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Theorem tanval3ap 11857
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 7967 . . . . . 6  |-  _i  e.  CC
2 simpl 109 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  A  e.  CC )
3 mulcl 7999 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 414 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
_i  x.  A )  e.  CC )
5 efcl 11807 . . . . 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 8220 . . . . . 6  |-  -u _i  e.  CC
8 mulcl 7999 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
97, 2, 8sylancr 414 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( -u _i  x.  A )  e.  CC )
10 efcl 11807 . . . . 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 8330 . . 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 8039 . . . 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 7999 . . . 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 414 . . 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 9345 . . . . . . . . . . 11  |-  2  e.  ZZ
17 efexp 11825 . . . . . . . . . . 11  |-  ( ( ( _i  x.  A
)  e.  CC  /\  2  e.  ZZ )  ->  ( exp `  (
2  x.  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) ) ^ 2 ) )
184, 16, 17sylancl 413 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( 2  x.  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
) ^ 2 ) )
196sqvald 10741 . . . . . . . . . 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 2226 . . . . . . . . 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 8414 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  -u ( _i  x.  A
) )
221, 2, 21sylancr 414 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( -u _i  x.  A )  =  -u ( _i  x.  A ) )
2322fveq2d 5558 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( exp `  ( -u _i  x.  A ) )  =  ( exp `  -u (
_i  x.  A )
) )
2423oveq2d 5934 . . . . . . . . . 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 11819 . . . . . . . . . . 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 2227 . . . . . . . . 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 5936 . . . . . . . 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 8044 . . . . . . . 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 2229 . . . . . . 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 5934 . . . . . 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 8171 . . . . . 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 2226 . . . . 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 9053 . . . . . . . . 9  |-  2  e.  CC
36 mulcl 7999 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 2  x.  ( _i  x.  A
) )  e.  CC )
3735, 4, 36sylancr 414 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
2  x.  ( _i  x.  A ) )  e.  CC )
38 efcl 11807 . . . . . . . 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 7965 . . . . . . 7  |-  1  e.  CC
41 addcl 7997 . . . . . . 7  |-  ( ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
4239, 40, 41sylancl 413 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 )  e.  CC )
43 iap0 9205 . . . . . . 7  |-  _i #  0
4443a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  _i #  0 )
45 simpr 110 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )
4632, 42, 44, 45mulap0d 8677 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
_i  x.  ( ( exp `  ( 2  x.  ( _i  x.  A
) ) )  +  1 ) ) #  0 )
4734, 46eqbrtrrd 4053 . . . 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 8679 . . 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 11820 . . . 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 8836 . 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 5936 . . . 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 8433 . . . 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 2229 . . 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 5936 . 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 11846 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5756adantr 276 . . . 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 9055 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  2  e.  CC )
5932, 13, 48mulap0bbd 8679 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) #  0 )
60 2ap0 9075 . . . . . 6  |-  2 #  0
6160a1i 9 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  2 #  0 )
6213, 58, 59, 61divap0d 8825 . . . 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 4051 . . 3  |-  ( ( A  e.  CC  /\  ( ( exp `  (
2  x.  ( _i  x.  A ) ) )  +  1 ) #  0 )  ->  ( cos `  A ) #  0 )
64 tanval2ap 11856 . . 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 282 . 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 2237 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 104    = wceq 1364    e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   _ici 7874    + caddc 7875    x. cmul 7877    - cmin 8190   -ucneg 8191   # cap 8600    / cdiv 8691   2c2 9033   ZZcz 9317   ^cexp 10609   expce 11785   cosccos 11788   tanctan 11789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-disj 4007  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-ico 9960  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-bc 10819  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497  df-ef 11791  df-sin 11793  df-cos 11794  df-tan 11795
This theorem is referenced by: (None)
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