Step |
Hyp |
Ref |
Expression |
1 |
|
elioore |
|- ( A e. ( 0 (,) _pi ) -> A e. RR ) |
2 |
1
|
recoscld |
|- ( A e. ( 0 (,) _pi ) -> ( cos ` A ) e. RR ) |
3 |
|
cospi |
|- ( cos ` _pi ) = -u 1 |
4 |
|
ioossicc |
|- ( 0 (,) _pi ) C_ ( 0 [,] _pi ) |
5 |
4
|
sseli |
|- ( A e. ( 0 (,) _pi ) -> A e. ( 0 [,] _pi ) ) |
6 |
|
0xr |
|- 0 e. RR* |
7 |
|
pire |
|- _pi e. RR |
8 |
7
|
rexri |
|- _pi e. RR* |
9 |
|
0re |
|- 0 e. RR |
10 |
|
pipos |
|- 0 < _pi |
11 |
9 7 10
|
ltleii |
|- 0 <_ _pi |
12 |
|
ubicc2 |
|- ( ( 0 e. RR* /\ _pi e. RR* /\ 0 <_ _pi ) -> _pi e. ( 0 [,] _pi ) ) |
13 |
6 8 11 12
|
mp3an |
|- _pi e. ( 0 [,] _pi ) |
14 |
13
|
a1i |
|- ( A e. ( 0 (,) _pi ) -> _pi e. ( 0 [,] _pi ) ) |
15 |
|
eliooord |
|- ( A e. ( 0 (,) _pi ) -> ( 0 < A /\ A < _pi ) ) |
16 |
15
|
simprd |
|- ( A e. ( 0 (,) _pi ) -> A < _pi ) |
17 |
5 14 16
|
cosordlem |
|- ( A e. ( 0 (,) _pi ) -> ( cos ` _pi ) < ( cos ` A ) ) |
18 |
3 17
|
eqbrtrrid |
|- ( A e. ( 0 (,) _pi ) -> -u 1 < ( cos ` A ) ) |
19 |
|
2re |
|- 2 e. RR |
20 |
19 7
|
remulcli |
|- ( 2 x. _pi ) e. RR |
21 |
20
|
rexri |
|- ( 2 x. _pi ) e. RR* |
22 |
|
1le2 |
|- 1 <_ 2 |
23 |
|
lemulge12 |
|- ( ( ( _pi e. RR /\ 2 e. RR ) /\ ( 0 <_ _pi /\ 1 <_ 2 ) ) -> _pi <_ ( 2 x. _pi ) ) |
24 |
7 19 11 22 23
|
mp4an |
|- _pi <_ ( 2 x. _pi ) |
25 |
|
iooss2 |
|- ( ( ( 2 x. _pi ) e. RR* /\ _pi <_ ( 2 x. _pi ) ) -> ( 0 (,) _pi ) C_ ( 0 (,) ( 2 x. _pi ) ) ) |
26 |
21 24 25
|
mp2an |
|- ( 0 (,) _pi ) C_ ( 0 (,) ( 2 x. _pi ) ) |
27 |
26
|
sseli |
|- ( A e. ( 0 (,) _pi ) -> A e. ( 0 (,) ( 2 x. _pi ) ) ) |
28 |
|
cos02pilt1 |
|- ( A e. ( 0 (,) ( 2 x. _pi ) ) -> ( cos ` A ) < 1 ) |
29 |
27 28
|
syl |
|- ( A e. ( 0 (,) _pi ) -> ( cos ` A ) < 1 ) |
30 |
|
neg1rr |
|- -u 1 e. RR |
31 |
30
|
rexri |
|- -u 1 e. RR* |
32 |
|
1re |
|- 1 e. RR |
33 |
32
|
rexri |
|- 1 e. RR* |
34 |
|
elioo2 |
|- ( ( -u 1 e. RR* /\ 1 e. RR* ) -> ( ( cos ` A ) e. ( -u 1 (,) 1 ) <-> ( ( cos ` A ) e. RR /\ -u 1 < ( cos ` A ) /\ ( cos ` A ) < 1 ) ) ) |
35 |
31 33 34
|
mp2an |
|- ( ( cos ` A ) e. ( -u 1 (,) 1 ) <-> ( ( cos ` A ) e. RR /\ -u 1 < ( cos ` A ) /\ ( cos ` A ) < 1 ) ) |
36 |
2 18 29 35
|
syl3anbrc |
|- ( A e. ( 0 (,) _pi ) -> ( cos ` A ) e. ( -u 1 (,) 1 ) ) |