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Theorem atanbndlem 23930
Description: Lemma for atanbnd 23931. (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 11331 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 atanrecl 23916 . . 3  |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
31, 2syl 17 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  RR )
4 picn 23493 . . . 4  |-  pi  e.  CC
5 2cn 10702 . . . 4  |-  2  e.  CC
6 2ne0 10724 . . . 4  |-  2  =/=  0
7 divneg 10324 . . . 4  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
pi  /  2 )  =  ( -u pi  /  2 ) )
84, 5, 6, 7mp3an 1390 . . 3  |-  -u (
pi  /  2 )  =  ( -u pi  /  2 )
9 ax-1cn 9615 . . . . . . . . . . . 12  |-  1  e.  CC
10 ax-icn 9616 . . . . . . . . . . . . 13  |-  _i  e.  CC
111recnd 9687 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e.  CC )
12 mulcl 9641 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
1310, 11, 12sylancr 676 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( _i  x.  A )  e.  CC )
14 addcl 9639 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
159, 13, 14sylancr 676 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
16 atanre 23890 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  dom arctan )
171, 16syl 17 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e. 
dom arctan )
18 atandm2 23882 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
1917, 18sylib 201 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  (
1  -  ( _i  x.  A ) )  =/=  0  /\  (
1  +  ( _i  x.  A ) )  =/=  0 ) )
2019simp3d 1044 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
2115, 20logcld 23599 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
22 subcl 9894 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
239, 13, 22sylancr 676 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
2419simp2d 1043 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2523, 24logcld 23599 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
2621, 25subcld 10005 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
27 imre 13248 . . . . . . . . 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 17 . . . . . . . 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 23889 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3017, 29syl 17 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3130oveq2d 6324 . . . . . . . . . . . 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 10372 . . . . . . . . . . . . . 14  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
3332oveq1i 6318 . . . . . . . . . . . . 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 10701 . . . . . . . . . . . . . . . 16  |-  2  e.  RR
3534a1i 11 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR+  ->  2  e.  RR )
3635recnd 9687 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  2  e.  CC )
37 halfcl 10861 . . . . . . . . . . . . . . 15  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
3810, 37mp1i 13 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( _i 
/  2 )  e.  CC )
3925, 21subcld 10005 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
4036, 38, 39mulassd 9684 . . . . . . . . . . . . 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 2519 . . . . . . . . . . . 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 2508 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
4321, 25negsubdi2d 10021 . . . . . . . . . . . 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 6324 . . . . . . . . . . 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 2508 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
46 mulneg12 10078 . . . . . . . . . . 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 676 . . . . . . . . . 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 2508 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  (
-u _i  x.  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
4948fveq2d 5883 . . . . . . . 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 9642 . . . . . . . . . 10  |-  ( ( 2  e.  RR  /\  (arctan `  A )  e.  RR )  ->  (
2  x.  (arctan `  A ) )  e.  RR )
5134, 3, 50sylancr 676 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  RR )
5251rered 13364 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Re
`  ( 2  x.  (arctan `  A )
) )  =  ( 2  x.  (arctan `  A ) ) )
5328, 49, 523eqtr2rd 2512 . . . . . . 7  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( Im `  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
54 rpgt0 11336 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  < 
A )
551rered 13364 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( Re
`  A )  =  A )
5654, 55breqtrrd 4422 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
( Re `  A
) )
57 atanlogsublem 23920 . . . . . . . 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 673 . . . . . . 7  |-  ( A  e.  RR+  ->  ( Im
`  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  e.  ( -u pi (,) pi ) )
5953, 58eqeltrd 2549 . . . . . 6  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  (
-u pi (,) pi ) )
60 eliooord 11719 . . . . . 6  |-  ( ( 2  x.  (arctan `  A ) )  e.  ( -u pi (,) pi )  ->  ( -u pi  <  ( 2  x.  (arctan `  A )
)  /\  ( 2  x.  (arctan `  A
) )  <  pi ) )
6159, 60syl 17 . . . . 5  |-  ( A  e.  RR+  ->  ( -u pi  <  ( 2  x.  (arctan `  A )
)  /\  ( 2  x.  (arctan `  A
) )  <  pi ) )
6261simpld 466 . . . 4  |-  ( A  e.  RR+  ->  -u pi  <  ( 2  x.  (arctan `  A ) ) )
63 pire 23492 . . . . . . 7  |-  pi  e.  RR
6463renegcli 9955 . . . . . 6  |-  -u pi  e.  RR
6564a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  -u pi  e.  RR )
66 2pos 10723 . . . . . 6  |-  0  <  2
6766a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  0  <  2 )
68 ltdivmul 10502 . . . . 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 1296 . . . 4  |-  ( A  e.  RR+  ->  ( (
-u pi  /  2
)  <  (arctan `  A
)  <->  -u pi  <  (
2  x.  (arctan `  A ) ) ) )
7062, 69mpbird 240 . . 3  |-  ( A  e.  RR+  ->  ( -u pi  /  2 )  < 
(arctan `  A )
)
718, 70syl5eqbr 4429 . 2  |-  ( A  e.  RR+  ->  -u (
pi  /  2 )  <  (arctan `  A
) )
7261simprd 470 . . 3  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  <  pi )
7363a1i 11 . . . 4  |-  ( A  e.  RR+  ->  pi  e.  RR )
74 ltmuldiv2 10501 . . . 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 1296 . . 3  |-  ( A  e.  RR+  ->  ( ( 2  x.  (arctan `  A ) )  < 
pi 
<->  (arctan `  A )  <  ( pi  /  2
) ) )
7672, 75mpbid 215 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  <  (
pi  /  2 ) )
77 halfpire 23498 . . . . 5  |-  ( pi 
/  2 )  e.  RR
7877renegcli 9955 . . . 4  |-  -u (
pi  /  2 )  e.  RR
7978rexri 9711 . . 3  |-  -u (
pi  /  2 )  e.  RR*
8077rexri 9711 . . 3  |-  ( pi 
/  2 )  e. 
RR*
81 elioo2 11702 . . 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 686 . 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 1214 1  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  (
-u ( pi  / 
2 ) (,) (
pi  /  2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   dom cdm 4839   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   _ici 9559    + caddc 9560    x. cmul 9562   RR*cxr 9692    < clt 9693    - cmin 9880   -ucneg 9881    / cdiv 10291   2c2 10681   RR+crp 11325   (,)cioo 11660   Recre 13237   Imcim 13238   picpi 14196   logclog 23583  arctancatan 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-atan 23872
This theorem is referenced by:  atanbnd  23931
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