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Theorem tanval3ap 11348
Description: Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
Assertion
Ref Expression
tanval3ap  |-  ( ( 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 tanval3ap
StepHypRef Expression
1 ax-icn 7683 . . . . . 6  |-  _i  e.  CC
2 simpl 108 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  A  e.  CC )
3 mulcl 7715 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 410 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
_i  x.  A )  e.  CC )
5 efcl 11297 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
64, 5syl 14 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
7 negicn 7931 . . . . . 6  |-  -u _i  e.  CC
8 mulcl 7715 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
97, 2, 8sylancr 410 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( -u _i  x.  A )  e.  CC )
10 efcl 11297 . . . . 5  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
119, 10syl 14 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
126, 11subcld 8041 . . 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 7753 . . . 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 7715 . . . 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 410 . . 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 9050 . . . . . . . . . . 11  |-  2  e.  ZZ
17 efexp 11315 . . . . . . . . . . 11  |-  ( ( ( _i  x.  A
)  e.  CC  /\  2  e.  ZZ )  ->  ( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
184, 16, 17sylancl 409 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( 2  x.  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
) ^ 2 ) )
196sqvald 10389 . . . . . . . . . 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 2150 . . . . . . . . 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 8125 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  -u ( _i  x.  A
) )
221, 2, 21sylancr 410 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( -u _i  x.  A )  =  -u ( _i  x.  A ) )
2322fveq2d 5393 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( -u _i  x.  A ) )  =  ( exp `  -u (
_i  x.  A )
) )
2423oveq2d 5758 . . . . . . . . . 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 11309 . . . . . . . . . . 11  |-  ( ( _i  x.  A )  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  x.  ( exp `  -u ( _i  x.  A ) ) )  =  1 )
264, 25syl 14 . . . . . . . . . 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 2151 . . . . . . . . 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 5760 . . . . . . . 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 7758 . . . . . . . 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 2153 . . . . . . 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 5758 . . . . . 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 9 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  _i  e.  CC )
3332, 6, 13mul12d 7882 . . . . . 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 2150 . . . . 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 8759 . . . . . . . . 9  |-  2  e.  CC
36 mulcl 7715 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 2  x.  ( _i  x.  A
) )  e.  CC )
3735, 4, 36sylancr 410 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
2  x.  ( _i  x.  A ) )  e.  CC )
38 efcl 11297 . . . . . . . 8  |-  ( ( 2  x.  ( _i  x.  A ) )  e.  CC  ->  ( exp `  ( 2  x.  ( _i  x.  A
) ) )  e.  CC )
3937, 38syl 14 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( 2  x.  ( _i  x.  A
) ) )  e.  CC )
40 ax-1cn 7681 . . . . . . 7  |-  1  e.  CC
41 addcl 7713 . . . . . . 7  |-  ( ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
4239, 40, 41sylancl 409 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
43 iap0 8911 . . . . . . 7  |-  _i #  0
4443a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  _i #  0 )
45 simpr 109 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )
4632, 42, 44, 45mulap0d 8387 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
_i  x.  ( ( exp `  ( 2  x.  ( _i  x.  A
) ) )  +  1 ) ) #  0 )
4734, 46eqbrtrrd 3922 . . . 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, 47mulap0bbd 8389 . . 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 efap0 11310 . . . 4  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) ) #  0 )
504, 49syl 14 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( _i  x.  A ) ) #  0 )
5112, 15, 6, 48, 50divcanap5d 8545 . 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 5760 . . . 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 8144 . . . 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 2153 . . 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 5760 . 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 11337 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5756adantr 274 . . . 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 8761 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  2  e.  CC )
5932, 13, 48mulap0bbd 8389 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) #  0 )
60 2ap0 8781 . . . . . 6  |-  2 #  0
6160a1i 9 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  2 #  0 )
6213, 58, 59, 61divap0d 8534 . . . 4  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) #  0 )
6357, 62eqbrtrd 3920 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( cos `  A ) #  0 )
64 tanval2ap 11347 . . 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 280 . 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 2161 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 103    = wceq 1316    e. wcel 1465   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   CCcc 7586   0cc0 7588   1c1 7589   _ici 7590    + caddc 7591    x. cmul 7593    - cmin 7901   -ucneg 7902   # cap 8311    / cdiv 8400   2c2 8739   ZZcz 9022   ^cexp 10260   expce 11275   cosccos 11278   tanctan 11279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-disj 3877  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-frec 6256  df-1o 6281  df-oadd 6285  df-er 6397  df-en 6603  df-dom 6604  df-fin 6605  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-ico 9645  df-fz 9759  df-fzo 9888  df-seqfrec 10187  df-exp 10261  df-fac 10440  df-bc 10462  df-ihash 10490  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-clim 11016  df-sumdc 11091  df-ef 11281  df-sin 11283  df-cos 11284  df-tan 11285
This theorem is referenced by: (None)
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