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Theorem tanadd 12760
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 9064 . . . 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 12721 . . 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 12757 . . . . 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 12758 . . . . 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 6091 . . 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 12724 . . . . . . 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 12724 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  B )  e.  CC )
1512, 14mulcld 9100 . . . . . 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 12725 . . . . . 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 12725 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  B )  e.  CC )
2015, 17, 19adddid 9104 . . . . 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 9268 . . . . . . 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 12721 . . . . . . . . . . 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 6089 . . . . . . . . 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 12723 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  A )  e.  CC )
2625, 12, 16divcan2d 9784 . . . . . . . . 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 2467 . . . . . . . 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 6088 . . . . . . 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 2467 . . . . . 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 9103 . . . . . . 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 12721 . . . . . . . . . . 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 6089 . . . . . . . . 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 12723 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  B )  e.  CC )
3534, 14, 18divcan2d 9784 . . . . . . . . 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 2467 . . . . . . . 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 6089 . . . . . . 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 2467 . . . . . 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 6091 . . . . 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 2467 . . . 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 9040 . . . . . . 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 9100 . . . . . 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 9481 . . . . 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 9097 . . . . . 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 9270 . . . . . . 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 6091 . . . . . . 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 2467 . . . . . 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 6091 . . . . 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 2467 . . . 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 6091 . . 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 9099 . . . 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 9297 . . . . 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 12759 . . . . . . . 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 2681 . . . . 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 9319 . . . . . . 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 2633 . . . . . 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 9666 . . . 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 9808 . . 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 2474 . 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 2470 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 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283    / cdiv 9669   sincsin 12658   cosccos 12659   tanctan 12660
This theorem is referenced by:  tanregt0  20433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-tan 12666
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