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Theorem 2efiatan 23836
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 23802 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6319 . . . 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 10682 . . . . . 6  |-  2  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5 ax-icn 9600 . . . . . 6  |-  _i  e.  CC
65a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
7 atancl 23799 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
84, 6, 7mulassd 9668 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
9 halfcl 10840 . . . . . . . . . 10  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
105, 9ax-mp 5 . . . . . . . . 9  |-  ( _i 
/  2 )  e.  CC
113, 5, 10mulassi 9654 . . . . . . . 8  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
123, 5, 10mul12i 9830 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
13 2ne0 10704 . . . . . . . . . . 11  |-  2  =/=  0
145, 3, 13divcan2i 10352 . . . . . . . . . 10  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
1514oveq2i 6314 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
16 ixi 10243 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
1715, 16eqtri 2452 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
1811, 12, 173eqtri 2456 . . . . . . 7  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
1918oveq1i 6313 . . . . . 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 9599 . . . . . . . . . 10  |-  1  e.  CC
21 atandm2 23795 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2221simp1bi 1021 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  A  e.  CC )
23 mulcl 9625 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
245, 22, 23sylancr 668 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
25 subcl 9876 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2620, 24, 25sylancr 668 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
2721simp2bi 1022 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2826, 27logcld 23512 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
29 addcl 9623 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3020, 24, 29sylancr 668 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
3121simp3bi 1023 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3230, 31logcld 23512 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3328, 32subcld 9988 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
3433mulm1d 10072 . . . . . 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 2476 . . . . 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 10838 . . . . . . 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 9668 . . . . 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 10004 . . . . 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 2472 . . . 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 2472 . . 3  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
4342fveq2d 5883 . 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 14147 . . 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 666 . 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 23518 . . . . 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 666 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
48 eflog 23518 . . . . 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 666 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
5047, 49oveq12d 6321 . . 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 9924 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
525, 22, 51sylancr 668 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
536mulid1d 9662 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  1 )  =  _i )
5416oveq1i 6313 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
556, 6, 22mulassd 9668 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  _i )  x.  A )  =  ( _i  x.  (
_i  x.  A )
) )
5622mulm1d 10072 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  A )  =  -u A )
5754, 55, 563eqtr3a 2488 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( _i  x.  A ) )  = 
-u A )
5853, 57oveq12d 6321 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  -u A ) )
596, 22, 6pnpcan2d 10026 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  +  _i )  -  ( A  +  _i ) )  =  ( _i  -  A ) )
6052, 58, 593eqtr4d 2474 . . . . . 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 9669 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A ) ) ) )
6362timesd 10857 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =  ( _i  +  _i ) )
6463oveq1d 6318 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( A  +  _i ) )  =  ( ( _i  +  _i )  -  ( A  +  _i ) ) )
6560, 62, 643eqtr4d 2474 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( 2  x.  _i )  -  ( A  +  _i )
) )
666, 61, 24subdid 10076 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  -  (
_i  x.  ( _i  x.  A ) ) ) )
6753, 57oveq12d 6321 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  -  -u A
) )
68 subneg 9925 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  -  -u A
)  =  ( _i  +  A ) )
695, 22, 68sylancr 668 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i 
-  -u A )  =  ( _i  +  A
) )
7067, 69eqtrd 2464 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  A
) )
71 addcom 9821 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  A
)  =  ( A  +  _i ) )
725, 22, 71sylancr 668 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  +  A )  =  ( A  +  _i ) )
7366, 70, 723eqtrd 2468 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( A  +  _i ) )
7465, 73oveq12d 6321 . . . 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 10056 . . . . . 6  |-  _i  =/=  0
7675a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  =/=  0 )
7730, 26, 6, 27, 76divcan5d 10411 . . . 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 9623 . . . . . 6  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
7922, 5, 78sylancl 667 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
80 subneg 9925 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
8122, 5, 80sylancl 667 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
82 atandm 23794 . . . . . . . 8  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
8382simp2bi 1022 . . . . . . 7  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
84 negicn 9878 . . . . . . . 8  |-  -u _i  e.  CC
85 subeq0 9902 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
8685necon3bid 2683 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8722, 84, 86sylancl 667 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8883, 87mpbird 236 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
8981, 88eqnetrrd 2719 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
9037, 79, 79, 89divsubdird 10424 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( A  +  _i ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( A  +  _i )  /  ( A  +  _i ) ) ) )
9174, 77, 903eqtr3d 2472 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  /  ( 1  -  ( _i  x.  A
) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) ) )
9279, 89dividd 10383 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( A  +  _i )  /  ( A  +  _i ) )  =  1 )
9392oveq2d 6319 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  1 ) )
9450, 91, 933eqtrd 2468 . 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 2468 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 188    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   dom cdm 4851   ` cfv 5599  (class class class)co 6303   CCcc 9539   0cc0 9541   1c1 9542   _ici 9543    + caddc 9544    x. cmul 9546    - cmin 9862   -ucneg 9863    / cdiv 10271   2c2 10661   expce 14107   logclog 23496  arctancatan 23782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-limc 22813  df-dv 22814  df-log 23498  df-atan 23785
This theorem is referenced by:  tanatan  23837
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