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Theorem tanval3 12735
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 9049 . . . . . 6  |-  _i  e.  CC
2 simpl 444 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  A  e.  CC )
3 mulcl 9074 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 645 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
5 efcl 12685 . . . . 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 )
71negcli 9368 . . . . . 6  |-  -u _i  e.  CC
8 mulcl 9074 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
97, 2, 8sylancr 645 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( -u _i  x.  A
)  e.  CC )
10 efcl 12685 . . . . 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 9411 . . 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 9107 . . . 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 9074 . . . 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 645 . . 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 10312 . . . . . . . . . . 11  |-  2  e.  ZZ
17 efexp 12702 . . . . . . . . . . 11  |-  ( ( ( _i  x.  A
)  e.  CC  /\  2  e.  ZZ )  ->  ( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
184, 16, 17sylancl 644 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
196sqvald 11520 . . . . . . . . . 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 2468 . . . . . . . . 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 9470 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  -u ( _i  x.  A
) )
221, 2, 21sylancr 645 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( -u _i  x.  A
)  =  -u (
_i  x.  A )
)
2322fveq2d 5732 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( exp `  ( -u _i  x.  A ) )  =  ( exp `  -u ( _i  x.  A ) ) )
2423oveq2d 6097 . . . . . . . . . 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 12697 . . . . . . . . . . 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 2469 . . . . . . . . 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 6099 . . . . . . . 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 9112 . . . . . . . 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 2471 . . . . . . 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 6097 . . . . . 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 9275 . . . . . 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 2468 . . . . 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 10070 . . . . . . . . 9  |-  2  e.  CC
36 mulcl 9074 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 2  x.  ( _i  x.  A
) )  e.  CC )
3735, 4, 36sylancr 645 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( 2  x.  (
_i  x.  A )
)  e.  CC )
38 efcl 12685 . . . . . . . 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 9048 . . . . . . 7  |-  1  e.  CC
41 addcl 9072 . . . . . . 7  |-  ( ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
4239, 40, 41sylancl 644 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
43 ine0 9469 . . . . . . 7  |-  _i  =/=  0
4443a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  ->  _i  =/=  0 )
45 simpr 448 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )
4632, 42, 44, 45mulne0d 9674 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) )  =/=  0 )
4734, 46eqnetrrd 2621 . . . 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 9676 . . 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 12698 . . . 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 9816 . 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 6099 . . . 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 9489 . . . 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 2471 . . 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 6099 . 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 12724 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5756adantr 452 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( cos `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5835a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
2  e.  CC )
5932, 13, 48mulne0bbd 9676 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )
60 2ne0 10083 . . . . . 6  |-  2  =/=  0
6160a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
2  =/=  0 )
6213, 58, 59, 61divne0d 9806 . . . 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 2619 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
64 tanval2 12734 . . 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 457 . 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 2479 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    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292    / cdiv 9677   2c2 10049   ZZcz 10282   ^cexp 11382   expce 12664   cosccos 12667   tanctan 12668
This theorem is referenced by:  tanarg  20514  tanatan  20759
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-tan 12674
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