Step |
Hyp |
Ref |
Expression |
1 |
|
atantayl.1 |
|- F = ( n e. NN |-> ( ( ( _i x. ( ( -u _i ^ n ) - ( _i ^ n ) ) ) / 2 ) x. ( ( A ^ n ) / n ) ) ) |
2 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
3 |
|
1zzd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> 1 e. ZZ ) |
4 |
|
ax-icn |
|- _i e. CC |
5 |
|
halfcl |
|- ( _i e. CC -> ( _i / 2 ) e. CC ) |
6 |
4 5
|
mp1i |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( _i / 2 ) e. CC ) |
7 |
|
simpl |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> A e. CC ) |
8 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
9 |
4 7 8
|
sylancr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( _i x. A ) e. CC ) |
10 |
9
|
negcld |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> -u ( _i x. A ) e. CC ) |
11 |
9
|
absnegd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` -u ( _i x. A ) ) = ( abs ` ( _i x. A ) ) ) |
12 |
|
absmul |
|- ( ( _i e. CC /\ A e. CC ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) ) |
13 |
4 7 12
|
sylancr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) ) |
14 |
|
absi |
|- ( abs ` _i ) = 1 |
15 |
14
|
oveq1i |
|- ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. ( abs ` A ) ) |
16 |
|
abscl |
|- ( A e. CC -> ( abs ` A ) e. RR ) |
17 |
16
|
adantr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` A ) e. RR ) |
18 |
17
|
recnd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` A ) e. CC ) |
19 |
18
|
mulid2d |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( 1 x. ( abs ` A ) ) = ( abs ` A ) ) |
20 |
15 19
|
syl5eq |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( ( abs ` _i ) x. ( abs ` A ) ) = ( abs ` A ) ) |
21 |
11 13 20
|
3eqtrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` -u ( _i x. A ) ) = ( abs ` A ) ) |
22 |
|
simpr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` A ) < 1 ) |
23 |
21 22
|
eqbrtrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` -u ( _i x. A ) ) < 1 ) |
24 |
|
logtayl |
|- ( ( -u ( _i x. A ) e. CC /\ ( abs ` -u ( _i x. A ) ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ) ~~> -u ( log ` ( 1 - -u ( _i x. A ) ) ) ) |
25 |
10 23 24
|
syl2anc |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ) ~~> -u ( log ` ( 1 - -u ( _i x. A ) ) ) ) |
26 |
|
ax-1cn |
|- 1 e. CC |
27 |
|
subneg |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - -u ( _i x. A ) ) = ( 1 + ( _i x. A ) ) ) |
28 |
26 9 27
|
sylancr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( 1 - -u ( _i x. A ) ) = ( 1 + ( _i x. A ) ) ) |
29 |
28
|
fveq2d |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( log ` ( 1 - -u ( _i x. A ) ) ) = ( log ` ( 1 + ( _i x. A ) ) ) ) |
30 |
29
|
negeqd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> -u ( log ` ( 1 - -u ( _i x. A ) ) ) = -u ( log ` ( 1 + ( _i x. A ) ) ) ) |
31 |
25 30
|
breqtrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ) ~~> -u ( log ` ( 1 + ( _i x. A ) ) ) ) |
32 |
|
seqex |
|- seq 1 ( + , ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ) e. _V |
33 |
32
|
a1i |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ) e. _V ) |
34 |
11 23
|
eqbrtrrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( abs ` ( _i x. A ) ) < 1 ) |
35 |
|
logtayl |
|- ( ( ( _i x. A ) e. CC /\ ( abs ` ( _i x. A ) ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ) ~~> -u ( log ` ( 1 - ( _i x. A ) ) ) ) |
36 |
9 34 35
|
syl2anc |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ) ~~> -u ( log ` ( 1 - ( _i x. A ) ) ) ) |
37 |
|
oveq2 |
|- ( n = m -> ( -u ( _i x. A ) ^ n ) = ( -u ( _i x. A ) ^ m ) ) |
38 |
|
id |
|- ( n = m -> n = m ) |
39 |
37 38
|
oveq12d |
|- ( n = m -> ( ( -u ( _i x. A ) ^ n ) / n ) = ( ( -u ( _i x. A ) ^ m ) / m ) ) |
40 |
|
eqid |
|- ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) = ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) |
41 |
|
ovex |
|- ( ( -u ( _i x. A ) ^ m ) / m ) e. _V |
42 |
39 40 41
|
fvmpt |
|- ( m e. NN -> ( ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ` m ) = ( ( -u ( _i x. A ) ^ m ) / m ) ) |
43 |
42
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ` m ) = ( ( -u ( _i x. A ) ^ m ) / m ) ) |
44 |
|
nnnn0 |
|- ( m e. NN -> m e. NN0 ) |
45 |
|
expcl |
|- ( ( -u ( _i x. A ) e. CC /\ m e. NN0 ) -> ( -u ( _i x. A ) ^ m ) e. CC ) |
46 |
10 44 45
|
syl2an |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( -u ( _i x. A ) ^ m ) e. CC ) |
47 |
|
nncn |
|- ( m e. NN -> m e. CC ) |
48 |
47
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> m e. CC ) |
49 |
|
nnne0 |
|- ( m e. NN -> m =/= 0 ) |
50 |
49
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> m =/= 0 ) |
51 |
46 48 50
|
divcld |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( -u ( _i x. A ) ^ m ) / m ) e. CC ) |
52 |
43 51
|
eqeltrd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ` m ) e. CC ) |
53 |
2 3 52
|
serf |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ) : NN --> CC ) |
54 |
53
|
ffvelrnda |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( seq 1 ( + , ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ) ` k ) e. CC ) |
55 |
|
oveq2 |
|- ( n = m -> ( ( _i x. A ) ^ n ) = ( ( _i x. A ) ^ m ) ) |
56 |
55 38
|
oveq12d |
|- ( n = m -> ( ( ( _i x. A ) ^ n ) / n ) = ( ( ( _i x. A ) ^ m ) / m ) ) |
57 |
|
eqid |
|- ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) = ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) |
58 |
|
ovex |
|- ( ( ( _i x. A ) ^ m ) / m ) e. _V |
59 |
56 57 58
|
fvmpt |
|- ( m e. NN -> ( ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ` m ) = ( ( ( _i x. A ) ^ m ) / m ) ) |
60 |
59
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ` m ) = ( ( ( _i x. A ) ^ m ) / m ) ) |
61 |
|
expcl |
|- ( ( ( _i x. A ) e. CC /\ m e. NN0 ) -> ( ( _i x. A ) ^ m ) e. CC ) |
62 |
9 44 61
|
syl2an |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( _i x. A ) ^ m ) e. CC ) |
63 |
62 48 50
|
divcld |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( _i x. A ) ^ m ) / m ) e. CC ) |
64 |
60 63
|
eqeltrd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ` m ) e. CC ) |
65 |
2 3 64
|
serf |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ) : NN --> CC ) |
66 |
65
|
ffvelrnda |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( seq 1 ( + , ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ) ` k ) e. CC ) |
67 |
|
simpr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> k e. NN ) |
68 |
67 2
|
eleqtrdi |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> k e. ( ZZ>= ` 1 ) ) |
69 |
|
simpl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( A e. CC /\ ( abs ` A ) < 1 ) ) |
70 |
|
elfznn |
|- ( m e. ( 1 ... k ) -> m e. NN ) |
71 |
69 70 52
|
syl2an |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ` m ) e. CC ) |
72 |
69 70 64
|
syl2an |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ` m ) e. CC ) |
73 |
39 56
|
oveq12d |
|- ( n = m -> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) = ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) |
74 |
|
eqid |
|- ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) = ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) |
75 |
|
ovex |
|- ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) e. _V |
76 |
73 74 75
|
fvmpt |
|- ( m e. NN -> ( ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ` m ) = ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) |
77 |
76
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ` m ) = ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) |
78 |
43 60
|
oveq12d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ` m ) - ( ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ` m ) ) = ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) |
79 |
77 78
|
eqtr4d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ` m ) = ( ( ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ` m ) - ( ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ` m ) ) ) |
80 |
69 70 79
|
syl2an |
|- ( ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ` m ) = ( ( ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ` m ) - ( ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ` m ) ) ) |
81 |
68 71 72 80
|
sersub |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ k e. NN ) -> ( seq 1 ( + , ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ) ` k ) = ( ( seq 1 ( + , ( n e. NN |-> ( ( -u ( _i x. A ) ^ n ) / n ) ) ) ` k ) - ( seq 1 ( + , ( n e. NN |-> ( ( ( _i x. A ) ^ n ) / n ) ) ) ` k ) ) ) |
82 |
2 3 31 33 36 54 66 81
|
climsub |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ) ~~> ( -u ( log ` ( 1 + ( _i x. A ) ) ) - -u ( log ` ( 1 - ( _i x. A ) ) ) ) ) |
83 |
|
addcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC ) |
84 |
26 9 83
|
sylancr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( 1 + ( _i x. A ) ) e. CC ) |
85 |
|
bndatandm |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> A e. dom arctan ) |
86 |
|
atandm2 |
|- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) |
87 |
85 86
|
sylib |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) |
88 |
87
|
simp3d |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( 1 + ( _i x. A ) ) =/= 0 ) |
89 |
84 88
|
logcld |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) |
90 |
|
subcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC ) |
91 |
26 9 90
|
sylancr |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( 1 - ( _i x. A ) ) e. CC ) |
92 |
87
|
simp2d |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( 1 - ( _i x. A ) ) =/= 0 ) |
93 |
91 92
|
logcld |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC ) |
94 |
89 93
|
neg2subd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( -u ( log ` ( 1 + ( _i x. A ) ) ) - -u ( log ` ( 1 - ( _i x. A ) ) ) ) = ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
95 |
82 94
|
breqtrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ) ~~> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
96 |
51 63
|
subcld |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) e. CC ) |
97 |
77 96
|
eqeltrd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ` m ) e. CC ) |
98 |
4
|
a1i |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> _i e. CC ) |
99 |
|
negicn |
|- -u _i e. CC |
100 |
44
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> m e. NN0 ) |
101 |
|
expcl |
|- ( ( -u _i e. CC /\ m e. NN0 ) -> ( -u _i ^ m ) e. CC ) |
102 |
99 100 101
|
sylancr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( -u _i ^ m ) e. CC ) |
103 |
|
expcl |
|- ( ( _i e. CC /\ m e. NN0 ) -> ( _i ^ m ) e. CC ) |
104 |
4 100 103
|
sylancr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( _i ^ m ) e. CC ) |
105 |
102 104
|
subcld |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( -u _i ^ m ) - ( _i ^ m ) ) e. CC ) |
106 |
|
2cnd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> 2 e. CC ) |
107 |
|
2ne0 |
|- 2 =/= 0 |
108 |
107
|
a1i |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> 2 =/= 0 ) |
109 |
98 105 106 108
|
div23d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) = ( ( _i / 2 ) x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) ) |
110 |
109
|
oveq1d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) x. ( ( A ^ m ) / m ) ) = ( ( ( _i / 2 ) x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) x. ( ( A ^ m ) / m ) ) ) |
111 |
6
|
adantr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( _i / 2 ) e. CC ) |
112 |
|
expcl |
|- ( ( A e. CC /\ m e. NN0 ) -> ( A ^ m ) e. CC ) |
113 |
7 44 112
|
syl2an |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( A ^ m ) e. CC ) |
114 |
113 48 50
|
divcld |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( A ^ m ) / m ) e. CC ) |
115 |
111 105 114
|
mulassd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( _i / 2 ) x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) x. ( ( A ^ m ) / m ) ) = ( ( _i / 2 ) x. ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( ( A ^ m ) / m ) ) ) ) |
116 |
102 104 113
|
subdird |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( A ^ m ) ) = ( ( ( -u _i ^ m ) x. ( A ^ m ) ) - ( ( _i ^ m ) x. ( A ^ m ) ) ) ) |
117 |
7
|
adantr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> A e. CC ) |
118 |
|
mulneg1 |
|- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = -u ( _i x. A ) ) |
119 |
4 117 118
|
sylancr |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( -u _i x. A ) = -u ( _i x. A ) ) |
120 |
119
|
oveq1d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( -u _i x. A ) ^ m ) = ( -u ( _i x. A ) ^ m ) ) |
121 |
99
|
a1i |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> -u _i e. CC ) |
122 |
121 117 100
|
mulexpd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( -u _i x. A ) ^ m ) = ( ( -u _i ^ m ) x. ( A ^ m ) ) ) |
123 |
120 122
|
eqtr3d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( -u ( _i x. A ) ^ m ) = ( ( -u _i ^ m ) x. ( A ^ m ) ) ) |
124 |
98 117 100
|
mulexpd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( _i x. A ) ^ m ) = ( ( _i ^ m ) x. ( A ^ m ) ) ) |
125 |
123 124
|
oveq12d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( -u ( _i x. A ) ^ m ) - ( ( _i x. A ) ^ m ) ) = ( ( ( -u _i ^ m ) x. ( A ^ m ) ) - ( ( _i ^ m ) x. ( A ^ m ) ) ) ) |
126 |
116 125
|
eqtr4d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( A ^ m ) ) = ( ( -u ( _i x. A ) ^ m ) - ( ( _i x. A ) ^ m ) ) ) |
127 |
126
|
oveq1d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( A ^ m ) ) / m ) = ( ( ( -u ( _i x. A ) ^ m ) - ( ( _i x. A ) ^ m ) ) / m ) ) |
128 |
105 113 48 50
|
divassd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( A ^ m ) ) / m ) = ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( ( A ^ m ) / m ) ) ) |
129 |
46 62 48 50
|
divsubdird |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( -u ( _i x. A ) ^ m ) - ( ( _i x. A ) ^ m ) ) / m ) = ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) |
130 |
127 128 129
|
3eqtr3d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( ( A ^ m ) / m ) ) = ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) |
131 |
130
|
oveq2d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( _i / 2 ) x. ( ( ( -u _i ^ m ) - ( _i ^ m ) ) x. ( ( A ^ m ) / m ) ) ) = ( ( _i / 2 ) x. ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) ) |
132 |
110 115 131
|
3eqtrd |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) x. ( ( A ^ m ) / m ) ) = ( ( _i / 2 ) x. ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) ) |
133 |
|
oveq2 |
|- ( n = m -> ( -u _i ^ n ) = ( -u _i ^ m ) ) |
134 |
|
oveq2 |
|- ( n = m -> ( _i ^ n ) = ( _i ^ m ) ) |
135 |
133 134
|
oveq12d |
|- ( n = m -> ( ( -u _i ^ n ) - ( _i ^ n ) ) = ( ( -u _i ^ m ) - ( _i ^ m ) ) ) |
136 |
135
|
oveq2d |
|- ( n = m -> ( _i x. ( ( -u _i ^ n ) - ( _i ^ n ) ) ) = ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) ) |
137 |
136
|
oveq1d |
|- ( n = m -> ( ( _i x. ( ( -u _i ^ n ) - ( _i ^ n ) ) ) / 2 ) = ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) ) |
138 |
|
oveq2 |
|- ( n = m -> ( A ^ n ) = ( A ^ m ) ) |
139 |
138 38
|
oveq12d |
|- ( n = m -> ( ( A ^ n ) / n ) = ( ( A ^ m ) / m ) ) |
140 |
137 139
|
oveq12d |
|- ( n = m -> ( ( ( _i x. ( ( -u _i ^ n ) - ( _i ^ n ) ) ) / 2 ) x. ( ( A ^ n ) / n ) ) = ( ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) x. ( ( A ^ m ) / m ) ) ) |
141 |
|
ovex |
|- ( ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) x. ( ( A ^ m ) / m ) ) e. _V |
142 |
140 1 141
|
fvmpt |
|- ( m e. NN -> ( F ` m ) = ( ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) x. ( ( A ^ m ) / m ) ) ) |
143 |
142
|
adantl |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( F ` m ) = ( ( ( _i x. ( ( -u _i ^ m ) - ( _i ^ m ) ) ) / 2 ) x. ( ( A ^ m ) / m ) ) ) |
144 |
77
|
oveq2d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( ( _i / 2 ) x. ( ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ` m ) ) = ( ( _i / 2 ) x. ( ( ( -u ( _i x. A ) ^ m ) / m ) - ( ( ( _i x. A ) ^ m ) / m ) ) ) ) |
145 |
132 143 144
|
3eqtr4d |
|- ( ( ( A e. CC /\ ( abs ` A ) < 1 ) /\ m e. NN ) -> ( F ` m ) = ( ( _i / 2 ) x. ( ( n e. NN |-> ( ( ( -u ( _i x. A ) ^ n ) / n ) - ( ( ( _i x. A ) ^ n ) / n ) ) ) ` m ) ) ) |
146 |
2 3 6 95 97 145
|
isermulc2 |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , F ) ~~> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
147 |
|
atanval |
|- ( A e. dom arctan -> ( arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
148 |
85 147
|
syl |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> ( arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
149 |
146 148
|
breqtrrd |
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> seq 1 ( + , F ) ~~> ( arctan ` A ) ) |