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Theorem tanval3 13437
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 9360 . . . . . 6  |-  _i  e.  CC
2 simpl 457 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  A  e.  CC )
3 mulcl 9385 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 663 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
5 efcl 13387 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
64, 5syl 16 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
_i  x.  A )
)  e.  CC )
7 negicn 9630 . . . . . 6  |-  -u _i  e.  CC
8 mulcl 9385 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
97, 2, 8sylancr 663 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( -u _i  x.  A
)  e.  CC )
10 efcl 13387 . . . . 5  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
119, 10syl 16 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  ( -u _i  x.  A ) )  e.  CC )
126, 11subcld 9738 . . 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 9424 . . . 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 9385 . . . 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 663 . . 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 10697 . . . . . . . . . . 11  |-  2  e.  ZZ
17 efexp 13404 . . . . . . . . . . 11  |-  ( ( ( _i  x.  A
)  e.  CC  /\  2  e.  ZZ )  ->  ( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
184, 16, 17sylancl 662 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
196sqvald 12024 . . . . . . . . . 10  |-  ( ( 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 2475 . . . . . . . . 9  |-  ( ( 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 9800 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  -u ( _i  x.  A
) )
221, 2, 21sylancr 663 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( -u _i  x.  A
)  =  -u (
_i  x.  A )
)
2322fveq2d 5714 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  ( -u _i  x.  A ) )  =  ( exp `  -u ( _i  x.  A ) ) )
2423oveq2d 6126 . . . . . . . . . 10  |-  ( ( 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 13399 . . . . . . . . . . 11  |-  ( ( _i  x.  A )  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  x.  ( exp `  -u ( _i  x.  A ) ) )  =  1 )
264, 25syl 16 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  -u ( _i  x.  A ) ) )  =  1 )
2724, 26eqtr2d 2476 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
1  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  ( -u _i  x.  A ) ) ) )
2820, 27oveq12d 6128 . . . . . . . 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 `  (
_i  x.  A )
)  x.  ( exp `  ( -u _i  x.  A ) ) ) ) )
296, 6, 11adddid 9429 . . . . . . . 8  |-  ( ( 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 2478 . . . . . . 7  |-  ( ( 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 6126 . . . . . 6  |-  ( ( 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 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  _i  e.  CC )
3332, 6, 13mul12d 9597 . . . . . 6  |-  ( ( 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 2475 . . . . 5  |-  ( ( 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 10411 . . . . . . . . 9  |-  2  e.  CC
36 mulcl 9385 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 2  x.  ( _i  x.  A
) )  e.  CC )
3735, 4, 36sylancr 663 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( 2  x.  (
_i  x.  A )
)  e.  CC )
38 efcl 13387 . . . . . . . 8  |-  ( ( 2  x.  ( _i  x.  A ) )  e.  CC  ->  ( exp `  ( 2  x.  ( _i  x.  A
) ) )  e.  CC )
3937, 38syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
2  x.  ( _i  x.  A ) ) )  e.  CC )
40 ax-1cn 9359 . . . . . . 7  |-  1  e.  CC
41 addcl 9383 . . . . . . 7  |-  ( ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
4239, 40, 41sylancl 662 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
43 ine0 9799 . . . . . . 7  |-  _i  =/=  0
4443a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  _i  =/=  0 )
45 simpr 461 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )
4632, 42, 44, 45mulne0d 10007 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) )  =/=  0 )
4734, 46eqnetrrd 2653 . . . 4  |-  ( ( 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, 15, 47mulne0bbd 10011 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) )  =/=  0 )
49 efne0 13400 . . . 4  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =/=  0 )
504, 49syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
_i  x.  A )
)  =/=  0 )
5112, 15, 6, 48, 50divcan5d 10152 . 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 ) ) ) ) ) )
5220, 27oveq12d 6128 . . . 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 ) ) ) ) )
536, 6, 11subdid 9819 . . . 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 ) ) ) ) )
5452, 53eqtr4d 2478 . . 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 ) ) ) ) )
5554, 34oveq12d 6128 . 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 ) ) ) ) ) ) )
56 cosval 13426 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5756adantr 465 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( cos `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
58 2cnd 10413 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
2  e.  CC )
5932, 13, 48mulne0bbd 10011 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )
60 2ne0 10433 . . . . . 6  |-  2  =/=  0
6160a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
2  =/=  0 )
6213, 58, 59, 61divne0d 10142 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =/=  0 )
6357, 62eqnetrd 2651 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
64 tanval2 13436 . . 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 ) ) ) ) ) )
6563, 64syldan 470 . 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 ) ) ) ) ) )
6651, 55, 653eqtr4rd 2486 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   ` cfv 5437  (class class class)co 6110   CCcc 9299   0cc0 9301   1c1 9302   _ici 9303    + caddc 9304    x. cmul 9306    - cmin 9614   -ucneg 9615    / cdiv 10012   2c2 10390   ZZcz 10665   ^cexp 11884   expce 13366   cosccos 13369   tanctan 13370
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379  ax-addf 9380  ax-mulf 9381
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 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-pm 7236  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-sup 7710  df-oi 7743  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-ico 11325  df-fz 11457  df-fzo 11568  df-fl 11661  df-seq 11826  df-exp 11885  df-fac 12071  df-bc 12098  df-hash 12123  df-shft 12575  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-limsup 12968  df-clim 12985  df-rlim 12986  df-sum 13183  df-ef 13372  df-sin 13374  df-cos 13375  df-tan 13376
This theorem is referenced by:  tanarg  22087  tanatan  22333
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