Step |
Hyp |
Ref |
Expression |
1 |
|
elioore |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> A e. RR ) |
2 |
1
|
recnd |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> A e. CC ) |
3 |
|
2cnd |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 2 e. CC ) |
4 |
|
picn |
|- _pi e. CC |
5 |
4
|
a1i |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> _pi e. CC ) |
6 |
|
2ne0 |
|- 2 =/= 0 |
7 |
6
|
a1i |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 2 =/= 0 ) |
8 |
|
pire |
|- _pi e. RR |
9 |
|
pipos |
|- 0 < _pi |
10 |
8 9
|
gt0ne0ii |
|- _pi =/= 0 |
11 |
10
|
a1i |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> _pi =/= 0 ) |
12 |
2 3 5 7 11
|
divdiv1d |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( ( A / 2 ) / _pi ) = ( A / ( 2 x. _pi ) ) ) |
13 |
|
0zd |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 0 e. ZZ ) |
14 |
|
2re |
|- 2 e. RR |
15 |
14 8
|
remulcli |
|- ( 2 x. _pi ) e. RR |
16 |
15
|
a1i |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( 2 x. _pi ) e. RR ) |
17 |
|
0xr |
|- 0 e. RR* |
18 |
17
|
a1i |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 0 e. RR* ) |
19 |
16
|
rexrd |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( 2 x. _pi ) e. RR* ) |
20 |
|
id |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> A e. ( 0 (,) ( 2 x. _pi ) ) ) |
21 |
|
ioogtlb |
|- ( ( 0 e. RR* /\ ( 2 x. _pi ) e. RR* /\ A e. ( 0 (,) ( 2 x. _pi ) ) ) -> 0 < A ) |
22 |
18 19 20 21
|
syl3anc |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 0 < A ) |
23 |
|
2pos |
|- 0 < 2 |
24 |
14 8 23 9
|
mulgt0ii |
|- 0 < ( 2 x. _pi ) |
25 |
24
|
a1i |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 0 < ( 2 x. _pi ) ) |
26 |
1 16 22 25
|
divgt0d |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 0 < ( A / ( 2 x. _pi ) ) ) |
27 |
|
1rp |
|- 1 e. RR+ |
28 |
27
|
a1i |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> 1 e. RR+ ) |
29 |
16 25
|
elrpd |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( 2 x. _pi ) e. RR+ ) |
30 |
2
|
div1d |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( A / 1 ) = A ) |
31 |
|
iooltub |
|- ( ( 0 e. RR* /\ ( 2 x. _pi ) e. RR* /\ A e. ( 0 (,) ( 2 x. _pi ) ) ) -> A < ( 2 x. _pi ) ) |
32 |
18 19 20 31
|
syl3anc |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> A < ( 2 x. _pi ) ) |
33 |
30 32
|
eqbrtrd |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( A / 1 ) < ( 2 x. _pi ) ) |
34 |
1 28 29 33
|
ltdiv23d |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( A / ( 2 x. _pi ) ) < 1 ) |
35 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
36 |
34 35
|
breqtrdi |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( A / ( 2 x. _pi ) ) < ( 0 + 1 ) ) |
37 |
|
btwnnz |
|- ( ( 0 e. ZZ /\ 0 < ( A / ( 2 x. _pi ) ) /\ ( A / ( 2 x. _pi ) ) < ( 0 + 1 ) ) -> -. ( A / ( 2 x. _pi ) ) e. ZZ ) |
38 |
13 26 36 37
|
syl3anc |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> -. ( A / ( 2 x. _pi ) ) e. ZZ ) |
39 |
12 38
|
eqneltrd |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> -. ( ( A / 2 ) / _pi ) e. ZZ ) |
40 |
2
|
halfcld |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( A / 2 ) e. CC ) |
41 |
|
sineq0 |
|- ( ( A / 2 ) e. CC -> ( ( sin ` ( A / 2 ) ) = 0 <-> ( ( A / 2 ) / _pi ) e. ZZ ) ) |
42 |
40 41
|
syl |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( ( sin ` ( A / 2 ) ) = 0 <-> ( ( A / 2 ) / _pi ) e. ZZ ) ) |
43 |
39 42
|
mtbird |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> -. ( sin ` ( A / 2 ) ) = 0 ) |
44 |
43
|
neqned |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( sin ` ( A / 2 ) ) =/= 0 ) |