Step |
Hyp |
Ref |
Expression |
1 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
2 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
3 |
1 2
|
eqtr2i |
|- ( 1 + 1 ) = ( 2 x. 1 ) |
4 |
|
zcn |
|- ( K e. ZZ -> K e. CC ) |
5 |
|
2cn |
|- 2 e. CC |
6 |
|
picn |
|- _pi e. CC |
7 |
|
mul12 |
|- ( ( K e. CC /\ 2 e. CC /\ _pi e. CC ) -> ( K x. ( 2 x. _pi ) ) = ( 2 x. ( K x. _pi ) ) ) |
8 |
5 6 7
|
mp3an23 |
|- ( K e. CC -> ( K x. ( 2 x. _pi ) ) = ( 2 x. ( K x. _pi ) ) ) |
9 |
4 8
|
syl |
|- ( K e. ZZ -> ( K x. ( 2 x. _pi ) ) = ( 2 x. ( K x. _pi ) ) ) |
10 |
9
|
fveq2d |
|- ( K e. ZZ -> ( cos ` ( K x. ( 2 x. _pi ) ) ) = ( cos ` ( 2 x. ( K x. _pi ) ) ) ) |
11 |
|
cos2kpi |
|- ( K e. ZZ -> ( cos ` ( K x. ( 2 x. _pi ) ) ) = 1 ) |
12 |
|
zre |
|- ( K e. ZZ -> K e. RR ) |
13 |
|
pire |
|- _pi e. RR |
14 |
|
remulcl |
|- ( ( K e. RR /\ _pi e. RR ) -> ( K x. _pi ) e. RR ) |
15 |
12 13 14
|
sylancl |
|- ( K e. ZZ -> ( K x. _pi ) e. RR ) |
16 |
15
|
recnd |
|- ( K e. ZZ -> ( K x. _pi ) e. CC ) |
17 |
|
cos2t |
|- ( ( K x. _pi ) e. CC -> ( cos ` ( 2 x. ( K x. _pi ) ) ) = ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) - 1 ) ) |
18 |
16 17
|
syl |
|- ( K e. ZZ -> ( cos ` ( 2 x. ( K x. _pi ) ) ) = ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) - 1 ) ) |
19 |
10 11 18
|
3eqtr3rd |
|- ( K e. ZZ -> ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) - 1 ) = 1 ) |
20 |
15
|
recoscld |
|- ( K e. ZZ -> ( cos ` ( K x. _pi ) ) e. RR ) |
21 |
20
|
recnd |
|- ( K e. ZZ -> ( cos ` ( K x. _pi ) ) e. CC ) |
22 |
21
|
sqcld |
|- ( K e. ZZ -> ( ( cos ` ( K x. _pi ) ) ^ 2 ) e. CC ) |
23 |
|
mulcl |
|- ( ( 2 e. CC /\ ( ( cos ` ( K x. _pi ) ) ^ 2 ) e. CC ) -> ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) e. CC ) |
24 |
5 22 23
|
sylancr |
|- ( K e. ZZ -> ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) e. CC ) |
25 |
|
ax-1cn |
|- 1 e. CC |
26 |
|
subadd |
|- ( ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) e. CC /\ 1 e. CC /\ 1 e. CC ) -> ( ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) - 1 ) = 1 <-> ( 1 + 1 ) = ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) ) ) |
27 |
25 25 26
|
mp3an23 |
|- ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) e. CC -> ( ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) - 1 ) = 1 <-> ( 1 + 1 ) = ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) ) ) |
28 |
24 27
|
syl |
|- ( K e. ZZ -> ( ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) - 1 ) = 1 <-> ( 1 + 1 ) = ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) ) ) |
29 |
19 28
|
mpbid |
|- ( K e. ZZ -> ( 1 + 1 ) = ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) ) |
30 |
3 29
|
syl5reqr |
|- ( K e. ZZ -> ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) = ( 2 x. 1 ) ) |
31 |
|
2cnne0 |
|- ( 2 e. CC /\ 2 =/= 0 ) |
32 |
|
mulcan |
|- ( ( ( ( cos ` ( K x. _pi ) ) ^ 2 ) e. CC /\ 1 e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) = ( 2 x. 1 ) <-> ( ( cos ` ( K x. _pi ) ) ^ 2 ) = 1 ) ) |
33 |
25 31 32
|
mp3an23 |
|- ( ( ( cos ` ( K x. _pi ) ) ^ 2 ) e. CC -> ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) = ( 2 x. 1 ) <-> ( ( cos ` ( K x. _pi ) ) ^ 2 ) = 1 ) ) |
34 |
22 33
|
syl |
|- ( K e. ZZ -> ( ( 2 x. ( ( cos ` ( K x. _pi ) ) ^ 2 ) ) = ( 2 x. 1 ) <-> ( ( cos ` ( K x. _pi ) ) ^ 2 ) = 1 ) ) |
35 |
30 34
|
mpbid |
|- ( K e. ZZ -> ( ( cos ` ( K x. _pi ) ) ^ 2 ) = 1 ) |
36 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
37 |
35 36
|
eqtr4di |
|- ( K e. ZZ -> ( ( cos ` ( K x. _pi ) ) ^ 2 ) = ( 1 ^ 2 ) ) |
38 |
|
1re |
|- 1 e. RR |
39 |
|
sqabs |
|- ( ( ( cos ` ( K x. _pi ) ) e. RR /\ 1 e. RR ) -> ( ( ( cos ` ( K x. _pi ) ) ^ 2 ) = ( 1 ^ 2 ) <-> ( abs ` ( cos ` ( K x. _pi ) ) ) = ( abs ` 1 ) ) ) |
40 |
20 38 39
|
sylancl |
|- ( K e. ZZ -> ( ( ( cos ` ( K x. _pi ) ) ^ 2 ) = ( 1 ^ 2 ) <-> ( abs ` ( cos ` ( K x. _pi ) ) ) = ( abs ` 1 ) ) ) |
41 |
37 40
|
mpbid |
|- ( K e. ZZ -> ( abs ` ( cos ` ( K x. _pi ) ) ) = ( abs ` 1 ) ) |
42 |
|
abs1 |
|- ( abs ` 1 ) = 1 |
43 |
41 42
|
eqtrdi |
|- ( K e. ZZ -> ( abs ` ( cos ` ( K x. _pi ) ) ) = 1 ) |