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Theorem 2efiatan 20748
Description: Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
2efiatan  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )

Proof of Theorem 2efiatan
StepHypRef Expression
1 atanval 20714 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6089 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( ( 2  x.  _i )  x.  ( (
_i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) ) )
3 2cn 10060 . . . . . 6  |-  2  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5 ax-icn 9039 . . . . . 6  |-  _i  e.  CC
65a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
7 atancl 20711 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
84, 6, 7mulassd 9101 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
9 halfcl 10183 . . . . . . . . . 10  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
105, 9ax-mp 8 . . . . . . . . 9  |-  ( _i 
/  2 )  e.  CC
113, 5, 10mulassi 9089 . . . . . . . 8  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
123, 5, 10mul12i 9251 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
13 2ne0 10073 . . . . . . . . . . 11  |-  2  =/=  0
145, 3, 13divcan2i 9747 . . . . . . . . . 10  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
1514oveq2i 6084 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
16 ixi 9641 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
1715, 16eqtri 2455 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
1811, 12, 173eqtri 2459 . . . . . . 7  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
1918oveq1i 6083 . . . . . 6  |-  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( -u 1  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
20 ax-1cn 9038 . . . . . . . . . 10  |-  1  e.  CC
21 atandm2 20707 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2221simp1bi 972 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  A  e.  CC )
23 mulcl 9064 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
245, 22, 23sylancr 645 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
25 subcl 9295 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2620, 24, 25sylancr 645 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
2721simp2bi 973 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2826, 27logcld 20458 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
29 addcl 9062 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3020, 24, 29sylancr 645 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
3121simp3bi 974 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3230, 31logcld 20458 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3328, 32subcld 9401 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
3433mulm1d 9475 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  = 
-u ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
3519, 34syl5eq 2479 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  -u ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
363, 5mulcli 9085 . . . . . . 7  |-  ( 2  x.  _i )  e.  CC
3736a1i 11 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
3810a1i 11 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
3937, 38, 33mulassd 9101 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( ( 2  x.  _i )  x.  ( ( _i  / 
2 )  x.  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
4028, 32negsubdi2d 9417 . . . . 5  |-  ( A  e.  dom arctan  ->  -u (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )
4135, 39, 403eqtr3d 2475 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
422, 8, 413eqtr3d 2475 . . 3  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
4342fveq2d 5724 . 2  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
44 efsub 12691 . . 3  |-  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC  /\  ( log `  ( 1  -  ( _i  x.  A ) ) )  e.  CC )  -> 
( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  /  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
4532, 28, 44syl2anc 643 . 2  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  /  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
46 eflog 20464 . . . . 5  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( 1  +  ( _i  x.  A
) ) )
4730, 31, 46syl2anc 643 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
48 eflog 20464 . . . . 5  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( 1  -  ( _i  x.  A
) ) )
4926, 27, 48syl2anc 643 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
5047, 49oveq12d 6091 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  / 
( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( ( 1  +  ( _i  x.  A ) )  /  ( 1  -  ( _i  x.  A
) ) ) )
51 negsub 9339 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
525, 22, 51sylancr 645 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
536mulid1d 9095 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  1 )  =  _i )
5416oveq1i 6083 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
556, 6, 22mulassd 9101 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  _i )  x.  A )  =  ( _i  x.  (
_i  x.  A )
) )
5622mulm1d 9475 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  A )  =  -u A )
5754, 55, 563eqtr3a 2491 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( _i  x.  A ) )  = 
-u A )
5853, 57oveq12d 6091 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  -u A ) )
596, 22, 6pnpcan2d 9439 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  +  _i )  -  ( A  +  _i ) )  =  ( _i  -  A ) )
6052, 58, 593eqtr4d 2477 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A
) ) )  =  ( ( _i  +  _i )  -  ( A  +  _i )
) )
6120a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  1  e.  CC )
626, 61, 24adddid 9102 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A ) ) ) )
6362timesd 10200 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =  ( _i  +  _i ) )
6463oveq1d 6088 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( A  +  _i ) )  =  ( ( _i  +  _i )  -  ( A  +  _i ) ) )
6560, 62, 643eqtr4d 2477 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( 2  x.  _i )  -  ( A  +  _i )
) )
666, 61, 24subdid 9479 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  -  (
_i  x.  ( _i  x.  A ) ) ) )
6753, 57oveq12d 6091 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  -  -u A
) )
68 subneg 9340 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  -  -u A
)  =  ( _i  +  A ) )
695, 22, 68sylancr 645 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i 
-  -u A )  =  ( _i  +  A
) )
7067, 69eqtrd 2467 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  A
) )
71 addcom 9242 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  A
)  =  ( A  +  _i ) )
725, 22, 71sylancr 645 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  +  A )  =  ( A  +  _i ) )
7366, 70, 723eqtrd 2471 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( A  +  _i ) )
7465, 73oveq12d 6091 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 1  +  ( _i  x.  A ) ) )  /  ( _i  x.  ( 1  -  (
_i  x.  A )
) ) )  =  ( ( ( 2  x.  _i )  -  ( A  +  _i ) )  /  ( A  +  _i )
) )
75 ine0 9459 . . . . . 6  |-  _i  =/=  0
7675a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  =/=  0 )
7730, 26, 6, 27, 76divcan5d 9806 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 1  +  ( _i  x.  A ) ) )  /  ( _i  x.  ( 1  -  (
_i  x.  A )
) ) )  =  ( ( 1  +  ( _i  x.  A
) )  /  (
1  -  ( _i  x.  A ) ) ) )
78 addcl 9062 . . . . . 6  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
7922, 5, 78sylancl 644 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
80 subneg 9340 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
8122, 5, 80sylancl 644 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
82 atandm 20706 . . . . . . . 8  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
8382simp2bi 973 . . . . . . 7  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
845negcli 9358 . . . . . . . 8  |-  -u _i  e.  CC
85 subeq0 9317 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
8685necon3bid 2633 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8722, 84, 86sylancl 644 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8883, 87mpbird 224 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
8981, 88eqnetrrd 2618 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
9037, 79, 79, 89divsubdird 9819 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( A  +  _i ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( A  +  _i )  /  ( A  +  _i ) ) ) )
9174, 77, 903eqtr3d 2475 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  /  ( 1  -  ( _i  x.  A
) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) ) )
9279, 89dividd 9778 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( A  +  _i )  /  ( A  +  _i ) )  =  1 )
9392oveq2d 6089 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  1 ) )
9450, 91, 933eqtrd 2471 . 2  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  / 
( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
9543, 45, 943eqtrd 2471 1  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   dom cdm 4870   ` cfv 5446  (class class class)co 6073   CCcc 8978   0cc0 8980   1c1 8981   _ici 8982    + caddc 8983    x. cmul 8985    - cmin 9281   -ucneg 9282    / cdiv 9667   2c2 10039   expce 12654   logclog 20442  arctancatan 20694
This theorem is referenced by:  tanatan  20749
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-addf 9059  ax-mulf 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7469  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-ioo 10910  df-ioc 10911  df-ico 10912  df-icc 10913  df-fz 11034  df-fzo 11126  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-fac 11557  df-bc 11584  df-hash 11609  df-shft 11872  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-limsup 12255  df-clim 12272  df-rlim 12273  df-sum 12470  df-ef 12660  df-sin 12662  df-cos 12663  df-pi 12665  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-starv 13534  df-sca 13535  df-vsca 13536  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-hom 13543  df-cco 13544  df-rest 13640  df-topn 13641  df-topgen 13657  df-pt 13658  df-prds 13661  df-xrs 13716  df-0g 13717  df-gsum 13718  df-qtop 13723  df-imas 13724  df-xps 13726  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-submnd 14729  df-mulg 14805  df-cntz 15106  df-cmn 15404  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-fbas 16689  df-fg 16690  df-cnfld 16694  df-top 16953  df-bases 16955  df-topon 16956  df-topsp 16957  df-cld 17073  df-ntr 17074  df-cls 17075  df-nei 17152  df-lp 17190  df-perf 17191  df-cn 17281  df-cnp 17282  df-haus 17369  df-tx 17584  df-hmeo 17777  df-fil 17868  df-fm 17960  df-flim 17961  df-flf 17962  df-xms 18340  df-ms 18341  df-tms 18342  df-cncf 18898  df-limc 19743  df-dv 19744  df-log 20444  df-atan 20697
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