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Theorem tanadd 12463
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
StepHypRef Expression
1 addcl 8835 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
21adantr 451 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( A  +  B )  e.  CC )
3 simpr3 963 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  ( A  +  B
) )  =/=  0
)
4 tanval 12424 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  ( cos `  ( A  +  B ) )  =/=  0 )  -> 
( tan `  ( A  +  B )
)  =  ( ( sin `  ( A  +  B ) )  /  ( cos `  ( A  +  B )
) ) )
52, 3, 4syl2anc 642 . 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 12460 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sin `  ( A  +  B )
)  =  ( ( ( sin `  A
)  x.  ( cos `  B ) )  +  ( ( cos `  A
)  x.  ( sin `  B ) ) ) )
76adantr 451 . . . 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 12461 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
98adantr 451 . . . 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 ) ) ) )
107, 9oveq12d 5892 . . 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 730 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  A  e.  CC )
1211coscld 12427 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  A )  e.  CC )
13 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  B  e.  CC )
1413coscld 12427 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  B )  e.  CC )
1512, 14mulcld 8871 . . . . . 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 961 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  A )  =/=  0
)
1711, 16tancld 12428 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  A )  e.  CC )
18 simpr2 962 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  B )  =/=  0
)
1913, 18tancld 12428 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  B )  e.  CC )
2015, 17, 19adddid 8875 . . . . 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
) ) ) )
2112, 14, 17mul32d 9038 . . . . . . 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 12424 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
2311, 16, 22syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
2423oveq2d 5890 . . . . . . . . 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
) ) ) )
2511sincld 12426 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  A )  e.  CC )
2625, 12, 16divcan2d 9554 . . . . . . . . 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
) )
2724, 26eqtrd 2328 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( cos `  A )  x.  ( tan `  A
) )  =  ( sin `  A ) )
2827oveq1d 5889 . . . . . . 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 ) ) )
2921, 28eqtrd 2328 . . . . . 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 ) ) )
3012, 14, 19mulassd 8874 . . . . . . 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 12424 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  ( cos `  B )  =/=  0 )  -> 
( tan `  B
)  =  ( ( sin `  B )  /  ( cos `  B
) ) )
3213, 18, 31syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  B )  =  ( ( sin `  B
)  /  ( cos `  B ) ) )
3332oveq2d 5890 . . . . . . . . 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
) ) ) )
3413sincld 12426 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  B )  e.  CC )
3534, 14, 18divcan2d 9554 . . . . . . . . 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
) )
3633, 35eqtrd 2328 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( cos `  B )  x.  ( tan `  B
) )  =  ( sin `  B ) )
3736oveq2d 5890 . . . . . . 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 ) ) )
3830, 37eqtrd 2328 . . . . . 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 ) ) )
3929, 38oveq12d 5892 . . . . 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 ) ) ) )
4020, 39eqtrd 2328 . . . 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 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
4241a1i 10 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  1  e.  CC )
4317, 19mulcld 8871 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( tan `  A )  x.  ( tan `  B
) )  e.  CC )
4415, 42, 43subdid 9251 . . . . 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 ) ) ) ) )
4515mulid1d 8868 . . . . . 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 ) ) )
4612, 14, 17, 19mul4d 9040 . . . . . . 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 ) ) ) )
4727, 36oveq12d 5892 . . . . . . 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 ) ) )
4846, 47eqtrd 2328 . . . . . 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 ) ) )
4945, 48oveq12d 5892 . . . . 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 ) ) ) )
5044, 49eqtrd 2328 . . . 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 ) ) ) )
5140, 50oveq12d 5892 . . 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 ) ) ) ) )
5217, 19addcld 8870 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( tan `  A )  +  ( tan `  B
) )  e.  CC )
53 subcl 9067 . . . . 5  |-  ( ( 1  e.  CC  /\  ( ( tan `  A
)  x.  ( tan `  B ) )  e.  CC )  ->  (
1  -  ( ( tan `  A )  x.  ( tan `  B
) ) )  e.  CC )
5441, 43, 53sylancr 644 . . . 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 12462 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( ( cos `  ( A  +  B ) )  =/=  0  <->  ( ( tan `  A )  x.  ( tan `  B ) )  =/=  1 ) )
56553adantr3 1116 . . . . . . 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 ) )
573, 56mpbid 201 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( tan `  A )  x.  ( tan `  B
) )  =/=  1
)
5857necomd 2542 . . . . 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 9089 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( ( tan `  A
)  x.  ( tan `  B ) )  e.  CC )  ->  (
( 1  -  (
( tan `  A
)  x.  ( tan `  B ) ) )  =  0  <->  1  =  ( ( tan `  A
)  x.  ( tan `  B ) ) ) )
6059necon3bid 2494 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( ( tan `  A
)  x.  ( tan `  B ) )  e.  CC )  ->  (
( 1  -  (
( tan `  A
)  x.  ( tan `  B ) ) )  =/=  0  <->  1  =/=  ( ( tan `  A
)  x.  ( tan `  B ) ) ) )
6141, 43, 60sylancr 644 . . . . 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 ) ) ) )
6258, 61mpbird 223 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( 1  -  ( ( tan `  A )  x.  ( tan `  B ) ) )  =/=  0 )
6312, 14, 16, 18mulne0d 9436 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B
) )  =/=  0
)
6452, 54, 15, 62, 63divcan5d 9578 . . 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 ) ) ) ) )
6510, 51, 643eqtr2rd 2335 . 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 )
) ) )
665, 65eqtr4d 2331 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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   sincsin 12361   cosccos 12362   tanctan 12363
This theorem is referenced by:  tanregt0  19917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-tan 12369
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