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Theorem atanbndlem 22320
Description: Lemma for atanbnd 22321. (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 10997 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 atanrecl 22306 . . 3  |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
31, 2syl 16 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  RR )
4 picn 21922 . . . 4  |-  pi  e.  CC
5 2cn 10392 . . . 4  |-  2  e.  CC
6 2ne0 10414 . . . 4  |-  2  =/=  0
7 divneg 10026 . . . 4  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
pi  /  2 )  =  ( -u pi  /  2 ) )
84, 5, 6, 7mp3an 1314 . . 3  |-  -u (
pi  /  2 )  =  ( -u pi  /  2 )
9 ax-1cn 9340 . . . . . . . . . . . 12  |-  1  e.  CC
10 ax-icn 9341 . . . . . . . . . . . . 13  |-  _i  e.  CC
111recnd 9412 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e.  CC )
12 mulcl 9366 . . . . . . . . . . . . 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 9364 . . . . . . . . . . . 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 22280 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  dom arctan )
171, 16syl 16 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e. 
dom arctan )
18 atandm2 22272 . . . . . . . . . . . . 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 1002 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
2115, 20logcld 22022 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
22 subcl 9609 . . . . . . . . . . . 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 1001 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2523, 24logcld 22022 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
2621, 25subcld 9719 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
27 imre 12597 . . . . . . . . 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 22279 . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . 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 10074 . . . . . . . . . . . . . 14  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
3332oveq1i 6101 . . . . . . . . . . . . 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 10391 . . . . . . . . . . . . . . . 16  |-  2  e.  RR
3534a1i 11 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR+  ->  2  e.  RR )
3635recnd 9412 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  2  e.  CC )
37 halfcl 10550 . . . . . . . . . . . . . . 15  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
3810, 37mp1i 12 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( _i 
/  2 )  e.  CC )
3925, 21subcld 9719 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
4036, 38, 39mulassd 9409 . . . . . . . . . . . . 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 2489 . . . . . . . . . . . 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 2478 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
4321, 25negsubdi2d 9735 . . . . . . . . . . . 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 6107 . . . . . . . . . . 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 2478 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
46 mulneg12 9783 . . . . . . . . . . 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 2478 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  (
-u _i  x.  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
4948fveq2d 5695 . . . . . . . 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 9367 . . . . . . . . . 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 12713 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Re
`  ( 2  x.  (arctan `  A )
) )  =  ( 2  x.  (arctan `  A ) ) )
5328, 49, 523eqtr2rd 2482 . . . . . . 7  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( Im `  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
54 rpgt0 11002 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  < 
A )
551rered 12713 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( Re
`  A )  =  A )
5654, 55breqtrrd 4318 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
( Re `  A
) )
57 atanlogsublem 22310 . . . . . . . 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 2517 . . . . . 6  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  (
-u pi (,) pi ) )
60 eliooord 11355 . . . . . 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 21921 . . . . . . 7  |-  pi  e.  RR
6463renegcli 9670 . . . . . 6  |-  -u pi  e.  RR
6564a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  -u pi  e.  RR )
66 2pos 10413 . . . . . 6  |-  0  <  2
6766a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  0  <  2 )
68 ltdivmul 10204 . . . . 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 1222 . . . 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 4325 . 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 10203 . . . 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 1222 . . 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 21926 . . . . 5  |-  ( pi 
/  2 )  e.  RR
7877renegcli 9670 . . . 4  |-  -u (
pi  /  2 )  e.  RR
7978rexri 9436 . . 3  |-  -u (
pi  /  2 )  e.  RR*
8077rexri 9436 . . 3  |-  ( pi 
/  2 )  e. 
RR*
81 elioo2 11341 . . 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 1172 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292   dom cdm 4840   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   _ici 9284    + caddc 9285    x. cmul 9287   RR*cxr 9417    < clt 9418    - cmin 9595   -ucneg 9596    / cdiv 9993   2c2 10371   RR+crp 10991   (,)cioo 11300   Recre 12586   Imcim 12587   picpi 13352   logclog 22006  arctancatan 22259
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-tan 13357  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-atan 22262
This theorem is referenced by:  atanbnd  22321
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