Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
|- ( ( A e. CC /\ B e. CC ) -> A e. CC ) |
2 |
1
|
sincld |
|- ( ( A e. CC /\ B e. CC ) -> ( sin ` A ) e. CC ) |
3 |
|
cosf |
|- cos : CC --> CC |
4 |
3
|
a1i |
|- ( A e. CC -> cos : CC --> CC ) |
5 |
4
|
ffvelrnda |
|- ( ( A e. CC /\ B e. CC ) -> ( cos ` B ) e. CC ) |
6 |
2 5
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) x. ( cos ` B ) ) e. CC ) |
7 |
1
|
coscld |
|- ( ( A e. CC /\ B e. CC ) -> ( cos ` A ) e. CC ) |
8 |
|
sinf |
|- sin : CC --> CC |
9 |
8
|
a1i |
|- ( A e. CC -> sin : CC --> CC ) |
10 |
9
|
ffvelrnda |
|- ( ( A e. CC /\ B e. CC ) -> ( sin ` B ) e. CC ) |
11 |
7 10
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( cos ` A ) x. ( sin ` B ) ) e. CC ) |
12 |
6 11 6
|
ppncand |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) + ( ( ( sin ` A ) x. ( cos ` B ) ) - ( ( cos ` A ) x. ( sin ` B ) ) ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( cos ` B ) ) ) ) |
13 |
|
sinadd |
|- ( ( A e. CC /\ B e. CC ) -> ( sin ` ( A + B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) ) |
14 |
|
sinsub |
|- ( ( A e. CC /\ B e. CC ) -> ( sin ` ( A - B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) - ( ( cos ` A ) x. ( sin ` B ) ) ) ) |
15 |
13 14
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` ( A + B ) ) + ( sin ` ( A - B ) ) ) = ( ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) + ( ( ( sin ` A ) x. ( cos ` B ) ) - ( ( cos ` A ) x. ( sin ` B ) ) ) ) ) |
16 |
6
|
2timesd |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( ( sin ` A ) x. ( cos ` B ) ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( cos ` B ) ) ) ) |
17 |
12 15 16
|
3eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` ( A + B ) ) + ( sin ` ( A - B ) ) ) = ( 2 x. ( ( sin ` A ) x. ( cos ` B ) ) ) ) |
18 |
17
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( sin ` ( A + B ) ) + ( sin ` ( A - B ) ) ) / 2 ) = ( ( 2 x. ( ( sin ` A ) x. ( cos ` B ) ) ) / 2 ) ) |
19 |
|
2cnd |
|- ( ( A e. CC /\ B e. CC ) -> 2 e. CC ) |
20 |
|
2ne0 |
|- 2 =/= 0 |
21 |
20
|
a1i |
|- ( ( A e. CC /\ B e. CC ) -> 2 =/= 0 ) |
22 |
6 19 21
|
divcan3d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 2 x. ( ( sin ` A ) x. ( cos ` B ) ) ) / 2 ) = ( ( sin ` A ) x. ( cos ` B ) ) ) |
23 |
18 22
|
eqtr2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) x. ( cos ` B ) ) = ( ( ( sin ` ( A + B ) ) + ( sin ` ( A - B ) ) ) / 2 ) ) |