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Theorem 2efiatan 22970
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 22936 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6291 . . . 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 10595 . . . . . 6  |-  2  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5 ax-icn 9540 . . . . . 6  |-  _i  e.  CC
65a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
7 atancl 22933 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
84, 6, 7mulassd 9608 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
9 halfcl 10753 . . . . . . . . . 10  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
105, 9ax-mp 5 . . . . . . . . 9  |-  ( _i 
/  2 )  e.  CC
113, 5, 10mulassi 9594 . . . . . . . 8  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
123, 5, 10mul12i 9763 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
13 2ne0 10617 . . . . . . . . . . 11  |-  2  =/=  0
145, 3, 13divcan2i 10276 . . . . . . . . . 10  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
1514oveq2i 6286 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
16 ixi 10167 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
1715, 16eqtri 2489 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
1811, 12, 173eqtri 2493 . . . . . . 7  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
1918oveq1i 6285 . . . . . 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 9539 . . . . . . . . . 10  |-  1  e.  CC
21 atandm2 22929 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2221simp1bi 1006 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  A  e.  CC )
23 mulcl 9565 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
245, 22, 23sylancr 663 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
25 subcl 9808 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2620, 24, 25sylancr 663 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
2721simp2bi 1007 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2826, 27logcld 22679 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
29 addcl 9563 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3020, 24, 29sylancr 663 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
3121simp3bi 1008 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3230, 31logcld 22679 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3328, 32subcld 9919 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
3433mulm1d 9997 . . . . . 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 2513 . . . . 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
) ) ) ) )
36 2mulicn 10751 . . . . . . 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 9608 . . . . 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 9935 . . . . 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 2509 . . . 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 2509 . . 3  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
4342fveq2d 5861 . 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 13685 . . 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 661 . 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 22685 . . . . 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 661 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
48 eflog 22685 . . . . 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 661 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
5047, 49oveq12d 6293 . . 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 9856 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
525, 22, 51sylancr 663 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
536mulid1d 9602 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  1 )  =  _i )
5416oveq1i 6285 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
556, 6, 22mulassd 9608 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  _i )  x.  A )  =  ( _i  x.  (
_i  x.  A )
) )
5622mulm1d 9997 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  A )  =  -u A )
5754, 55, 563eqtr3a 2525 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( _i  x.  A ) )  = 
-u A )
5853, 57oveq12d 6293 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  -u A ) )
596, 22, 6pnpcan2d 9957 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  +  _i )  -  ( A  +  _i ) )  =  ( _i  -  A ) )
6052, 58, 593eqtr4d 2511 . . . . . 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 9609 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A ) ) ) )
6362timesd 10770 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =  ( _i  +  _i ) )
6463oveq1d 6290 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( A  +  _i ) )  =  ( ( _i  +  _i )  -  ( A  +  _i ) ) )
6560, 62, 643eqtr4d 2511 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( 2  x.  _i )  -  ( A  +  _i )
) )
666, 61, 24subdid 10001 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  -  (
_i  x.  ( _i  x.  A ) ) ) )
6753, 57oveq12d 6293 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  -  -u A
) )
68 subneg 9857 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  -  -u A
)  =  ( _i  +  A ) )
695, 22, 68sylancr 663 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i 
-  -u A )  =  ( _i  +  A
) )
7067, 69eqtrd 2501 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  A
) )
71 addcom 9754 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  A
)  =  ( A  +  _i ) )
725, 22, 71sylancr 663 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  +  A )  =  ( A  +  _i ) )
7366, 70, 723eqtrd 2505 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( A  +  _i ) )
7465, 73oveq12d 6293 . . . 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 9981 . . . . . 6  |-  _i  =/=  0
7675a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  =/=  0 )
7730, 26, 6, 27, 76divcan5d 10335 . . . 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 9563 . . . . . 6  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
7922, 5, 78sylancl 662 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
80 subneg 9857 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
8122, 5, 80sylancl 662 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
82 atandm 22928 . . . . . . . 8  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
8382simp2bi 1007 . . . . . . 7  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
84 negicn 9810 . . . . . . . 8  |-  -u _i  e.  CC
85 subeq0 9834 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
8685necon3bid 2718 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8722, 84, 86sylancl 662 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8883, 87mpbird 232 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
8981, 88eqnetrrd 2754 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
9037, 79, 79, 89divsubdird 10348 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( A  +  _i ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( A  +  _i )  /  ( A  +  _i ) ) ) )
9174, 77, 903eqtr3d 2509 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  /  ( 1  -  ( _i  x.  A
) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) ) )
9279, 89dividd 10307 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( A  +  _i )  /  ( A  +  _i ) )  =  1 )
9392oveq2d 6291 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  1 ) )
9450, 91, 933eqtrd 2505 . 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 2505 1  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   dom cdm 4992   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486    - cmin 9794   -ucneg 9795    / cdiv 10195   2c2 10574   expce 13648   logclog 22663  arctancatan 22916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-ef 13654  df-sin 13656  df-cos 13657  df-pi 13659  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999  df-log 22665  df-atan 22919
This theorem is referenced by:  tanatan  22971
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