Step |
Hyp |
Ref |
Expression |
1 |
|
tanval |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) ) |
2 |
|
2cn |
|- 2 e. CC |
3 |
|
ax-icn |
|- _i e. CC |
4 |
2 3
|
mulcomi |
|- ( 2 x. _i ) = ( _i x. 2 ) |
5 |
4
|
oveq2i |
|- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( _i x. 2 ) ) |
6 |
|
sinval |
|- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) |
7 |
6
|
adantr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) |
8 |
|
simpl |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> A e. CC ) |
9 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
10 |
3 8 9
|
sylancr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( _i x. A ) e. CC ) |
11 |
|
efcl |
|- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC ) |
12 |
10 11
|
syl |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( exp ` ( _i x. A ) ) e. CC ) |
13 |
|
negicn |
|- -u _i e. CC |
14 |
|
mulcl |
|- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC ) |
15 |
13 8 14
|
sylancr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( -u _i x. A ) e. CC ) |
16 |
|
efcl |
|- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC ) |
17 |
15 16
|
syl |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( exp ` ( -u _i x. A ) ) e. CC ) |
18 |
12 17
|
subcld |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC ) |
19 |
3
|
a1i |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> _i e. CC ) |
20 |
2
|
a1i |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> 2 e. CC ) |
21 |
|
ine0 |
|- _i =/= 0 |
22 |
21
|
a1i |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> _i =/= 0 ) |
23 |
|
2ne0 |
|- 2 =/= 0 |
24 |
23
|
a1i |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> 2 =/= 0 ) |
25 |
18 19 20 22 24
|
divdiv1d |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( _i x. 2 ) ) ) |
26 |
5 7 25
|
3eqtr4a |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( sin ` A ) = ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) ) |
27 |
|
cosval |
|- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) ) |
28 |
27
|
adantr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) ) |
29 |
26 28
|
oveq12d |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( sin ` A ) / ( cos ` A ) ) = ( ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) / ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) ) ) |
30 |
1 29
|
eqtrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) / ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) ) ) |
31 |
18 19 22
|
divcld |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) e. CC ) |
32 |
12 17
|
addcld |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC ) |
33 |
|
simpr |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( cos ` A ) =/= 0 ) |
34 |
28 33
|
eqnetrrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) =/= 0 ) |
35 |
32 20 24
|
diveq0ad |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) = 0 <-> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) = 0 ) ) |
36 |
35
|
necon3bid |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) =/= 0 <-> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) =/= 0 ) ) |
37 |
34 36
|
mpbid |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) =/= 0 ) |
38 |
31 32 20 37 24
|
divcan7d |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) / ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) ) = ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) |
39 |
18 19 32 22 37
|
divdiv1d |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) ) |
40 |
30 38 39
|
3eqtrd |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) ) |