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Theorem tanadd 12696
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 9006 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
21adantr 452 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( A  +  B )  e.  CC )
3 simpr3 965 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  ( A  +  B
) )  =/=  0
)
4 tanval 12657 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  ( cos `  ( A  +  B ) )  =/=  0 )  -> 
( tan `  ( A  +  B )
)  =  ( ( sin `  ( A  +  B ) )  /  ( cos `  ( A  +  B )
) ) )
52, 3, 4syl2anc 643 . 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 12693 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sin `  ( A  +  B )
)  =  ( ( ( sin `  A
)  x.  ( cos `  B ) )  +  ( ( cos `  A
)  x.  ( sin `  B ) ) ) )
76adantr 452 . . . 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 12694 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
98adantr 452 . . . 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 6039 . . 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 731 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  A  e.  CC )
1211coscld 12660 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  A )  e.  CC )
13 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  B  e.  CC )
1413coscld 12660 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  B )  e.  CC )
1512, 14mulcld 9042 . . . . . 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 963 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  A )  =/=  0
)
1711, 16tancld 12661 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  A )  e.  CC )
18 simpr2 964 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  B )  =/=  0
)
1913, 18tancld 12661 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  B )  e.  CC )
2015, 17, 19adddid 9046 . . . . 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 9209 . . . . . . 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 12657 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
2311, 16, 22syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
2423oveq2d 6037 . . . . . . . . 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 12659 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  A )  e.  CC )
2625, 12, 16divcan2d 9725 . . . . . . . . 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 2420 . . . . . . . 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 6036 . . . . . . 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 2420 . . . . . 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 9045 . . . . . . 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 12657 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  ( cos `  B )  =/=  0 )  -> 
( tan `  B
)  =  ( ( sin `  B )  /  ( cos `  B
) ) )
3213, 18, 31syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  B )  =  ( ( sin `  B
)  /  ( cos `  B ) ) )
3332oveq2d 6037 . . . . . . . . 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 12659 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  B )  e.  CC )
3534, 14, 18divcan2d 9725 . . . . . . . . 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 2420 . . . . . . . 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 6037 . . . . . . 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 2420 . . . . . 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 6039 . . . . 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 2420 . . . 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 8982 . . . . . . 7  |-  1  e.  CC
4241a1i 11 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  1  e.  CC )
4317, 19mulcld 9042 . . . . . 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 9422 . . . . 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 9039 . . . . . 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 9211 . . . . . . 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 6039 . . . . . . 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 2420 . . . . . 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 6039 . . . . 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 2420 . . . 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 6039 . . 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 9041 . . . 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 9238 . . . . 5  |-  ( ( 1  e.  CC  /\  ( ( tan `  A
)  x.  ( tan `  B ) )  e.  CC )  ->  (
1  -  ( ( tan `  A )  x.  ( tan `  B
) ) )  e.  CC )
5441, 43, 53sylancr 645 . . . 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 12695 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( ( cos `  ( A  +  B ) )  =/=  0  <->  ( ( tan `  A )  x.  ( tan `  B ) )  =/=  1 ) )
56553adantr3 1118 . . . . . . 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 202 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( tan `  A )  x.  ( tan `  B
) )  =/=  1
)
5857necomd 2634 . . . . 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 9260 . . . . . . 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 2586 . . . . . 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 645 . . . . 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 224 . . . 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 9607 . . . 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 9749 . . 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 2427 . 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 2423 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    - cmin 9224    / cdiv 9610   sincsin 12594   cosccos 12595   tanctan 12596
This theorem is referenced by:  tanregt0  20309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-ico 10855  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-tan 12602
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