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Theorem atanbndlem 23128
Description: Lemma for atanbnd 23129. (Contributed by Mario Carneiro, 5-Apr-2015.)
Assertion
Ref Expression
atanbndlem  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  (
-u ( pi  / 
2 ) (,) (
pi  /  2 ) ) )

Proof of Theorem atanbndlem
StepHypRef Expression
1 rpre 11235 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 atanrecl 23114 . . 3  |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
31, 2syl 16 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  RR )
4 picn 22724 . . . 4  |-  pi  e.  CC
5 2cn 10612 . . . 4  |-  2  e.  CC
6 2ne0 10634 . . . 4  |-  2  =/=  0
7 divneg 10245 . . . 4  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
pi  /  2 )  =  ( -u pi  /  2 ) )
84, 5, 6, 7mp3an 1325 . . 3  |-  -u (
pi  /  2 )  =  ( -u pi  /  2 )
9 ax-1cn 9553 . . . . . . . . . . . 12  |-  1  e.  CC
10 ax-icn 9554 . . . . . . . . . . . . 13  |-  _i  e.  CC
111recnd 9625 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e.  CC )
12 mulcl 9579 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
1310, 11, 12sylancr 663 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( _i  x.  A )  e.  CC )
14 addcl 9577 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
159, 13, 14sylancr 663 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
16 atanre 23088 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  dom arctan )
171, 16syl 16 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e. 
dom arctan )
18 atandm2 23080 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
1917, 18sylib 196 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  (
1  -  ( _i  x.  A ) )  =/=  0  /\  (
1  +  ( _i  x.  A ) )  =/=  0 ) )
2019simp3d 1011 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
2115, 20logcld 22830 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
22 subcl 9824 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
239, 13, 22sylancr 663 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
2419simp2d 1010 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2523, 24logcld 22830 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
2621, 25subcld 9936 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
27 imre 12920 . . . . . . . . 9  |-  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) )  e.  CC  ->  ( Im `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( Re `  ( -u _i  x.  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) ) )
2826, 27syl 16 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Im
`  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( Re
`  ( -u _i  x.  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) ) )
29 atanval 23087 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3017, 29syl 16 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3130oveq2d 6297 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( 2  x.  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) ) )
3210, 5, 6divcan2i 10293 . . . . . . . . . . . . . 14  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
3332oveq1i 6291 . . . . . . . . . . . . 13  |-  ( ( 2  x.  ( _i 
/  2 ) )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( _i  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
34 2re 10611 . . . . . . . . . . . . . . . 16  |-  2  e.  RR
3534a1i 11 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR+  ->  2  e.  RR )
3635recnd 9625 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  2  e.  CC )
37 halfcl 10770 . . . . . . . . . . . . . . 15  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
3810, 37mp1i 12 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( _i 
/  2 )  e.  CC )
3925, 21subcld 9936 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
4036, 38, 39mulassd 9622 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  ( ( 2  x.  ( _i 
/  2 ) )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( 2  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
4133, 40syl5eqr 2498 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( _i  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( 2  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
4231, 41eqtr4d 2487 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
4321, 25negsubdi2d 9952 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) )  =  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
4443oveq2d 6297 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( _i  x.  -u ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( _i  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) )
4542, 44eqtr4d 2487 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
46 mulneg12 10001 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) )  e.  CC )  ->  ( -u _i  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )  =  ( _i  x.  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
4710, 26, 46sylancr 663 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( -u _i  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( _i  x.  -u ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) ) )
4845, 47eqtr4d 2487 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  (
-u _i  x.  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
4948fveq2d 5860 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Re
`  ( 2  x.  (arctan `  A )
) )  =  ( Re `  ( -u _i  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) ) ) )
50 remulcl 9580 . . . . . . . . . 10  |-  ( ( 2  e.  RR  /\  (arctan `  A )  e.  RR )  ->  (
2  x.  (arctan `  A ) )  e.  RR )
5134, 3, 50sylancr 663 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  RR )
5251rered 13036 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Re
`  ( 2  x.  (arctan `  A )
) )  =  ( 2  x.  (arctan `  A ) ) )
5328, 49, 523eqtr2rd 2491 . . . . . . 7  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( Im `  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
54 rpgt0 11240 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  < 
A )
551rered 13036 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( Re
`  A )  =  A )
5654, 55breqtrrd 4463 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
( Re `  A
) )
57 atanlogsublem 23118 . . . . . . . 8  |-  ( ( A  e.  dom arctan  /\  0  <  ( Re `  A
) )  ->  (
Im `  ( ( log `  ( 1  +  ( _i  x.  A
) ) )  -  ( log `  ( 1  -  ( _i  x.  A ) ) ) ) )  e.  (
-u pi (,) pi ) )
5817, 56, 57syl2anc 661 . . . . . . 7  |-  ( A  e.  RR+  ->  ( Im
`  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  e.  ( -u pi (,) pi ) )
5953, 58eqeltrd 2531 . . . . . 6  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  (
-u pi (,) pi ) )
60 eliooord 11593 . . . . . 6  |-  ( ( 2  x.  (arctan `  A ) )  e.  ( -u pi (,) pi )  ->  ( -u pi  <  ( 2  x.  (arctan `  A )
)  /\  ( 2  x.  (arctan `  A
) )  <  pi ) )
6159, 60syl 16 . . . . 5  |-  ( A  e.  RR+  ->  ( -u pi  <  ( 2  x.  (arctan `  A )
)  /\  ( 2  x.  (arctan `  A
) )  <  pi ) )
6261simpld 459 . . . 4  |-  ( A  e.  RR+  ->  -u pi  <  ( 2  x.  (arctan `  A ) ) )
63 pire 22723 . . . . . . 7  |-  pi  e.  RR
6463renegcli 9885 . . . . . 6  |-  -u pi  e.  RR
6564a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  -u pi  e.  RR )
66 2pos 10633 . . . . . 6  |-  0  <  2
6766a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  0  <  2 )
68 ltdivmul 10423 . . . . 5  |-  ( (
-u pi  e.  RR  /\  (arctan `  A )  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( ( -u pi  /  2 )  <  (arctan `  A )  <->  -u pi  <  ( 2  x.  (arctan `  A ) ) ) )
6965, 3, 35, 67, 68syl112anc 1233 . . . 4  |-  ( A  e.  RR+  ->  ( (
-u pi  /  2
)  <  (arctan `  A
)  <->  -u pi  <  (
2  x.  (arctan `  A ) ) ) )
7062, 69mpbird 232 . . 3  |-  ( A  e.  RR+  ->  ( -u pi  /  2 )  < 
(arctan `  A )
)
718, 70syl5eqbr 4470 . 2  |-  ( A  e.  RR+  ->  -u (
pi  /  2 )  <  (arctan `  A
) )
7261simprd 463 . . 3  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  <  pi )
7363a1i 11 . . . 4  |-  ( A  e.  RR+  ->  pi  e.  RR )
74 ltmuldiv2 10422 . . . 4  |-  ( ( (arctan `  A )  e.  RR  /\  pi  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( ( 2  x.  (arctan `  A )
)  <  pi  <->  (arctan `  A
)  <  ( pi  /  2 ) ) )
753, 73, 35, 67, 74syl112anc 1233 . . 3  |-  ( A  e.  RR+  ->  ( ( 2  x.  (arctan `  A ) )  < 
pi 
<->  (arctan `  A )  <  ( pi  /  2
) ) )
7672, 75mpbid 210 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  <  (
pi  /  2 ) )
77 halfpire 22729 . . . . 5  |-  ( pi 
/  2 )  e.  RR
7877renegcli 9885 . . . 4  |-  -u (
pi  /  2 )  e.  RR
7978rexri 9649 . . 3  |-  -u (
pi  /  2 )  e.  RR*
8077rexri 9649 . . 3  |-  ( pi 
/  2 )  e. 
RR*
81 elioo2 11579 . . 3  |-  ( (
-u ( pi  / 
2 )  e.  RR*  /\  ( pi  /  2
)  e.  RR* )  ->  ( (arctan `  A
)  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  <-> 
( (arctan `  A
)  e.  RR  /\  -u ( pi  /  2
)  <  (arctan `  A
)  /\  (arctan `  A
)  <  ( pi  /  2 ) ) ) )
8279, 80, 81mp2an 672 . 2  |-  ( (arctan `  A )  e.  (
-u ( pi  / 
2 ) (,) (
pi  /  2 ) )  <->  ( (arctan `  A )  e.  RR  /\  -u ( pi  /  2
)  <  (arctan `  A
)  /\  (arctan `  A
)  <  ( pi  /  2 ) ) )
833, 71, 76, 82syl3anbrc 1181 1  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  (
-u ( pi  / 
2 ) (,) (
pi  /  2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437   dom cdm 4989   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   _ici 9497    + caddc 9498    x. cmul 9500   RR*cxr 9630    < clt 9631    - cmin 9810   -ucneg 9811    / cdiv 10212   2c2 10591   RR+crp 11229   (,)cioo 11538   Recre 12909   Imcim 12910   picpi 13680   logclog 22814  arctancatan 23067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-tan 13685  df-pi 13686  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-limc 22143  df-dv 22144  df-log 22816  df-atan 23070
This theorem is referenced by:  atanbnd  23129
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