Step |
Hyp |
Ref |
Expression |
1 |
|
atanre |
|- ( A e. RR -> A e. dom arctan ) |
2 |
1
|
adantr |
|- ( ( A e. RR /\ A < 0 ) -> A e. dom arctan ) |
3 |
|
atanneg |
|- ( A e. dom arctan -> ( arctan ` -u A ) = -u ( arctan ` A ) ) |
4 |
2 3
|
syl |
|- ( ( A e. RR /\ A < 0 ) -> ( arctan ` -u A ) = -u ( arctan ` A ) ) |
5 |
|
renegcl |
|- ( A e. RR -> -u A e. RR ) |
6 |
5
|
adantr |
|- ( ( A e. RR /\ A < 0 ) -> -u A e. RR ) |
7 |
|
lt0neg1 |
|- ( A e. RR -> ( A < 0 <-> 0 < -u A ) ) |
8 |
7
|
biimpa |
|- ( ( A e. RR /\ A < 0 ) -> 0 < -u A ) |
9 |
6 8
|
elrpd |
|- ( ( A e. RR /\ A < 0 ) -> -u A e. RR+ ) |
10 |
|
atanbndlem |
|- ( -u A e. RR+ -> ( arctan ` -u A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |
11 |
9 10
|
syl |
|- ( ( A e. RR /\ A < 0 ) -> ( arctan ` -u A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |
12 |
4 11
|
eqeltrrd |
|- ( ( A e. RR /\ A < 0 ) -> -u ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |
13 |
|
halfpire |
|- ( _pi / 2 ) e. RR |
14 |
13
|
recni |
|- ( _pi / 2 ) e. CC |
15 |
14
|
negnegi |
|- -u -u ( _pi / 2 ) = ( _pi / 2 ) |
16 |
15
|
oveq2i |
|- ( -u ( _pi / 2 ) (,) -u -u ( _pi / 2 ) ) = ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) |
17 |
12 16
|
eleqtrrdi |
|- ( ( A e. RR /\ A < 0 ) -> -u ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) -u -u ( _pi / 2 ) ) ) |
18 |
|
neghalfpire |
|- -u ( _pi / 2 ) e. RR |
19 |
|
atanrecl |
|- ( A e. RR -> ( arctan ` A ) e. RR ) |
20 |
19
|
adantr |
|- ( ( A e. RR /\ A < 0 ) -> ( arctan ` A ) e. RR ) |
21 |
|
iooneg |
|- ( ( -u ( _pi / 2 ) e. RR /\ ( _pi / 2 ) e. RR /\ ( arctan ` A ) e. RR ) -> ( ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) <-> -u ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) -u -u ( _pi / 2 ) ) ) ) |
22 |
18 13 20 21
|
mp3an12i |
|- ( ( A e. RR /\ A < 0 ) -> ( ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) <-> -u ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) -u -u ( _pi / 2 ) ) ) ) |
23 |
17 22
|
mpbird |
|- ( ( A e. RR /\ A < 0 ) -> ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |
24 |
|
simpr |
|- ( ( A e. RR /\ A = 0 ) -> A = 0 ) |
25 |
24
|
fveq2d |
|- ( ( A e. RR /\ A = 0 ) -> ( arctan ` A ) = ( arctan ` 0 ) ) |
26 |
|
atan0 |
|- ( arctan ` 0 ) = 0 |
27 |
25 26
|
eqtrdi |
|- ( ( A e. RR /\ A = 0 ) -> ( arctan ` A ) = 0 ) |
28 |
|
0re |
|- 0 e. RR |
29 |
|
pirp |
|- _pi e. RR+ |
30 |
|
rphalfcl |
|- ( _pi e. RR+ -> ( _pi / 2 ) e. RR+ ) |
31 |
|
rpgt0 |
|- ( ( _pi / 2 ) e. RR+ -> 0 < ( _pi / 2 ) ) |
32 |
29 30 31
|
mp2b |
|- 0 < ( _pi / 2 ) |
33 |
|
lt0neg2 |
|- ( ( _pi / 2 ) e. RR -> ( 0 < ( _pi / 2 ) <-> -u ( _pi / 2 ) < 0 ) ) |
34 |
13 33
|
ax-mp |
|- ( 0 < ( _pi / 2 ) <-> -u ( _pi / 2 ) < 0 ) |
35 |
32 34
|
mpbi |
|- -u ( _pi / 2 ) < 0 |
36 |
18
|
rexri |
|- -u ( _pi / 2 ) e. RR* |
37 |
13
|
rexri |
|- ( _pi / 2 ) e. RR* |
38 |
|
elioo2 |
|- ( ( -u ( _pi / 2 ) e. RR* /\ ( _pi / 2 ) e. RR* ) -> ( 0 e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) <-> ( 0 e. RR /\ -u ( _pi / 2 ) < 0 /\ 0 < ( _pi / 2 ) ) ) ) |
39 |
36 37 38
|
mp2an |
|- ( 0 e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) <-> ( 0 e. RR /\ -u ( _pi / 2 ) < 0 /\ 0 < ( _pi / 2 ) ) ) |
40 |
28 35 32 39
|
mpbir3an |
|- 0 e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) |
41 |
27 40
|
eqeltrdi |
|- ( ( A e. RR /\ A = 0 ) -> ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |
42 |
|
elrp |
|- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) ) |
43 |
|
atanbndlem |
|- ( A e. RR+ -> ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |
44 |
42 43
|
sylbir |
|- ( ( A e. RR /\ 0 < A ) -> ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |
45 |
|
lttri4 |
|- ( ( A e. RR /\ 0 e. RR ) -> ( A < 0 \/ A = 0 \/ 0 < A ) ) |
46 |
28 45
|
mpan2 |
|- ( A e. RR -> ( A < 0 \/ A = 0 \/ 0 < A ) ) |
47 |
23 41 44 46
|
mpjao3dan |
|- ( A e. RR -> ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |