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Theorem 2efiatan 22311
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 22277 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6105 . . . 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 10390 . . . . . 6  |-  2  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5 ax-icn 9339 . . . . . 6  |-  _i  e.  CC
65a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
7 atancl 22274 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
84, 6, 7mulassd 9407 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
9 halfcl 10548 . . . . . . . . . 10  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
105, 9ax-mp 5 . . . . . . . . 9  |-  ( _i 
/  2 )  e.  CC
113, 5, 10mulassi 9393 . . . . . . . 8  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
123, 5, 10mul12i 9562 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
13 2ne0 10412 . . . . . . . . . . 11  |-  2  =/=  0
145, 3, 13divcan2i 10072 . . . . . . . . . 10  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
1514oveq2i 6100 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
16 ixi 9963 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
1715, 16eqtri 2461 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
1811, 12, 173eqtri 2465 . . . . . . 7  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
1918oveq1i 6099 . . . . . 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 9338 . . . . . . . . . 10  |-  1  e.  CC
21 atandm2 22270 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2221simp1bi 1003 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  A  e.  CC )
23 mulcl 9364 . . . . . . . . . . 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 9607 . . . . . . . . . 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 1004 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2826, 27logcld 22020 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
29 addcl 9362 . . . . . . . . . 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 1005 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3230, 31logcld 22020 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3328, 32subcld 9717 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
3433mulm1d 9794 . . . . . 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 2485 . . . . 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 10546 . . . . . . 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 9407 . . . . 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 9733 . . . . 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 2481 . . . 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 2481 . . 3  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
4342fveq2d 5693 . 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 13382 . . 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 22026 . . . . 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 22026 . . . . 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 6107 . . 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 9655 . . . . . . . 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 9401 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  1 )  =  _i )
5416oveq1i 6099 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
556, 6, 22mulassd 9407 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  _i )  x.  A )  =  ( _i  x.  (
_i  x.  A )
) )
5622mulm1d 9794 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  A )  =  -u A )
5754, 55, 563eqtr3a 2497 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( _i  x.  A ) )  = 
-u A )
5853, 57oveq12d 6107 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  -u A ) )
596, 22, 6pnpcan2d 9755 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  +  _i )  -  ( A  +  _i ) )  =  ( _i  -  A ) )
6052, 58, 593eqtr4d 2483 . . . . . 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 9408 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A ) ) ) )
6362timesd 10565 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =  ( _i  +  _i ) )
6463oveq1d 6104 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( A  +  _i ) )  =  ( ( _i  +  _i )  -  ( A  +  _i ) ) )
6560, 62, 643eqtr4d 2483 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( 2  x.  _i )  -  ( A  +  _i )
) )
666, 61, 24subdid 9798 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  -  (
_i  x.  ( _i  x.  A ) ) ) )
6753, 57oveq12d 6107 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  -  -u A
) )
68 subneg 9656 . . . . . . . 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 2473 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  A
) )
71 addcom 9553 . . . . . . 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 2477 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( A  +  _i ) )
7465, 73oveq12d 6107 . . . 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 9778 . . . . . 6  |-  _i  =/=  0
7675a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  =/=  0 )
7730, 26, 6, 27, 76divcan5d 10131 . . . 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 9362 . . . . . 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 9656 . . . . . . 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 22269 . . . . . . . 8  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
8382simp2bi 1004 . . . . . . 7  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
84 negicn 9609 . . . . . . . 8  |-  -u _i  e.  CC
85 subeq0 9633 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
8685necon3bid 2641 . . . . . . . 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 2626 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
9037, 79, 79, 89divsubdird 10144 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( A  +  _i ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( A  +  _i )  /  ( A  +  _i ) ) ) )
9174, 77, 903eqtr3d 2481 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  /  ( 1  -  ( _i  x.  A
) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) ) )
9279, 89dividd 10103 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( A  +  _i )  /  ( A  +  _i ) )  =  1 )
9392oveq2d 6105 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  1 ) )
9450, 91, 933eqtrd 2477 . 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 2477 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 1369    e. wcel 1756    =/= wne 2604   dom cdm 4838   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281   _ici 9282    + caddc 9283    x. cmul 9285    - cmin 9593   -ucneg 9594    / cdiv 9991   2c2 10369   expce 13345   logclog 22004  arctancatan 22257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-fi 7659  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ioo 11302  df-ioc 11303  df-ico 11304  df-icc 11305  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-seq 11805  df-exp 11864  df-fac 12050  df-bc 12077  df-hash 12102  df-shft 12554  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-limsup 12947  df-clim 12964  df-rlim 12965  df-sum 13162  df-ef 13351  df-sin 13353  df-cos 13354  df-pi 13356  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-rest 14359  df-topn 14360  df-0g 14378  df-gsum 14379  df-topgen 14380  df-pt 14381  df-prds 14384  df-xrs 14438  df-qtop 14443  df-imas 14444  df-xps 14446  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-submnd 15463  df-mulg 15546  df-cntz 15833  df-cmn 16277  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-mopn 17811  df-fbas 17812  df-fg 17813  df-cnfld 17817  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-cld 18621  df-ntr 18622  df-cls 18623  df-nei 18700  df-lp 18738  df-perf 18739  df-cn 18829  df-cnp 18830  df-haus 18917  df-tx 19133  df-hmeo 19326  df-fil 19417  df-fm 19509  df-flim 19510  df-flf 19511  df-xms 19893  df-ms 19894  df-tms 19895  df-cncf 20452  df-limc 21339  df-dv 21340  df-log 22006  df-atan 22260
This theorem is referenced by:  tanatan  22312
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