Step |
Hyp |
Ref |
Expression |
1 |
|
tancl |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) e. CC ) |
2 |
|
tanval |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) ) |
3 |
2
|
oveq1d |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( tan ` A ) ^ 2 ) = ( ( ( sin ` A ) / ( cos ` A ) ) ^ 2 ) ) |
4 |
|
sincl |
|- ( A e. CC -> ( sin ` A ) e. CC ) |
5 |
4
|
adantr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( sin ` A ) e. CC ) |
6 |
|
coscl |
|- ( A e. CC -> ( cos ` A ) e. CC ) |
7 |
6
|
adantr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( cos ` A ) e. CC ) |
8 |
|
simpr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( cos ` A ) =/= 0 ) |
9 |
5 7 8
|
sqdivd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( sin ` A ) / ( cos ` A ) ) ^ 2 ) = ( ( ( sin ` A ) ^ 2 ) / ( ( cos ` A ) ^ 2 ) ) ) |
10 |
3 9
|
eqtrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( tan ` A ) ^ 2 ) = ( ( ( sin ` A ) ^ 2 ) / ( ( cos ` A ) ^ 2 ) ) ) |
11 |
5
|
sqcld |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( sin ` A ) ^ 2 ) e. CC ) |
12 |
7
|
sqcld |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( cos ` A ) ^ 2 ) e. CC ) |
13 |
12
|
negcld |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> -u ( ( cos ` A ) ^ 2 ) e. CC ) |
14 |
11 12
|
subnegd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( sin ` A ) ^ 2 ) - -u ( ( cos ` A ) ^ 2 ) ) = ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) ) |
15 |
|
sincossq |
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 ) |
16 |
15
|
adantr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 ) |
17 |
14 16
|
eqtrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( sin ` A ) ^ 2 ) - -u ( ( cos ` A ) ^ 2 ) ) = 1 ) |
18 |
|
ax-1ne0 |
|- 1 =/= 0 |
19 |
18
|
a1i |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> 1 =/= 0 ) |
20 |
17 19
|
eqnetrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( sin ` A ) ^ 2 ) - -u ( ( cos ` A ) ^ 2 ) ) =/= 0 ) |
21 |
11 13 20
|
subne0ad |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( sin ` A ) ^ 2 ) =/= -u ( ( cos ` A ) ^ 2 ) ) |
22 |
12
|
mulm1d |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( -u 1 x. ( ( cos ` A ) ^ 2 ) ) = -u ( ( cos ` A ) ^ 2 ) ) |
23 |
21 22
|
neeqtrrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( sin ` A ) ^ 2 ) =/= ( -u 1 x. ( ( cos ` A ) ^ 2 ) ) ) |
24 |
|
neg1cn |
|- -u 1 e. CC |
25 |
24
|
a1i |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> -u 1 e. CC ) |
26 |
|
sqne0 |
|- ( ( cos ` A ) e. CC -> ( ( ( cos ` A ) ^ 2 ) =/= 0 <-> ( cos ` A ) =/= 0 ) ) |
27 |
6 26
|
syl |
|- ( A e. CC -> ( ( ( cos ` A ) ^ 2 ) =/= 0 <-> ( cos ` A ) =/= 0 ) ) |
28 |
27
|
biimpar |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( cos ` A ) ^ 2 ) =/= 0 ) |
29 |
11 25 12 28
|
divmul3d |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( ( sin ` A ) ^ 2 ) / ( ( cos ` A ) ^ 2 ) ) = -u 1 <-> ( ( sin ` A ) ^ 2 ) = ( -u 1 x. ( ( cos ` A ) ^ 2 ) ) ) ) |
30 |
29
|
necon3bid |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( ( sin ` A ) ^ 2 ) / ( ( cos ` A ) ^ 2 ) ) =/= -u 1 <-> ( ( sin ` A ) ^ 2 ) =/= ( -u 1 x. ( ( cos ` A ) ^ 2 ) ) ) ) |
31 |
23 30
|
mpbird |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( sin ` A ) ^ 2 ) / ( ( cos ` A ) ^ 2 ) ) =/= -u 1 ) |
32 |
10 31
|
eqnetrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( tan ` A ) ^ 2 ) =/= -u 1 ) |
33 |
|
atandm3 |
|- ( ( tan ` A ) e. dom arctan <-> ( ( tan ` A ) e. CC /\ ( ( tan ` A ) ^ 2 ) =/= -u 1 ) ) |
34 |
1 32 33
|
sylanbrc |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) e. dom arctan ) |