Step |
Hyp |
Ref |
Expression |
1 |
|
pire |
|- _pi e. RR |
2 |
|
3re |
|- 3 e. RR |
3 |
|
halfpire |
|- ( _pi / 2 ) e. RR |
4 |
2 3
|
remulcli |
|- ( 3 x. ( _pi / 2 ) ) e. RR |
5 |
|
rexr |
|- ( _pi e. RR -> _pi e. RR* ) |
6 |
|
rexr |
|- ( ( 3 x. ( _pi / 2 ) ) e. RR -> ( 3 x. ( _pi / 2 ) ) e. RR* ) |
7 |
|
elioo2 |
|- ( ( _pi e. RR* /\ ( 3 x. ( _pi / 2 ) ) e. RR* ) -> ( A e. ( _pi (,) ( 3 x. ( _pi / 2 ) ) ) <-> ( A e. RR /\ _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) ) ) |
8 |
5 6 7
|
syl2an |
|- ( ( _pi e. RR /\ ( 3 x. ( _pi / 2 ) ) e. RR ) -> ( A e. ( _pi (,) ( 3 x. ( _pi / 2 ) ) ) <-> ( A e. RR /\ _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) ) ) |
9 |
1 4 8
|
mp2an |
|- ( A e. ( _pi (,) ( 3 x. ( _pi / 2 ) ) ) <-> ( A e. RR /\ _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) ) |
10 |
|
pidiv2halves |
|- ( ( _pi / 2 ) + ( _pi / 2 ) ) = _pi |
11 |
10
|
breq1i |
|- ( ( ( _pi / 2 ) + ( _pi / 2 ) ) < A <-> _pi < A ) |
12 |
|
ltaddsub |
|- ( ( ( _pi / 2 ) e. RR /\ ( _pi / 2 ) e. RR /\ A e. RR ) -> ( ( ( _pi / 2 ) + ( _pi / 2 ) ) < A <-> ( _pi / 2 ) < ( A - ( _pi / 2 ) ) ) ) |
13 |
3 3 12
|
mp3an12 |
|- ( A e. RR -> ( ( ( _pi / 2 ) + ( _pi / 2 ) ) < A <-> ( _pi / 2 ) < ( A - ( _pi / 2 ) ) ) ) |
14 |
11 13
|
bitr3id |
|- ( A e. RR -> ( _pi < A <-> ( _pi / 2 ) < ( A - ( _pi / 2 ) ) ) ) |
15 |
|
ltsubadd |
|- ( ( A e. RR /\ ( _pi / 2 ) e. RR /\ _pi e. RR ) -> ( ( A - ( _pi / 2 ) ) < _pi <-> A < ( _pi + ( _pi / 2 ) ) ) ) |
16 |
3 1 15
|
mp3an23 |
|- ( A e. RR -> ( ( A - ( _pi / 2 ) ) < _pi <-> A < ( _pi + ( _pi / 2 ) ) ) ) |
17 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
18 |
17
|
oveq1i |
|- ( 3 x. ( _pi / 2 ) ) = ( ( 2 + 1 ) x. ( _pi / 2 ) ) |
19 |
|
2cn |
|- 2 e. CC |
20 |
|
ax-1cn |
|- 1 e. CC |
21 |
3
|
recni |
|- ( _pi / 2 ) e. CC |
22 |
19 20 21
|
adddiri |
|- ( ( 2 + 1 ) x. ( _pi / 2 ) ) = ( ( 2 x. ( _pi / 2 ) ) + ( 1 x. ( _pi / 2 ) ) ) |
23 |
1
|
recni |
|- _pi e. CC |
24 |
|
2ne0 |
|- 2 =/= 0 |
25 |
23 19 24
|
divcan2i |
|- ( 2 x. ( _pi / 2 ) ) = _pi |
26 |
21
|
mulid2i |
|- ( 1 x. ( _pi / 2 ) ) = ( _pi / 2 ) |
27 |
25 26
|
oveq12i |
|- ( ( 2 x. ( _pi / 2 ) ) + ( 1 x. ( _pi / 2 ) ) ) = ( _pi + ( _pi / 2 ) ) |
28 |
18 22 27
|
3eqtrri |
|- ( _pi + ( _pi / 2 ) ) = ( 3 x. ( _pi / 2 ) ) |
29 |
28
|
breq2i |
|- ( A < ( _pi + ( _pi / 2 ) ) <-> A < ( 3 x. ( _pi / 2 ) ) ) |
30 |
16 29
|
bitr2di |
|- ( A e. RR -> ( A < ( 3 x. ( _pi / 2 ) ) <-> ( A - ( _pi / 2 ) ) < _pi ) ) |
31 |
14 30
|
anbi12d |
|- ( A e. RR -> ( ( _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) <-> ( ( _pi / 2 ) < ( A - ( _pi / 2 ) ) /\ ( A - ( _pi / 2 ) ) < _pi ) ) ) |
32 |
|
resubcl |
|- ( ( A e. RR /\ ( _pi / 2 ) e. RR ) -> ( A - ( _pi / 2 ) ) e. RR ) |
33 |
3 32
|
mpan2 |
|- ( A e. RR -> ( A - ( _pi / 2 ) ) e. RR ) |
34 |
|
sincosq2sgn |
|- ( ( A - ( _pi / 2 ) ) e. ( ( _pi / 2 ) (,) _pi ) -> ( 0 < ( sin ` ( A - ( _pi / 2 ) ) ) /\ ( cos ` ( A - ( _pi / 2 ) ) ) < 0 ) ) |
35 |
|
rexr |
|- ( ( _pi / 2 ) e. RR -> ( _pi / 2 ) e. RR* ) |
36 |
|
elioo2 |
|- ( ( ( _pi / 2 ) e. RR* /\ _pi e. RR* ) -> ( ( A - ( _pi / 2 ) ) e. ( ( _pi / 2 ) (,) _pi ) <-> ( ( A - ( _pi / 2 ) ) e. RR /\ ( _pi / 2 ) < ( A - ( _pi / 2 ) ) /\ ( A - ( _pi / 2 ) ) < _pi ) ) ) |
37 |
35 5 36
|
syl2an |
|- ( ( ( _pi / 2 ) e. RR /\ _pi e. RR ) -> ( ( A - ( _pi / 2 ) ) e. ( ( _pi / 2 ) (,) _pi ) <-> ( ( A - ( _pi / 2 ) ) e. RR /\ ( _pi / 2 ) < ( A - ( _pi / 2 ) ) /\ ( A - ( _pi / 2 ) ) < _pi ) ) ) |
38 |
3 1 37
|
mp2an |
|- ( ( A - ( _pi / 2 ) ) e. ( ( _pi / 2 ) (,) _pi ) <-> ( ( A - ( _pi / 2 ) ) e. RR /\ ( _pi / 2 ) < ( A - ( _pi / 2 ) ) /\ ( A - ( _pi / 2 ) ) < _pi ) ) |
39 |
|
ancom |
|- ( ( 0 < ( sin ` ( A - ( _pi / 2 ) ) ) /\ ( cos ` ( A - ( _pi / 2 ) ) ) < 0 ) <-> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( _pi / 2 ) ) ) ) ) |
40 |
34 38 39
|
3imtr3i |
|- ( ( ( A - ( _pi / 2 ) ) e. RR /\ ( _pi / 2 ) < ( A - ( _pi / 2 ) ) /\ ( A - ( _pi / 2 ) ) < _pi ) -> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( _pi / 2 ) ) ) ) ) |
41 |
33 40
|
syl3an1 |
|- ( ( A e. RR /\ ( _pi / 2 ) < ( A - ( _pi / 2 ) ) /\ ( A - ( _pi / 2 ) ) < _pi ) -> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( _pi / 2 ) ) ) ) ) |
42 |
41
|
3expib |
|- ( A e. RR -> ( ( ( _pi / 2 ) < ( A - ( _pi / 2 ) ) /\ ( A - ( _pi / 2 ) ) < _pi ) -> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( _pi / 2 ) ) ) ) ) ) |
43 |
31 42
|
sylbid |
|- ( A e. RR -> ( ( _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) -> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( _pi / 2 ) ) ) ) ) ) |
44 |
33
|
resincld |
|- ( A e. RR -> ( sin ` ( A - ( _pi / 2 ) ) ) e. RR ) |
45 |
44
|
lt0neg2d |
|- ( A e. RR -> ( 0 < ( sin ` ( A - ( _pi / 2 ) ) ) <-> -u ( sin ` ( A - ( _pi / 2 ) ) ) < 0 ) ) |
46 |
45
|
anbi2d |
|- ( A e. RR -> ( ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ 0 < ( sin ` ( A - ( _pi / 2 ) ) ) ) <-> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ -u ( sin ` ( A - ( _pi / 2 ) ) ) < 0 ) ) ) |
47 |
43 46
|
sylibd |
|- ( A e. RR -> ( ( _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) -> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ -u ( sin ` ( A - ( _pi / 2 ) ) ) < 0 ) ) ) |
48 |
|
recn |
|- ( A e. RR -> A e. CC ) |
49 |
|
pncan3 |
|- ( ( ( _pi / 2 ) e. CC /\ A e. CC ) -> ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) = A ) |
50 |
21 48 49
|
sylancr |
|- ( A e. RR -> ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) = A ) |
51 |
50
|
fveq2d |
|- ( A e. RR -> ( sin ` ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) ) = ( sin ` A ) ) |
52 |
33
|
recnd |
|- ( A e. RR -> ( A - ( _pi / 2 ) ) e. CC ) |
53 |
|
sinhalfpip |
|- ( ( A - ( _pi / 2 ) ) e. CC -> ( sin ` ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) ) = ( cos ` ( A - ( _pi / 2 ) ) ) ) |
54 |
52 53
|
syl |
|- ( A e. RR -> ( sin ` ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) ) = ( cos ` ( A - ( _pi / 2 ) ) ) ) |
55 |
51 54
|
eqtr3d |
|- ( A e. RR -> ( sin ` A ) = ( cos ` ( A - ( _pi / 2 ) ) ) ) |
56 |
55
|
breq1d |
|- ( A e. RR -> ( ( sin ` A ) < 0 <-> ( cos ` ( A - ( _pi / 2 ) ) ) < 0 ) ) |
57 |
50
|
fveq2d |
|- ( A e. RR -> ( cos ` ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) ) = ( cos ` A ) ) |
58 |
|
coshalfpip |
|- ( ( A - ( _pi / 2 ) ) e. CC -> ( cos ` ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) ) = -u ( sin ` ( A - ( _pi / 2 ) ) ) ) |
59 |
52 58
|
syl |
|- ( A e. RR -> ( cos ` ( ( _pi / 2 ) + ( A - ( _pi / 2 ) ) ) ) = -u ( sin ` ( A - ( _pi / 2 ) ) ) ) |
60 |
57 59
|
eqtr3d |
|- ( A e. RR -> ( cos ` A ) = -u ( sin ` ( A - ( _pi / 2 ) ) ) ) |
61 |
60
|
breq1d |
|- ( A e. RR -> ( ( cos ` A ) < 0 <-> -u ( sin ` ( A - ( _pi / 2 ) ) ) < 0 ) ) |
62 |
56 61
|
anbi12d |
|- ( A e. RR -> ( ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) <-> ( ( cos ` ( A - ( _pi / 2 ) ) ) < 0 /\ -u ( sin ` ( A - ( _pi / 2 ) ) ) < 0 ) ) ) |
63 |
47 62
|
sylibrd |
|- ( A e. RR -> ( ( _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) ) ) |
64 |
63
|
3impib |
|- ( ( A e. RR /\ _pi < A /\ A < ( 3 x. ( _pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) ) |
65 |
9 64
|
sylbi |
|- ( A e. ( _pi (,) ( 3 x. ( _pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) ) |