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Theorem tanval3 12461
Description: Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
Assertion
Ref Expression
tanval3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( tan `  A
)  =  ( ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  -  1 )  /  ( _i  x.  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) ) ) )

Proof of Theorem tanval3
StepHypRef Expression
1 ax-icn 8841 . . . . . 6  |-  _i  e.  CC
2 simpl 443 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  A  e.  CC )
3 mulcl 8866 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 644 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
5 efcl 12411 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
64, 5syl 15 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
_i  x.  A )
)  e.  CC )
71negcli 9159 . . . . . 6  |-  -u _i  e.  CC
8 mulcl 8866 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
97, 2, 8sylancr 644 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( -u _i  x.  A
)  e.  CC )
10 efcl 12411 . . . . 5  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
119, 10syl 15 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  ( -u _i  x.  A ) )  e.  CC )
126, 11subcld 9202 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
136, 11addcld 8899 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
14 mulcl 8866 . . . 4  |-  ( ( _i  e.  CC  /\  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )  -> 
( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  e.  CC )
151, 13, 14sylancr 644 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  e.  CC )
16 2z 10101 . . . . . . . . . . . 12  |-  2  e.  ZZ
17 efexp 12428 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  A
)  e.  CC  /\  2  e.  ZZ )  ->  ( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
184, 16, 17sylancl 643 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
196sqvald 11289 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
) ^ 2 )  =  ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( _i  x.  A ) ) ) )
2018, 19eqtrd 2348 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  (
_i  x.  A )
) ) )
21 mulneg1 9261 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  -u ( _i  x.  A
) )
221, 2, 21sylancr 644 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( -u _i  x.  A
)  =  -u (
_i  x.  A )
)
2322fveq2d 5567 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  ( -u _i  x.  A ) )  =  ( exp `  -u ( _i  x.  A ) ) )
2423oveq2d 5916 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( -u _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  -u ( _i  x.  A ) ) ) )
25 efcan 12423 . . . . . . . . . . . 12  |-  ( ( _i  x.  A )  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  x.  ( exp `  -u ( _i  x.  A ) ) )  =  1 )
264, 25syl 15 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  -u ( _i  x.  A ) ) )  =  1 )
2724, 26eqtr2d 2349 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
1  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  ( -u _i  x.  A ) ) ) )
2820, 27oveq12d 5918 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =  ( ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  (
_i  x.  A )
) )  +  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( -u _i  x.  A ) ) ) ) )
296, 6, 11adddid 8904 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( -u _i  x.  A ) ) ) ) )
3028, 29eqtr4d 2351 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =  ( ( exp `  ( _i  x.  A
) )  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) )
3130oveq2d 5916 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) )  =  ( _i  x.  ( ( exp `  ( _i  x.  A
) )  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
321a1i 10 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  _i  e.  CC )
3332, 6, 13mul12d 9066 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
_i  x.  A )
)  x.  ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) ) ) )  =  ( ( exp `  ( _i  x.  A
) )  x.  (
_i  x.  ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
3431, 33eqtrd 2348 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
35 2cn 9861 . . . . . . . . . 10  |-  2  e.  CC
36 mulcl 8866 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 2  x.  ( _i  x.  A
) )  e.  CC )
3735, 4, 36sylancr 644 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( 2  x.  (
_i  x.  A )
)  e.  CC )
38 efcl 12411 . . . . . . . . 9  |-  ( ( 2  x.  ( _i  x.  A ) )  e.  CC  ->  ( exp `  ( 2  x.  ( _i  x.  A
) ) )  e.  CC )
3937, 38syl 15 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
2  x.  ( _i  x.  A ) ) )  e.  CC )
40 ax-1cn 8840 . . . . . . . 8  |-  1  e.  CC
41 addcl 8864 . . . . . . . 8  |-  ( ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
4239, 40, 41sylancl 643 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
43 ine0 9260 . . . . . . . 8  |-  _i  =/=  0
4443a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  _i  =/=  0 )
45 simpr 447 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )
4632, 42, 44, 45mulne0d 9465 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) )  =/=  0 )
4734, 46eqnetrrd 2499 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( _i  x.  ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) ) ) )  =/=  0 )
486, 15mulne0bd 9464 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  =/=  0  /\  ( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  =/=  0 )  <-> 
( ( exp `  (
_i  x.  A )
)  x.  ( _i  x.  ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) ) ) )  =/=  0 ) )
4947, 48mpbird 223 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  =/=  0  /\  ( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  =/=  0 ) )
5049simprd 449 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  =/=  0 )
51 efne0 12424 . . . 4  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =/=  0 )
524, 51syl 15 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
_i  x.  A )
)  =/=  0 )
5312, 15, 6, 50, 52divcan5d 9607 . 2  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  x.  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  ( ( exp `  ( _i  x.  A ) )  x.  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
5420, 27oveq12d 5918 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  -  1 )  =  ( ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  (
_i  x.  A )
) )  -  (
( exp `  (
_i  x.  A )
)  x.  ( exp `  ( -u _i  x.  A ) ) ) ) )
556, 6, 11subdid 9280 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( _i  x.  A ) ) )  -  ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( -u _i  x.  A ) ) ) ) )
5654, 55eqtr4d 2351 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  -  1 )  =  ( ( exp `  ( _i  x.  A
) )  x.  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) ) )
5756, 34oveq12d 5918 . 2  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( ( exp `  ( 2  x.  (
_i  x.  A )
) )  -  1 )  /  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  A )
) )  +  1 ) ) )  =  ( ( ( exp `  ( _i  x.  A
) )  x.  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  ( ( exp `  ( _i  x.  A ) )  x.  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) ) )
58 cosval 12450 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5958adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( cos `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
6035a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
2  e.  CC )
61 mulne0b 9454 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )  -> 
( ( _i  =/=  0  /\  ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )  <-> 
( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  =/=  0 ) )
621, 13, 61sylancr 644 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( _i  =/=  0  /\  ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )  <-> 
( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  =/=  0 ) )
6350, 62mpbird 223 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  =/=  0  /\  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 ) )
6463simprd 449 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )
65 2ne0 9874 . . . . . 6  |-  2  =/=  0
6665a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
2  =/=  0 )
6713, 60, 64, 66divne0d 9597 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =/=  0 )
6859, 67eqnetrd 2497 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
69 tanval2 12460 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
7068, 69syldan 456 . 2  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( tan `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
7153, 57, 703eqtr4rd 2359 1  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( tan `  A
)  =  ( ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  -  1 )  /  ( _i  x.  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   ` cfv 5292  (class class class)co 5900   CCcc 8780   0cc0 8782   1c1 8783   _ici 8784    + caddc 8785    x. cmul 8787    - cmin 9082   -ucneg 9083    / cdiv 9468   2c2 9840   ZZcz 10071   ^cexp 11151   expce 12390   cosccos 12393   tanctan 12394
This theorem is referenced by:  tanarg  20023  tanatan  20268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-pm 6818  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-ico 10709  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-fac 11336  df-bc 11363  df-hash 11385  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206  df-ef 12396  df-sin 12398  df-cos 12399  df-tan 12400
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