Theorem tanadd 14897

Description: Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)

Assertion

Ref	Expression
tanadd	|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` ( A + B ) ) = ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) )

Proof of Theorem tanadd

Step	Hyp	Ref	Expression
1		addcl 10018	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
2	1	adantr 481	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( A + B ) e. CC )
3		simpr3 1069	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` ( A + B ) ) =/= 0 )
4		tanval 14858	. . 3 |- ( ( ( A + B ) e. CC /\ ( cos ` ( A + B ) ) =/= 0 ) -> ( tan ` ( A + B ) ) = ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) )
5	2, 3, 4	syl2anc 693	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` ( A + B ) ) = ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) )
6		sinadd 14894	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( sin ` ( A + B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) )
7	6	adantr 481	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( sin ` ( A + B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) )
8		cosadd 14895	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
9	8	adantr 481	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
10	7, 9	oveq12d 6668	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) = ( ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) )
11		simpll 790	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> A e. CC )
12	11	coscld 14861	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` A ) e. CC )
13		simplr 792	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> B e. CC )
14	13	coscld 14861	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` B ) e. CC )
15	12, 14	mulcld 10060	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC )
16		simpr1 1067	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` A ) =/= 0 )
17	11, 16	tancld 14862	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` A ) e. CC )
18		simpr2 1068	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` B ) =/= 0 )
19	13, 18	tancld 14862	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` B ) e. CC )
20	15, 17, 19	adddid 10064	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) = ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) + ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) ) )
21	12, 14, 17	mul32d 10246	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) = ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( cos ` B ) ) )
22		tanval 14858	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
23	11, 16, 22	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
24	23	oveq2d 6666	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( tan ` A ) ) = ( ( cos ` A ) x. ( ( sin ` A ) / ( cos ` A ) ) ) )
25	11	sincld 14860	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( sin ` A ) e. CC )
26	25, 12, 16	divcan2d 10803	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( ( sin ` A ) / ( cos ` A ) ) ) = ( sin ` A ) )
27	24, 26	eqtrd 2656	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( tan ` A ) ) = ( sin ` A ) )
28	27	oveq1d 6665	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( cos ` B ) ) = ( ( sin ` A ) x. ( cos ` B ) ) )
29	21, 28	eqtrd 2656	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) = ( ( sin ` A ) x. ( cos ` B ) ) )
30	12, 14, 19	mulassd 10063	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) = ( ( cos ` A ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) )
31		tanval 14858	. . . . . . . . . . 11 |- ( ( B e. CC /\ ( cos ` B ) =/= 0 ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
32	13, 18, 31	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
33	32	oveq2d 6666	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` B ) x. ( tan ` B ) ) = ( ( cos ` B ) x. ( ( sin ` B ) / ( cos ` B ) ) ) )
34	13	sincld 14860	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( sin ` B ) e. CC )
35	34, 14, 18	divcan2d 10803	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` B ) x. ( ( sin ` B ) / ( cos ` B ) ) ) = ( sin ` B ) )
36	33, 35	eqtrd 2656	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` B ) x. ( tan ` B ) ) = ( sin ` B ) )
37	36	oveq2d 6666	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) = ( ( cos ` A ) x. ( sin ` B ) ) )
38	30, 37	eqtrd 2656	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) = ( ( cos ` A ) x. ( sin ` B ) ) )
39	29, 38	oveq12d 6668	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) + ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) )
40	20, 39	eqtrd 2656	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) )
41		1cnd 10056	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> 1 e. CC )
42	17, 19	mulcld 10060	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( tan ` A ) x. ( tan ` B ) ) e. CC )
43	15, 41, 42	subdid 10486	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. 1 ) - ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) ) )
44	15	mulid1d 10057	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. 1 ) = ( ( cos ` A ) x. ( cos ` B ) ) )
45	12, 14, 17, 19	mul4d 10248	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) = ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) )
46	27, 36	oveq12d 6668	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) = ( ( sin ` A ) x. ( sin ` B ) ) )
47	45, 46	eqtrd 2656	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) = ( ( sin ` A ) x. ( sin ` B ) ) )
48	44, 47	oveq12d 6668	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. 1 ) - ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
49	43, 48	eqtrd 2656	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
50	40, 49	oveq12d 6668	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) ) = ( ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) )
51	17, 19	addcld 10059	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( tan ` A ) + ( tan ` B ) ) e. CC )
52		ax-1cn 9994	. . . . 5 |- 1 e. CC
53		subcl 10280	. . . . 5 |- ( ( 1 e. CC /\ ( ( tan ` A ) x. ( tan ` B ) ) e. CC ) -> ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) e. CC )
54	52, 42, 53	sylancr 695	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) e. CC )
55		tanaddlem 14896	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 ) ) -> ( ( cos ` ( A + B ) ) =/= 0 <-> ( ( tan ` A ) x. ( tan ` B ) ) =/= 1 ) )
56	55	3adantr3 1222	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` ( A + B ) ) =/= 0 <-> ( ( tan ` A ) x. ( tan ` B ) ) =/= 1 ) )
57	3, 56	mpbid 222	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( tan ` A ) x. ( tan ` B ) ) =/= 1 )
58	57	necomd 2849	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> 1 =/= ( ( tan ` A ) x. ( tan ` B ) ) )
59		subeq0 10307	. . . . . . 7 |- ( ( 1 e. CC /\ ( ( tan ` A ) x. ( tan ` B ) ) e. CC ) -> ( ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) = 0 <-> 1 = ( ( tan ` A ) x. ( tan ` B ) ) ) )
60	59	necon3bid 2838	. . . . . 6 |- ( ( 1 e. CC /\ ( ( tan ` A ) x. ( tan ` B ) ) e. CC ) -> ( ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) =/= 0 <-> 1 =/= ( ( tan ` A ) x. ( tan ` B ) ) ) )
61	52, 42, 60	sylancr 695	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) =/= 0 <-> 1 =/= ( ( tan ` A ) x. ( tan ` B ) ) ) )
62	58, 61	mpbird 247	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) =/= 0 )
63	12, 14, 16, 18	mulne0d 10679	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) =/= 0 )
64	51, 54, 15, 62, 63	divcan5d 10827	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) ) = ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) )
65	10, 50, 64	3eqtr2rd 2663	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) )
66	5, 65	eqtr4d 2659	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` ( A + B ) ) = ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   sincsin 14794   cosccos 14795   tanctan 14796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802

This theorem is referenced by:  tanregt0  24285