Step |
Hyp |
Ref |
Expression |
1 |
|
neg1cn |
|- -u 1 e. CC |
2 |
|
2re |
|- 2 e. RR |
3 |
|
simpl |
|- ( ( N e. NN /\ K e. ZZ ) -> N e. NN ) |
4 |
|
nndivre |
|- ( ( 2 e. RR /\ N e. NN ) -> ( 2 / N ) e. RR ) |
5 |
2 3 4
|
sylancr |
|- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. RR ) |
6 |
5
|
recnd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. CC ) |
7 |
|
cxpcl |
|- ( ( -u 1 e. CC /\ ( 2 / N ) e. CC ) -> ( -u 1 ^c ( 2 / N ) ) e. CC ) |
8 |
1 6 7
|
sylancr |
|- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) e. CC ) |
9 |
1
|
a1i |
|- ( ( N e. NN /\ K e. ZZ ) -> -u 1 e. CC ) |
10 |
|
neg1ne0 |
|- -u 1 =/= 0 |
11 |
10
|
a1i |
|- ( ( N e. NN /\ K e. ZZ ) -> -u 1 =/= 0 ) |
12 |
9 11 6
|
cxpne0d |
|- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) =/= 0 ) |
13 |
|
simpr |
|- ( ( N e. NN /\ K e. ZZ ) -> K e. ZZ ) |
14 |
|
nnz |
|- ( N e. NN -> N e. ZZ ) |
15 |
14
|
adantr |
|- ( ( N e. NN /\ K e. ZZ ) -> N e. ZZ ) |
16 |
8 12 13 15
|
expsubd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ ( N - K ) ) = ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) / ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ) |
17 |
|
root1id |
|- ( N e. NN -> ( ( -u 1 ^c ( 2 / N ) ) ^ N ) = 1 ) |
18 |
17
|
adantr |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ N ) = 1 ) |
19 |
18
|
oveq1d |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ N ) / ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( 1 / ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ) |
20 |
8 12 13
|
expclzd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ K ) e. CC ) |
21 |
8 12 13
|
expne0d |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ K ) =/= 0 ) |
22 |
|
recval |
|- ( ( ( ( -u 1 ^c ( 2 / N ) ) ^ K ) e. CC /\ ( ( -u 1 ^c ( 2 / N ) ) ^ K ) =/= 0 ) -> ( 1 / ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) / ( ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ^ 2 ) ) ) |
23 |
20 21 22
|
syl2anc |
|- ( ( N e. NN /\ K e. ZZ ) -> ( 1 / ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) / ( ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ^ 2 ) ) ) |
24 |
|
absexpz |
|- ( ( ( -u 1 ^c ( 2 / N ) ) e. CC /\ ( -u 1 ^c ( 2 / N ) ) =/= 0 /\ K e. ZZ ) -> ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( ( abs ` ( -u 1 ^c ( 2 / N ) ) ) ^ K ) ) |
25 |
8 12 13 24
|
syl3anc |
|- ( ( N e. NN /\ K e. ZZ ) -> ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( ( abs ` ( -u 1 ^c ( 2 / N ) ) ) ^ K ) ) |
26 |
|
abscxp2 |
|- ( ( -u 1 e. CC /\ ( 2 / N ) e. RR ) -> ( abs ` ( -u 1 ^c ( 2 / N ) ) ) = ( ( abs ` -u 1 ) ^c ( 2 / N ) ) ) |
27 |
1 5 26
|
sylancr |
|- ( ( N e. NN /\ K e. ZZ ) -> ( abs ` ( -u 1 ^c ( 2 / N ) ) ) = ( ( abs ` -u 1 ) ^c ( 2 / N ) ) ) |
28 |
|
ax-1cn |
|- 1 e. CC |
29 |
28
|
absnegi |
|- ( abs ` -u 1 ) = ( abs ` 1 ) |
30 |
|
abs1 |
|- ( abs ` 1 ) = 1 |
31 |
29 30
|
eqtri |
|- ( abs ` -u 1 ) = 1 |
32 |
31
|
oveq1i |
|- ( ( abs ` -u 1 ) ^c ( 2 / N ) ) = ( 1 ^c ( 2 / N ) ) |
33 |
27 32
|
eqtrdi |
|- ( ( N e. NN /\ K e. ZZ ) -> ( abs ` ( -u 1 ^c ( 2 / N ) ) ) = ( 1 ^c ( 2 / N ) ) ) |
34 |
6
|
1cxpd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( 1 ^c ( 2 / N ) ) = 1 ) |
35 |
33 34
|
eqtrd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( abs ` ( -u 1 ^c ( 2 / N ) ) ) = 1 ) |
36 |
35
|
oveq1d |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( abs ` ( -u 1 ^c ( 2 / N ) ) ) ^ K ) = ( 1 ^ K ) ) |
37 |
|
1exp |
|- ( K e. ZZ -> ( 1 ^ K ) = 1 ) |
38 |
37
|
adantl |
|- ( ( N e. NN /\ K e. ZZ ) -> ( 1 ^ K ) = 1 ) |
39 |
25 36 38
|
3eqtrd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = 1 ) |
40 |
39
|
oveq1d |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ^ 2 ) = ( 1 ^ 2 ) ) |
41 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
42 |
40 41
|
eqtrdi |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ^ 2 ) = 1 ) |
43 |
42
|
oveq2d |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) / ( ( abs ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ^ 2 ) ) = ( ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) / 1 ) ) |
44 |
20
|
cjcld |
|- ( ( N e. NN /\ K e. ZZ ) -> ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) e. CC ) |
45 |
44
|
div1d |
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) / 1 ) = ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ) |
46 |
23 43 45
|
3eqtrd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( 1 / ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) ) |
47 |
16 19 46
|
3eqtrrd |
|- ( ( N e. NN /\ K e. ZZ ) -> ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ ( N - K ) ) ) |