Step |
Hyp |
Ref |
Expression |
1 |
|
0xr |
|- 0 e. RR* |
2 |
|
pire |
|- _pi e. RR |
3 |
2
|
rexri |
|- _pi e. RR* |
4 |
|
elioo2 |
|- ( ( 0 e. RR* /\ _pi e. RR* ) -> ( A e. ( 0 (,) _pi ) <-> ( A e. RR /\ 0 < A /\ A < _pi ) ) ) |
5 |
1 3 4
|
mp2an |
|- ( A e. ( 0 (,) _pi ) <-> ( A e. RR /\ 0 < A /\ A < _pi ) ) |
6 |
|
rehalfcl |
|- ( A e. RR -> ( A / 2 ) e. RR ) |
7 |
6
|
3ad2ant1 |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> ( A / 2 ) e. RR ) |
8 |
|
halfpos2 |
|- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) ) |
9 |
8
|
biimpa |
|- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) ) |
10 |
9
|
3adant3 |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> 0 < ( A / 2 ) ) |
11 |
|
2re |
|- 2 e. RR |
12 |
|
2pos |
|- 0 < 2 |
13 |
11 12
|
pm3.2i |
|- ( 2 e. RR /\ 0 < 2 ) |
14 |
|
ltdiv1 |
|- ( ( A e. RR /\ _pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A < _pi <-> ( A / 2 ) < ( _pi / 2 ) ) ) |
15 |
2 13 14
|
mp3an23 |
|- ( A e. RR -> ( A < _pi <-> ( A / 2 ) < ( _pi / 2 ) ) ) |
16 |
15
|
adantr |
|- ( ( A e. RR /\ 0 < A ) -> ( A < _pi <-> ( A / 2 ) < ( _pi / 2 ) ) ) |
17 |
16
|
biimp3a |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> ( A / 2 ) < ( _pi / 2 ) ) |
18 |
|
sincosq1lem |
|- ( ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) < ( _pi / 2 ) ) -> 0 < ( sin ` ( A / 2 ) ) ) |
19 |
7 10 17 18
|
syl3anc |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> 0 < ( sin ` ( A / 2 ) ) ) |
20 |
|
resubcl |
|- ( ( _pi e. RR /\ A e. RR ) -> ( _pi - A ) e. RR ) |
21 |
2 20
|
mpan |
|- ( A e. RR -> ( _pi - A ) e. RR ) |
22 |
|
rehalfcl |
|- ( ( _pi - A ) e. RR -> ( ( _pi - A ) / 2 ) e. RR ) |
23 |
21 22
|
syl |
|- ( A e. RR -> ( ( _pi - A ) / 2 ) e. RR ) |
24 |
23
|
3ad2ant1 |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> ( ( _pi - A ) / 2 ) e. RR ) |
25 |
|
posdif |
|- ( ( A e. RR /\ _pi e. RR ) -> ( A < _pi <-> 0 < ( _pi - A ) ) ) |
26 |
2 25
|
mpan2 |
|- ( A e. RR -> ( A < _pi <-> 0 < ( _pi - A ) ) ) |
27 |
|
halfpos2 |
|- ( ( _pi - A ) e. RR -> ( 0 < ( _pi - A ) <-> 0 < ( ( _pi - A ) / 2 ) ) ) |
28 |
21 27
|
syl |
|- ( A e. RR -> ( 0 < ( _pi - A ) <-> 0 < ( ( _pi - A ) / 2 ) ) ) |
29 |
26 28
|
bitrd |
|- ( A e. RR -> ( A < _pi <-> 0 < ( ( _pi - A ) / 2 ) ) ) |
30 |
29
|
adantr |
|- ( ( A e. RR /\ 0 < A ) -> ( A < _pi <-> 0 < ( ( _pi - A ) / 2 ) ) ) |
31 |
30
|
biimp3a |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> 0 < ( ( _pi - A ) / 2 ) ) |
32 |
|
ltsubpos |
|- ( ( A e. RR /\ _pi e. RR ) -> ( 0 < A <-> ( _pi - A ) < _pi ) ) |
33 |
2 32
|
mpan2 |
|- ( A e. RR -> ( 0 < A <-> ( _pi - A ) < _pi ) ) |
34 |
|
ltdiv1 |
|- ( ( ( _pi - A ) e. RR /\ _pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( _pi - A ) < _pi <-> ( ( _pi - A ) / 2 ) < ( _pi / 2 ) ) ) |
35 |
2 13 34
|
mp3an23 |
|- ( ( _pi - A ) e. RR -> ( ( _pi - A ) < _pi <-> ( ( _pi - A ) / 2 ) < ( _pi / 2 ) ) ) |
36 |
21 35
|
syl |
|- ( A e. RR -> ( ( _pi - A ) < _pi <-> ( ( _pi - A ) / 2 ) < ( _pi / 2 ) ) ) |
37 |
33 36
|
bitrd |
|- ( A e. RR -> ( 0 < A <-> ( ( _pi - A ) / 2 ) < ( _pi / 2 ) ) ) |
38 |
37
|
biimpa |
|- ( ( A e. RR /\ 0 < A ) -> ( ( _pi - A ) / 2 ) < ( _pi / 2 ) ) |
39 |
38
|
3adant3 |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> ( ( _pi - A ) / 2 ) < ( _pi / 2 ) ) |
40 |
|
sincosq1lem |
|- ( ( ( ( _pi - A ) / 2 ) e. RR /\ 0 < ( ( _pi - A ) / 2 ) /\ ( ( _pi - A ) / 2 ) < ( _pi / 2 ) ) -> 0 < ( sin ` ( ( _pi - A ) / 2 ) ) ) |
41 |
24 31 39 40
|
syl3anc |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> 0 < ( sin ` ( ( _pi - A ) / 2 ) ) ) |
42 |
|
recn |
|- ( A e. RR -> A e. CC ) |
43 |
|
picn |
|- _pi e. CC |
44 |
|
2cnne0 |
|- ( 2 e. CC /\ 2 =/= 0 ) |
45 |
|
divsubdir |
|- ( ( _pi e. CC /\ A e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( _pi - A ) / 2 ) = ( ( _pi / 2 ) - ( A / 2 ) ) ) |
46 |
43 44 45
|
mp3an13 |
|- ( A e. CC -> ( ( _pi - A ) / 2 ) = ( ( _pi / 2 ) - ( A / 2 ) ) ) |
47 |
42 46
|
syl |
|- ( A e. RR -> ( ( _pi - A ) / 2 ) = ( ( _pi / 2 ) - ( A / 2 ) ) ) |
48 |
47
|
fveq2d |
|- ( A e. RR -> ( sin ` ( ( _pi - A ) / 2 ) ) = ( sin ` ( ( _pi / 2 ) - ( A / 2 ) ) ) ) |
49 |
6
|
recnd |
|- ( A e. RR -> ( A / 2 ) e. CC ) |
50 |
|
sinhalfpim |
|- ( ( A / 2 ) e. CC -> ( sin ` ( ( _pi / 2 ) - ( A / 2 ) ) ) = ( cos ` ( A / 2 ) ) ) |
51 |
49 50
|
syl |
|- ( A e. RR -> ( sin ` ( ( _pi / 2 ) - ( A / 2 ) ) ) = ( cos ` ( A / 2 ) ) ) |
52 |
48 51
|
eqtrd |
|- ( A e. RR -> ( sin ` ( ( _pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) ) |
53 |
52
|
3ad2ant1 |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> ( sin ` ( ( _pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) ) |
54 |
41 53
|
breqtrd |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> 0 < ( cos ` ( A / 2 ) ) ) |
55 |
|
resincl |
|- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR ) |
56 |
|
recoscl |
|- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR ) |
57 |
55 56
|
jca |
|- ( ( A / 2 ) e. RR -> ( ( sin ` ( A / 2 ) ) e. RR /\ ( cos ` ( A / 2 ) ) e. RR ) ) |
58 |
|
axmulgt0 |
|- ( ( ( sin ` ( A / 2 ) ) e. RR /\ ( cos ` ( A / 2 ) ) e. RR ) -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) |
59 |
6 57 58
|
3syl |
|- ( A e. RR -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) |
60 |
|
remulcl |
|- ( ( ( sin ` ( A / 2 ) ) e. RR /\ ( cos ` ( A / 2 ) ) e. RR ) -> ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR ) |
61 |
6 57 60
|
3syl |
|- ( A e. RR -> ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR ) |
62 |
|
axmulgt0 |
|- ( ( 2 e. RR /\ ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR ) -> ( ( 0 < 2 /\ 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) ) |
63 |
11 61 62
|
sylancr |
|- ( A e. RR -> ( ( 0 < 2 /\ 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) ) |
64 |
12 63
|
mpani |
|- ( A e. RR -> ( 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) ) |
65 |
59 64
|
syld |
|- ( A e. RR -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) ) |
66 |
65
|
3ad2ant1 |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) ) |
67 |
19 54 66
|
mp2and |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) |
68 |
|
2cn |
|- 2 e. CC |
69 |
|
2ne0 |
|- 2 =/= 0 |
70 |
|
divcan2 |
|- ( ( A e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( 2 x. ( A / 2 ) ) = A ) |
71 |
68 69 70
|
mp3an23 |
|- ( A e. CC -> ( 2 x. ( A / 2 ) ) = A ) |
72 |
42 71
|
syl |
|- ( A e. RR -> ( 2 x. ( A / 2 ) ) = A ) |
73 |
72
|
fveq2d |
|- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) ) |
74 |
|
sin2t |
|- ( ( A / 2 ) e. CC -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) |
75 |
49 74
|
syl |
|- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) |
76 |
73 75
|
eqtr3d |
|- ( A e. RR -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) |
77 |
76
|
3ad2ant1 |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) |
78 |
67 77
|
breqtrrd |
|- ( ( A e. RR /\ 0 < A /\ A < _pi ) -> 0 < ( sin ` A ) ) |
79 |
5 78
|
sylbi |
|- ( A e. ( 0 (,) _pi ) -> 0 < ( sin ` A ) ) |