Step |
Hyp |
Ref |
Expression |
1 |
|
simpl2 |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> N e. NN ) |
2 |
|
nnm1nn0 |
|- ( N e. NN -> ( N - 1 ) e. NN0 ) |
3 |
1 2
|
syl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( N - 1 ) e. NN0 ) |
4 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
5 |
3 4
|
eleqtrdi |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( N - 1 ) e. ( ZZ>= ` 0 ) ) |
6 |
|
eluzfz1 |
|- ( ( N - 1 ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... ( N - 1 ) ) ) |
7 |
5 6
|
syl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> 0 e. ( 0 ... ( N - 1 ) ) ) |
8 |
|
neg1cn |
|- -u 1 e. CC |
9 |
|
2re |
|- 2 e. RR |
10 |
|
simp2 |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> N e. NN ) |
11 |
|
nndivre |
|- ( ( 2 e. RR /\ N e. NN ) -> ( 2 / N ) e. RR ) |
12 |
9 10 11
|
sylancr |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( 2 / N ) e. RR ) |
13 |
12
|
recnd |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( 2 / N ) e. CC ) |
14 |
|
cxpcl |
|- ( ( -u 1 e. CC /\ ( 2 / N ) e. CC ) -> ( -u 1 ^c ( 2 / N ) ) e. CC ) |
15 |
8 13 14
|
sylancr |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( -u 1 ^c ( 2 / N ) ) e. CC ) |
16 |
15
|
adantr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( -u 1 ^c ( 2 / N ) ) e. CC ) |
17 |
|
0nn0 |
|- 0 e. NN0 |
18 |
|
expcl |
|- ( ( ( -u 1 ^c ( 2 / N ) ) e. CC /\ 0 e. NN0 ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) e. CC ) |
19 |
16 17 18
|
sylancl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) e. CC ) |
20 |
19
|
mul02d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( 0 x. ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) ) = 0 ) |
21 |
|
simprl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> A = 0 ) |
22 |
21
|
oveq1d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( A ^ N ) = ( 0 ^ N ) ) |
23 |
|
simprr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( A ^ N ) = B ) |
24 |
1
|
0expd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( 0 ^ N ) = 0 ) |
25 |
22 23 24
|
3eqtr3d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> B = 0 ) |
26 |
25
|
oveq1d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( B ^c ( 1 / N ) ) = ( 0 ^c ( 1 / N ) ) ) |
27 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
28 |
|
nnne0 |
|- ( N e. NN -> N =/= 0 ) |
29 |
|
reccl |
|- ( ( N e. CC /\ N =/= 0 ) -> ( 1 / N ) e. CC ) |
30 |
|
recne0 |
|- ( ( N e. CC /\ N =/= 0 ) -> ( 1 / N ) =/= 0 ) |
31 |
29 30
|
0cxpd |
|- ( ( N e. CC /\ N =/= 0 ) -> ( 0 ^c ( 1 / N ) ) = 0 ) |
32 |
27 28 31
|
syl2anc |
|- ( N e. NN -> ( 0 ^c ( 1 / N ) ) = 0 ) |
33 |
1 32
|
syl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( 0 ^c ( 1 / N ) ) = 0 ) |
34 |
26 33
|
eqtrd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( B ^c ( 1 / N ) ) = 0 ) |
35 |
34
|
oveq1d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) ) = ( 0 x. ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) ) ) |
36 |
20 35 21
|
3eqtr4rd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) ) ) |
37 |
|
oveq2 |
|- ( n = 0 -> ( ( -u 1 ^c ( 2 / N ) ) ^ n ) = ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) ) |
38 |
37
|
oveq2d |
|- ( n = 0 -> ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) ) ) |
39 |
38
|
rspceeqv |
|- ( ( 0 e. ( 0 ... ( N - 1 ) ) /\ A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ 0 ) ) ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) |
40 |
7 36 39
|
syl2anc |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ ( A = 0 /\ ( A ^ N ) = B ) ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) |
41 |
40
|
expr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A = 0 ) -> ( ( A ^ N ) = B -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
42 |
|
simpl1 |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> A e. CC ) |
43 |
|
simpr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> A =/= 0 ) |
44 |
|
simpl2 |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> N e. NN ) |
45 |
44
|
nnzd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> N e. ZZ ) |
46 |
|
explog |
|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) ) |
47 |
42 43 45 46
|
syl3anc |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) ) |
48 |
47
|
eqcomd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( exp ` ( N x. ( log ` A ) ) ) = ( A ^ N ) ) |
49 |
10
|
nncnd |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> N e. CC ) |
50 |
49
|
adantr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> N e. CC ) |
51 |
42 43
|
logcld |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( log ` A ) e. CC ) |
52 |
50 51
|
mulcld |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( N x. ( log ` A ) ) e. CC ) |
53 |
44
|
nnnn0d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> N e. NN0 ) |
54 |
42 53
|
expcld |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( A ^ N ) e. CC ) |
55 |
42 43 45
|
expne0d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( A ^ N ) =/= 0 ) |
56 |
|
eflogeq |
|- ( ( ( N x. ( log ` A ) ) e. CC /\ ( A ^ N ) e. CC /\ ( A ^ N ) =/= 0 ) -> ( ( exp ` ( N x. ( log ` A ) ) ) = ( A ^ N ) <-> E. m e. ZZ ( N x. ( log ` A ) ) = ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) ) ) |
57 |
52 54 55 56
|
syl3anc |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( ( exp ` ( N x. ( log ` A ) ) ) = ( A ^ N ) <-> E. m e. ZZ ( N x. ( log ` A ) ) = ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) ) ) |
58 |
48 57
|
mpbid |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> E. m e. ZZ ( N x. ( log ` A ) ) = ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) ) |
59 |
54 55
|
logcld |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( log ` ( A ^ N ) ) e. CC ) |
60 |
59
|
adantr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( log ` ( A ^ N ) ) e. CC ) |
61 |
|
ax-icn |
|- _i e. CC |
62 |
|
2cn |
|- 2 e. CC |
63 |
|
picn |
|- _pi e. CC |
64 |
62 63
|
mulcli |
|- ( 2 x. _pi ) e. CC |
65 |
61 64
|
mulcli |
|- ( _i x. ( 2 x. _pi ) ) e. CC |
66 |
|
zcn |
|- ( m e. ZZ -> m e. CC ) |
67 |
66
|
adantl |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> m e. CC ) |
68 |
|
mulcl |
|- ( ( ( _i x. ( 2 x. _pi ) ) e. CC /\ m e. CC ) -> ( ( _i x. ( 2 x. _pi ) ) x. m ) e. CC ) |
69 |
65 67 68
|
sylancr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( _i x. ( 2 x. _pi ) ) x. m ) e. CC ) |
70 |
60 69
|
addcld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) e. CC ) |
71 |
50
|
adantr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> N e. CC ) |
72 |
51
|
adantr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( log ` A ) e. CC ) |
73 |
10
|
nnne0d |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> N =/= 0 ) |
74 |
73
|
ad2antrr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> N =/= 0 ) |
75 |
70 71 72 74
|
divmuld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) = ( log ` A ) <-> ( N x. ( log ` A ) ) = ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) ) ) |
76 |
|
fveq2 |
|- ( ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) = ( log ` A ) -> ( exp ` ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) ) = ( exp ` ( log ` A ) ) ) |
77 |
71 74
|
reccld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( 1 / N ) e. CC ) |
78 |
77 60
|
mulcld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) e. CC ) |
79 |
13
|
ad2antrr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( 2 / N ) e. CC ) |
80 |
79 67
|
mulcld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( 2 / N ) x. m ) e. CC ) |
81 |
61 63
|
mulcli |
|- ( _i x. _pi ) e. CC |
82 |
|
mulcl |
|- ( ( ( ( 2 / N ) x. m ) e. CC /\ ( _i x. _pi ) e. CC ) -> ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) e. CC ) |
83 |
80 81 82
|
sylancl |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) e. CC ) |
84 |
|
efadd |
|- ( ( ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) e. CC /\ ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) e. CC ) -> ( exp ` ( ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) + ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) = ( ( exp ` ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) ) x. ( exp ` ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) ) |
85 |
78 83 84
|
syl2anc |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( exp ` ( ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) + ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) = ( ( exp ` ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) ) x. ( exp ` ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) ) |
86 |
60 69 71 74
|
divdird |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) = ( ( ( log ` ( A ^ N ) ) / N ) + ( ( ( _i x. ( 2 x. _pi ) ) x. m ) / N ) ) ) |
87 |
60 71 74
|
divrec2d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( log ` ( A ^ N ) ) / N ) = ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) ) |
88 |
65
|
a1i |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( _i x. ( 2 x. _pi ) ) e. CC ) |
89 |
88 67 71 74
|
div23d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. m ) / N ) = ( ( ( _i x. ( 2 x. _pi ) ) / N ) x. m ) ) |
90 |
61 62 63
|
mul12i |
|- ( _i x. ( 2 x. _pi ) ) = ( 2 x. ( _i x. _pi ) ) |
91 |
90
|
oveq1i |
|- ( ( _i x. ( 2 x. _pi ) ) / N ) = ( ( 2 x. ( _i x. _pi ) ) / N ) |
92 |
62
|
a1i |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> 2 e. CC ) |
93 |
81
|
a1i |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( _i x. _pi ) e. CC ) |
94 |
92 93 71 74
|
div23d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( 2 x. ( _i x. _pi ) ) / N ) = ( ( 2 / N ) x. ( _i x. _pi ) ) ) |
95 |
91 94
|
syl5eq |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( _i x. ( 2 x. _pi ) ) / N ) = ( ( 2 / N ) x. ( _i x. _pi ) ) ) |
96 |
95
|
oveq1d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( _i x. ( 2 x. _pi ) ) / N ) x. m ) = ( ( ( 2 / N ) x. ( _i x. _pi ) ) x. m ) ) |
97 |
79 93 67
|
mul32d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( 2 / N ) x. ( _i x. _pi ) ) x. m ) = ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) |
98 |
89 96 97
|
3eqtrd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( _i x. ( 2 x. _pi ) ) x. m ) / N ) = ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) |
99 |
87 98
|
oveq12d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( log ` ( A ^ N ) ) / N ) + ( ( ( _i x. ( 2 x. _pi ) ) x. m ) / N ) ) = ( ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) + ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) |
100 |
86 99
|
eqtrd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) = ( ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) + ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) |
101 |
100
|
fveq2d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( exp ` ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) ) = ( exp ` ( ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) + ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) ) |
102 |
54
|
adantr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( A ^ N ) e. CC ) |
103 |
55
|
adantr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( A ^ N ) =/= 0 ) |
104 |
102 103 77
|
cxpefd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( A ^ N ) ^c ( 1 / N ) ) = ( exp ` ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) ) ) |
105 |
8
|
a1i |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> -u 1 e. CC ) |
106 |
|
neg1ne0 |
|- -u 1 =/= 0 |
107 |
106
|
a1i |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> -u 1 =/= 0 ) |
108 |
|
simpr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> m e. ZZ ) |
109 |
105 107 79 108
|
cxpmul2zd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( -u 1 ^c ( ( 2 / N ) x. m ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ m ) ) |
110 |
105 107 80
|
cxpefd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( -u 1 ^c ( ( 2 / N ) x. m ) ) = ( exp ` ( ( ( 2 / N ) x. m ) x. ( log ` -u 1 ) ) ) ) |
111 |
|
logm1 |
|- ( log ` -u 1 ) = ( _i x. _pi ) |
112 |
111
|
oveq2i |
|- ( ( ( 2 / N ) x. m ) x. ( log ` -u 1 ) ) = ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) |
113 |
112
|
fveq2i |
|- ( exp ` ( ( ( 2 / N ) x. m ) x. ( log ` -u 1 ) ) ) = ( exp ` ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) |
114 |
110 113
|
eqtrdi |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( -u 1 ^c ( ( 2 / N ) x. m ) ) = ( exp ` ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) |
115 |
105 79
|
cxpcld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) e. CC ) |
116 |
8
|
a1i |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> -u 1 e. CC ) |
117 |
106
|
a1i |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> -u 1 =/= 0 ) |
118 |
116 117 13
|
cxpne0d |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( -u 1 ^c ( 2 / N ) ) =/= 0 ) |
119 |
118
|
ad2antrr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) =/= 0 ) |
120 |
115 119 108
|
expclzd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ m ) e. CC ) |
121 |
44
|
adantr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> N e. NN ) |
122 |
108 121
|
zmodcld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( m mod N ) e. NN0 ) |
123 |
115 122
|
expcld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) e. CC ) |
124 |
122
|
nn0zd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( m mod N ) e. ZZ ) |
125 |
115 119 124
|
expne0d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) =/= 0 ) |
126 |
115 119 124 108
|
expsubd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( m - ( m mod N ) ) ) = ( ( ( -u 1 ^c ( 2 / N ) ) ^ m ) / ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) ) |
127 |
121
|
nnzd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> N e. ZZ ) |
128 |
|
zre |
|- ( m e. ZZ -> m e. RR ) |
129 |
121
|
nnrpd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> N e. RR+ ) |
130 |
|
moddifz |
|- ( ( m e. RR /\ N e. RR+ ) -> ( ( m - ( m mod N ) ) / N ) e. ZZ ) |
131 |
128 129 130
|
syl2an2 |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( m - ( m mod N ) ) / N ) e. ZZ ) |
132 |
|
expmulz |
|- ( ( ( ( -u 1 ^c ( 2 / N ) ) e. CC /\ ( -u 1 ^c ( 2 / N ) ) =/= 0 ) /\ ( N e. ZZ /\ ( ( m - ( m mod N ) ) / N ) e. ZZ ) ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( N x. ( ( m - ( m mod N ) ) / N ) ) ) = ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) ^ ( ( m - ( m mod N ) ) / N ) ) ) |
133 |
115 119 127 131 132
|
syl22anc |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( N x. ( ( m - ( m mod N ) ) / N ) ) ) = ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) ^ ( ( m - ( m mod N ) ) / N ) ) ) |
134 |
122
|
nn0cnd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( m mod N ) e. CC ) |
135 |
67 134
|
subcld |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( m - ( m mod N ) ) e. CC ) |
136 |
135 71 74
|
divcan2d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( N x. ( ( m - ( m mod N ) ) / N ) ) = ( m - ( m mod N ) ) ) |
137 |
136
|
oveq2d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( N x. ( ( m - ( m mod N ) ) / N ) ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ ( m - ( m mod N ) ) ) ) |
138 |
|
root1id |
|- ( N e. NN -> ( ( -u 1 ^c ( 2 / N ) ) ^ N ) = 1 ) |
139 |
121 138
|
syl |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ N ) = 1 ) |
140 |
139
|
oveq1d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) ^ ( ( m - ( m mod N ) ) / N ) ) = ( 1 ^ ( ( m - ( m mod N ) ) / N ) ) ) |
141 |
|
1exp |
|- ( ( ( m - ( m mod N ) ) / N ) e. ZZ -> ( 1 ^ ( ( m - ( m mod N ) ) / N ) ) = 1 ) |
142 |
131 141
|
syl |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( 1 ^ ( ( m - ( m mod N ) ) / N ) ) = 1 ) |
143 |
140 142
|
eqtrd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) ^ ( ( m - ( m mod N ) ) / N ) ) = 1 ) |
144 |
133 137 143
|
3eqtr3d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( m - ( m mod N ) ) ) = 1 ) |
145 |
126 144
|
eqtr3d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ m ) / ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = 1 ) |
146 |
120 123 125 145
|
diveq1d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ m ) = ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) |
147 |
109 114 146
|
3eqtr3rd |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) = ( exp ` ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) |
148 |
104 147
|
oveq12d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = ( ( exp ` ( ( 1 / N ) x. ( log ` ( A ^ N ) ) ) ) x. ( exp ` ( ( ( 2 / N ) x. m ) x. ( _i x. _pi ) ) ) ) ) |
149 |
85 101 148
|
3eqtr4d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( exp ` ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) ) = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) ) |
150 |
|
eflog |
|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = A ) |
151 |
42 43 150
|
syl2anc |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = A ) |
152 |
151
|
adantr |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( exp ` ( log ` A ) ) = A ) |
153 |
149 152
|
eqeq12d |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( exp ` ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) ) = ( exp ` ( log ` A ) ) <-> ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = A ) ) |
154 |
|
zmodfz |
|- ( ( m e. ZZ /\ N e. NN ) -> ( m mod N ) e. ( 0 ... ( N - 1 ) ) ) |
155 |
108 121 154
|
syl2anc |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( m mod N ) e. ( 0 ... ( N - 1 ) ) ) |
156 |
|
eqcom |
|- ( A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) <-> ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) = A ) |
157 |
|
oveq2 |
|- ( n = ( m mod N ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ n ) = ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) |
158 |
157
|
oveq2d |
|- ( n = ( m mod N ) -> ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) ) |
159 |
158
|
eqeq1d |
|- ( n = ( m mod N ) -> ( ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) = A <-> ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = A ) ) |
160 |
156 159
|
syl5bb |
|- ( n = ( m mod N ) -> ( A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) <-> ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = A ) ) |
161 |
160
|
rspcev |
|- ( ( ( m mod N ) e. ( 0 ... ( N - 1 ) ) /\ ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = A ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) |
162 |
161
|
ex |
|- ( ( m mod N ) e. ( 0 ... ( N - 1 ) ) -> ( ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = A -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
163 |
155 162
|
syl |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ ( m mod N ) ) ) = A -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
164 |
153 163
|
sylbid |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( exp ` ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) ) = ( exp ` ( log ` A ) ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
165 |
76 164
|
syl5 |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) / N ) = ( log ` A ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
166 |
75 165
|
sylbird |
|- ( ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) /\ m e. ZZ ) -> ( ( N x. ( log ` A ) ) = ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
167 |
166
|
rexlimdva |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( E. m e. ZZ ( N x. ( log ` A ) ) = ( ( log ` ( A ^ N ) ) + ( ( _i x. ( 2 x. _pi ) ) x. m ) ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
168 |
58 167
|
mpd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) |
169 |
|
oveq1 |
|- ( ( A ^ N ) = B -> ( ( A ^ N ) ^c ( 1 / N ) ) = ( B ^c ( 1 / N ) ) ) |
170 |
169
|
oveq1d |
|- ( ( A ^ N ) = B -> ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) |
171 |
170
|
eqeq2d |
|- ( ( A ^ N ) = B -> ( A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) <-> A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
172 |
171
|
rexbidv |
|- ( ( A ^ N ) = B -> ( E. n e. ( 0 ... ( N - 1 ) ) A = ( ( ( A ^ N ) ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) <-> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
173 |
168 172
|
syl5ibcom |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ A =/= 0 ) -> ( ( A ^ N ) = B -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
174 |
41 173
|
pm2.61dane |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( ( A ^ N ) = B -> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |
175 |
|
simp3 |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> B e. CC ) |
176 |
|
nnrecre |
|- ( N e. NN -> ( 1 / N ) e. RR ) |
177 |
176
|
3ad2ant2 |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( 1 / N ) e. RR ) |
178 |
177
|
recnd |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( 1 / N ) e. CC ) |
179 |
175 178
|
cxpcld |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( B ^c ( 1 / N ) ) e. CC ) |
180 |
179
|
adantr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( B ^c ( 1 / N ) ) e. CC ) |
181 |
|
elfznn0 |
|- ( n e. ( 0 ... ( N - 1 ) ) -> n e. NN0 ) |
182 |
|
expcl |
|- ( ( ( -u 1 ^c ( 2 / N ) ) e. CC /\ n e. NN0 ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ n ) e. CC ) |
183 |
15 181 182
|
syl2an |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ n ) e. CC ) |
184 |
10
|
adantr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. NN ) |
185 |
184
|
nnnn0d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. NN0 ) |
186 |
180 183 185
|
mulexpd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ^ N ) = ( ( ( B ^c ( 1 / N ) ) ^ N ) x. ( ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ^ N ) ) ) |
187 |
175
|
adantr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> B e. CC ) |
188 |
|
cxproot |
|- ( ( B e. CC /\ N e. NN ) -> ( ( B ^c ( 1 / N ) ) ^ N ) = B ) |
189 |
187 184 188
|
syl2anc |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( B ^c ( 1 / N ) ) ^ N ) = B ) |
190 |
181
|
adantl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> n e. NN0 ) |
191 |
190
|
nn0cnd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> n e. CC ) |
192 |
184
|
nncnd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. CC ) |
193 |
191 192
|
mulcomd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( n x. N ) = ( N x. n ) ) |
194 |
193
|
oveq2d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( n x. N ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ ( N x. n ) ) ) |
195 |
15
|
adantr |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( -u 1 ^c ( 2 / N ) ) e. CC ) |
196 |
195 185 190
|
expmuld |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( n x. N ) ) = ( ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ^ N ) ) |
197 |
195 190 185
|
expmuld |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( N x. n ) ) = ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) ^ n ) ) |
198 |
194 196 197
|
3eqtr3d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ^ N ) = ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) ^ n ) ) |
199 |
184 138
|
syl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ N ) = 1 ) |
200 |
199
|
oveq1d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) ^ n ) = ( 1 ^ n ) ) |
201 |
|
elfzelz |
|- ( n e. ( 0 ... ( N - 1 ) ) -> n e. ZZ ) |
202 |
201
|
adantl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> n e. ZZ ) |
203 |
|
1exp |
|- ( n e. ZZ -> ( 1 ^ n ) = 1 ) |
204 |
202 203
|
syl |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( 1 ^ n ) = 1 ) |
205 |
198 200 204
|
3eqtrd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ^ N ) = 1 ) |
206 |
189 205
|
oveq12d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( B ^c ( 1 / N ) ) ^ N ) x. ( ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ^ N ) ) = ( B x. 1 ) ) |
207 |
187
|
mulid1d |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( B x. 1 ) = B ) |
208 |
186 206 207
|
3eqtrd |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ^ N ) = B ) |
209 |
|
oveq1 |
|- ( A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) -> ( A ^ N ) = ( ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ^ N ) ) |
210 |
209
|
eqeq1d |
|- ( A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) -> ( ( A ^ N ) = B <-> ( ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ^ N ) = B ) ) |
211 |
208 210
|
syl5ibrcom |
|- ( ( ( A e. CC /\ N e. NN /\ B e. CC ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) -> ( A ^ N ) = B ) ) |
212 |
211
|
rexlimdva |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) -> ( A ^ N ) = B ) ) |
213 |
174 212
|
impbid |
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( ( A ^ N ) = B <-> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) ) |