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Theorem tanadd 13472
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 9385 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
21adantr 465 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( A  +  B )  e.  CC )
3 simpr3 996 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  ( A  +  B
) )  =/=  0
)
4 tanval 13433 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  ( cos `  ( A  +  B ) )  =/=  0 )  -> 
( tan `  ( A  +  B )
)  =  ( ( sin `  ( A  +  B ) )  /  ( cos `  ( A  +  B )
) ) )
52, 3, 4syl2anc 661 . 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 13469 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( sin `  ( A  +  B )
)  =  ( ( ( sin `  A
)  x.  ( cos `  B ) )  +  ( ( cos `  A
)  x.  ( sin `  B ) ) ) )
76adantr 465 . . . 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 13470 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
98adantr 465 . . . 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 6130 . . 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 753 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  A  e.  CC )
1211coscld 13436 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  A )  e.  CC )
13 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  B  e.  CC )
1413coscld 13436 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  B )  e.  CC )
1512, 14mulcld 9427 . . . . . 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 994 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  A )  =/=  0
)
1711, 16tancld 13437 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  A )  e.  CC )
18 simpr2 995 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( cos `  B )  =/=  0
)
1913, 18tancld 13437 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  B )  e.  CC )
2015, 17, 19adddid 9431 . . . . 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 9600 . . . . . . 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 13433 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
2311, 16, 22syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
2423oveq2d 6128 . . . . . . . . 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 13435 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  A )  e.  CC )
2625, 12, 16divcan2d 10130 . . . . . . . . 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 2475 . . . . . . . 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 6127 . . . . . . 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 2475 . . . . . 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 9430 . . . . . . 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 13433 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  ( cos `  B )  =/=  0 )  -> 
( tan `  B
)  =  ( ( sin `  B )  /  ( cos `  B
) ) )
3213, 18, 31syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( tan `  B )  =  ( ( sin `  B
)  /  ( cos `  B ) ) )
3332oveq2d 6128 . . . . . . . . 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 13435 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( sin `  B )  e.  CC )
3534, 14, 18divcan2d 10130 . . . . . . . . 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 2475 . . . . . . . 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 6128 . . . . . . 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 2475 . . . . . 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 6130 . . . . 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 2475 . . . 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 9423 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  1  e.  CC )
4217, 19mulcld 9427 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( tan `  A )  x.  ( tan `  B
) )  e.  CC )
4315, 41, 42subdid 9821 . . . . 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 ) ) ) ) )
4415mulid1d 9424 . . . . . 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 ) ) )
4512, 14, 17, 19mul4d 9602 . . . . . . 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 ) ) ) )
4627, 36oveq12d 6130 . . . . . . 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 ) ) )
4745, 46eqtrd 2475 . . . . . 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 ) ) )
4844, 47oveq12d 6130 . . . . 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 ) ) ) )
4943, 48eqtrd 2475 . . . 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 ) ) ) )
5040, 49oveq12d 6130 . . 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 ) ) ) ) )
5117, 19addcld 9426 . . . 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 9361 . . . . 5  |-  1  e.  CC
53 subcl 9630 . . . . 5  |-  ( ( 1  e.  CC  /\  ( ( tan `  A
)  x.  ( tan `  B ) )  e.  CC )  ->  (
1  -  ( ( tan `  A )  x.  ( tan `  B
) ) )  e.  CC )
5452, 42, 53sylancr 663 . . . 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 13471 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( ( cos `  ( A  +  B ) )  =/=  0  <->  ( ( tan `  A )  x.  ( tan `  B ) )  =/=  1 ) )
56553adantr3 1149 . . . . . . 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 210 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( tan `  A )  x.  ( tan `  B
) )  =/=  1
)
5857necomd 2640 . . . . 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 9656 . . . . . . 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 2637 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( ( tan `  A
)  x.  ( tan `  B ) )  e.  CC )  ->  (
( 1  -  (
( tan `  A
)  x.  ( tan `  B ) ) )  =/=  0  <->  1  =/=  ( ( tan `  A
)  x.  ( tan `  B ) ) ) )
6152, 42, 60sylancr 663 . . . . 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 232 . . . 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 10009 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0  /\  ( cos `  ( A  +  B )
)  =/=  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B
) )  =/=  0
)
6451, 54, 15, 62, 63divcan5d 10154 . . 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, 50, 643eqtr2rd 2482 . 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 2478 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   ` cfv 5439  (class class class)co 6112   CCcc 9301   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308    - cmin 9616    / cdiv 10014   sincsin 13370   cosccos 13371   tanctan 13372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-ico 11327  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377  df-tan 13378
This theorem is referenced by:  tanregt0  22017
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