336 |
334 335
|
eqtrdi |
|- ( y = 0 -> ( 1 - y ) = 1 ) |
337 |
336
|
fveq2d |
|- ( y = 0 -> ( log ` ( 1 - y ) ) = ( log ` 1 ) ) |
338 |
|
log1 |
|- ( log ` 1 ) = 0 |
339 |
337 338
|
eqtrdi |
|- ( y = 0 -> ( log ` ( 1 - y ) ) = 0 ) |
340 |
339
|
negeqd |
|- ( y = 0 -> -u ( log ` ( 1 - y ) ) = -u 0 ) |
341 |
|
neg0 |
|- -u 0 = 0 |
342 |
340 341
|
eqtrdi |
|- ( y = 0 -> -u ( log ` ( 1 - y ) ) = 0 ) |
343 |
|
eqid |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) |
344 |
342 343 90
|
fvmpt |
|- ( 0 e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ` 0 ) = 0 ) |
345 |
332 344
|
mp1i |
|- ( T. -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ` 0 ) = 0 ) |
346 |
|
oveq1 |
|- ( 0 = if ( n = 0 , 0 , ( 1 / n ) ) -> ( 0 x. ( y ^ n ) ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) |
347 |
346
|
eqeq1d |
|- ( 0 = if ( n = 0 , 0 , ( 1 / n ) ) -> ( ( 0 x. ( y ^ n ) ) = 0 <-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = 0 ) ) |
348 |
|
oveq1 |
|- ( ( 1 / n ) = if ( n = 0 , 0 , ( 1 / n ) ) -> ( ( 1 / n ) x. ( y ^ n ) ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) |
349 |
348
|
eqeq1d |
|- ( ( 1 / n ) = if ( n = 0 , 0 , ( 1 / n ) ) -> ( ( ( 1 / n ) x. ( y ^ n ) ) = 0 <-> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = 0 ) ) |
350 |
|
simpll |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ n = 0 ) -> y = 0 ) |
351 |
350 28
|
eqeltrdi |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ n = 0 ) -> y e. CC ) |
352 |
|
simplr |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ n = 0 ) -> n e. NN0 ) |
353 |
351 352
|
expcld |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ n = 0 ) -> ( y ^ n ) e. CC ) |
354 |
353
|
mul02d |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ n = 0 ) -> ( 0 x. ( y ^ n ) ) = 0 ) |
355 |
|
simpll |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> y = 0 ) |
356 |
355
|
oveq1d |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( y ^ n ) = ( 0 ^ n ) ) |
357 |
|
simpr |
|- ( ( y = 0 /\ n e. NN0 ) -> n e. NN0 ) |
358 |
357 14
|
sylib |
|- ( ( y = 0 /\ n e. NN0 ) -> ( n e. NN \/ n = 0 ) ) |
359 |
358
|
ord |
|- ( ( y = 0 /\ n e. NN0 ) -> ( -. n e. NN -> n = 0 ) ) |
360 |
359
|
con1d |
|- ( ( y = 0 /\ n e. NN0 ) -> ( -. n = 0 -> n e. NN ) ) |
361 |
360
|
imp |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN ) |
362 |
361
|
0expd |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( 0 ^ n ) = 0 ) |
363 |
356 362
|
eqtrd |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( y ^ n ) = 0 ) |
364 |
363
|
oveq2d |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( ( 1 / n ) x. ( y ^ n ) ) = ( ( 1 / n ) x. 0 ) ) |
365 |
361
|
nnrecred |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( 1 / n ) e. RR ) |
366 |
365
|
recnd |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( 1 / n ) e. CC ) |
367 |
366
|
mul01d |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( ( 1 / n ) x. 0 ) = 0 ) |
368 |
364 367
|
eqtrd |
|- ( ( ( y = 0 /\ n e. NN0 ) /\ -. n = 0 ) -> ( ( 1 / n ) x. ( y ^ n ) ) = 0 ) |
369 |
347 349 354 368
|
ifbothda |
|- ( ( y = 0 /\ n e. NN0 ) -> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = 0 ) |
370 |
369
|
sumeq2dv |
|- ( y = 0 -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = sum_ n e. NN0 0 ) |
371 |
1
|
eqimssi |
|- NN0 C_ ( ZZ>= ` 0 ) |
372 |
371
|
orci |
|- ( NN0 C_ ( ZZ>= ` 0 ) \/ NN0 e. Fin ) |
373 |
|
sumz |
|- ( ( NN0 C_ ( ZZ>= ` 0 ) \/ NN0 e. Fin ) -> sum_ n e. NN0 0 = 0 ) |
374 |
372 373
|
ax-mp |
|- sum_ n e. NN0 0 = 0 |
375 |
370 374
|
eqtrdi |
|- ( y = 0 -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = 0 ) |
376 |
|
eqid |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) |
377 |
375 376 90
|
fvmpt |
|- ( 0 e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ` 0 ) = 0 ) |
378 |
332 377
|
mp1i |
|- ( T. -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ` 0 ) = 0 ) |
379 |
345 378
|
eqtr4d |
|- ( T. -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ` 0 ) = ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ` 0 ) ) |
380 |
45 46 47 78 190 272 329 333 379
|
dv11cn |
|- ( T. -> ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ) |
381 |
380
|
fveq1d |
|- ( T. -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ` A ) = ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ` A ) ) |
382 |
44 381
|
mp1i |
|- ( A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ` A ) = ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ` A ) ) |
383 |
|
oveq2 |
|- ( y = A -> ( 1 - y ) = ( 1 - A ) ) |
384 |
383
|
fveq2d |
|- ( y = A -> ( log ` ( 1 - y ) ) = ( log ` ( 1 - A ) ) ) |
385 |
384
|
negeqd |
|- ( y = A -> -u ( log ` ( 1 - y ) ) = -u ( log ` ( 1 - A ) ) ) |
386 |
|
negex |
|- -u ( log ` ( 1 - A ) ) e. _V |
387 |
385 343 386
|
fvmpt |
|- ( A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> -u ( log ` ( 1 - y ) ) ) ` A ) = -u ( log ` ( 1 - A ) ) ) |
388 |
|
oveq1 |
|- ( y = A -> ( y ^ n ) = ( A ^ n ) ) |
389 |
388
|
oveq2d |
|- ( y = A -> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
390 |
389
|
sumeq2sdv |
|- ( y = A -> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) = sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
391 |
|
sumex |
|- sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) e. _V |
392 |
390 376 391
|
fvmpt |
|- ( A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> ( ( y e. ( 0 ( ball ` ( abs o. - ) ) 1 ) |-> sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( y ^ n ) ) ) ` A ) = sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
393 |
382 387 392
|
3eqtr3d |
|- ( A e. ( 0 ( ball ` ( abs o. - ) ) 1 ) -> -u ( log ` ( 1 - A ) ) = sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
394 |
43 393
|
syl |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> -u ( log ` ( 1 - A ) ) = sum_ n e. NN0 ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
395 |
26 394
|
breqtrrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ~~> -u ( log ` ( 1 - A ) ) ) |
396 |
|
seqex |
|- seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) e. _V |
397 |
396
|
a1i |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) e. _V ) |
398 |
|
seqex |
|- seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) e. _V |
399 |
398
|
a1i |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) e. _V ) |
400 |
|
1zzd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> 1 e. ZZ ) |
401 |
|
elnnuz |
|- ( n e. NN <-> n e. ( ZZ>= ` 1 ) ) |
402 |
|
fvres |
|- ( n e. ( ZZ>= ` 1 ) -> ( ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) |` ( ZZ>= ` 1 ) ) ` n ) = ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ` n ) ) |
403 |
401 402
|
sylbi |
|- ( n e. NN -> ( ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) |` ( ZZ>= ` 1 ) ) ` n ) = ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ` n ) ) |
404 |
403
|
eqcomd |
|- ( n e. NN -> ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ` n ) = ( ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) |` ( ZZ>= ` 1 ) ) ` n ) ) |
405 |
|
addid2 |
|- ( n e. CC -> ( 0 + n ) = n ) |
406 |
405
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. CC ) -> ( 0 + n ) = n ) |
407 |
|
0cnd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> 0 e. CC ) |
408 |
|
1eluzge0 |
|- 1 e. ( ZZ>= ` 0 ) |
409 |
408
|
a1i |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> 1 e. ( ZZ>= ` 0 ) ) |
410 |
|
0cnd |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) /\ k = 0 ) -> 0 e. CC ) |
411 |
|
nn0cn |
|- ( k e. NN0 -> k e. CC ) |
412 |
411
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> k e. CC ) |
413 |
|
neqne |
|- ( -. k = 0 -> k =/= 0 ) |
414 |
|
reccl |
|- ( ( k e. CC /\ k =/= 0 ) -> ( 1 / k ) e. CC ) |
415 |
412 413 414
|
syl2an |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) /\ -. k = 0 ) -> ( 1 / k ) e. CC ) |
416 |
410 415
|
ifclda |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> if ( k = 0 , 0 , ( 1 / k ) ) e. CC ) |
417 |
|
expcl |
|- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC ) |
418 |
417
|
adantlr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( A ^ k ) e. CC ) |
419 |
416 418
|
mulcld |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN0 ) -> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) e. CC ) |
420 |
419
|
fmpttd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) : NN0 --> CC ) |
421 |
|
1nn0 |
|- 1 e. NN0 |
422 |
|
ffvelrn |
|- ( ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) : NN0 --> CC /\ 1 e. NN0 ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` 1 ) e. CC ) |
423 |
420 421 422
|
sylancl |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` 1 ) e. CC ) |
424 |
|
elfz1eq |
|- ( n e. ( 0 ... 0 ) -> n = 0 ) |
425 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
426 |
425
|
oveq2i |
|- ( 0 ... ( 1 - 1 ) ) = ( 0 ... 0 ) |
427 |
424 426
|
eleq2s |
|- ( n e. ( 0 ... ( 1 - 1 ) ) -> n = 0 ) |
428 |
427
|
fveq2d |
|- ( n e. ( 0 ... ( 1 - 1 ) ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` n ) = ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` 0 ) ) |
429 |
|
0nn0 |
|- 0 e. NN0 |
430 |
|
iftrue |
|- ( k = 0 -> if ( k = 0 , 0 , ( 1 / k ) ) = 0 ) |
431 |
|
oveq2 |
|- ( k = 0 -> ( A ^ k ) = ( A ^ 0 ) ) |
432 |
430 431
|
oveq12d |
|- ( k = 0 -> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) = ( 0 x. ( A ^ 0 ) ) ) |
433 |
|
ovex |
|- ( 0 x. ( A ^ 0 ) ) e. _V |
434 |
432 8 433
|
fvmpt |
|- ( 0 e. NN0 -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` 0 ) = ( 0 x. ( A ^ 0 ) ) ) |
435 |
429 434
|
ax-mp |
|- ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` 0 ) = ( 0 x. ( A ^ 0 ) ) |
436 |
|
expcl |
|- ( ( A e. CC /\ 0 e. NN0 ) -> ( A ^ 0 ) e. CC ) |
437 |
27 429 436
|
sylancl |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( A ^ 0 ) e. CC ) |
438 |
437
|
mul02d |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( 0 x. ( A ^ 0 ) ) = 0 ) |
439 |
435 438
|
syl5eq |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` 0 ) = 0 ) |
440 |
428 439
|
sylan9eqr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` n ) = 0 ) |
441 |
406 407 409 423 440
|
seqid |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ) |
442 |
293
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> n =/= 0 ) |
443 |
442
|
neneqd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> -. n = 0 ) |
444 |
443
|
iffalsed |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> if ( n = 0 , 0 , ( 1 / n ) ) = ( 1 / n ) ) |
445 |
444
|
oveq1d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) = ( ( 1 / n ) x. ( A ^ n ) ) ) |
446 |
284 23
|
sylan2 |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( A ^ n ) e. CC ) |
447 |
299
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> n e. CC ) |
448 |
446 447 442
|
divrec2d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( ( A ^ n ) / n ) = ( ( 1 / n ) x. ( A ^ n ) ) ) |
449 |
445 448
|
eqtr4d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) = ( ( A ^ n ) / n ) ) |
450 |
284 11
|
sylan2 |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` n ) = ( if ( n = 0 , 0 , ( 1 / n ) ) x. ( A ^ n ) ) ) |
451 |
|
id |
|- ( k = n -> k = n ) |
452 |
6 451
|
oveq12d |
|- ( k = n -> ( ( A ^ k ) / k ) = ( ( A ^ n ) / n ) ) |
453 |
|
eqid |
|- ( k e. NN |-> ( ( A ^ k ) / k ) ) = ( k e. NN |-> ( ( A ^ k ) / k ) ) |
454 |
|
ovex |
|- ( ( A ^ n ) / n ) e. _V |
455 |
452 453 454
|
fvmpt |
|- ( n e. NN -> ( ( k e. NN |-> ( ( A ^ k ) / k ) ) ` n ) = ( ( A ^ n ) / n ) ) |
456 |
455
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( ( k e. NN |-> ( ( A ^ k ) / k ) ) ` n ) = ( ( A ^ n ) / n ) ) |
457 |
449 450 456
|
3eqtr4d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` n ) = ( ( k e. NN |-> ( ( A ^ k ) / k ) ) ` n ) ) |
458 |
401 457
|
sylan2br |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. ( ZZ>= ` 1 ) ) -> ( ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ` n ) = ( ( k e. NN |-> ( ( A ^ k ) / k ) ) ` n ) ) |
459 |
400 458
|
seqfeq |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) = seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) ) |
460 |
441 459
|
eqtrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) ) |
461 |
460
|
fveq1d |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) |` ( ZZ>= ` 1 ) ) ` n ) = ( seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) ` n ) ) |
462 |
404 461
|
sylan9eqr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ n e. NN ) -> ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ` n ) = ( seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) ` n ) ) |
463 |
310 397 399 400 462
|
climeq |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( seq 0 ( + , ( k e. NN0 |-> ( if ( k = 0 , 0 , ( 1 / k ) ) x. ( A ^ k ) ) ) ) ~~> -u ( log ` ( 1 - A ) ) <-> seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) ~~> -u ( log ` ( 1 - A ) ) ) ) |
464 |
395 463
|
mpbid |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( k e. NN |-> ( ( A ^ k ) / k ) ) ) ~~> -u ( log ` ( 1 - A ) ) ) |