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Theorem atanbndlem 22977
Description: Lemma for atanbnd 22978. (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 11215 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 atanrecl 22963 . . 3  |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
31, 2syl 16 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  RR )
4 picn 22579 . . . 4  |-  pi  e.  CC
5 2cn 10595 . . . 4  |-  2  e.  CC
6 2ne0 10617 . . . 4  |-  2  =/=  0
7 divneg 10228 . . . 4  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
pi  /  2 )  =  ( -u pi  /  2 ) )
84, 5, 6, 7mp3an 1319 . . 3  |-  -u (
pi  /  2 )  =  ( -u pi  /  2 )
9 ax-1cn 9539 . . . . . . . . . . . 12  |-  1  e.  CC
10 ax-icn 9540 . . . . . . . . . . . . 13  |-  _i  e.  CC
111recnd 9611 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e.  CC )
12 mulcl 9565 . . . . . . . . . . . . 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 9563 . . . . . . . . . . . 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 22937 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  dom arctan )
171, 16syl 16 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e. 
dom arctan )
18 atandm2 22929 . . . . . . . . . . . . 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 1005 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
2115, 20logcld 22679 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
22 subcl 9808 . . . . . . . . . . . 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 1004 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2523, 24logcld 22679 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
2621, 25subcld 9919 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
27 imre 12891 . . . . . . . . 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 22936 . . . . . . . . . . . . . 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 6291 . . . . . . . . . . . 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 10276 . . . . . . . . . . . . . 14  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
3332oveq1i 6285 . . . . . . . . . . . . 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 10594 . . . . . . . . . . . . . . . 16  |-  2  e.  RR
3534a1i 11 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR+  ->  2  e.  RR )
3635recnd 9611 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  2  e.  CC )
37 halfcl 10753 . . . . . . . . . . . . . . 15  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
3810, 37mp1i 12 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( _i 
/  2 )  e.  CC )
3925, 21subcld 9919 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
4036, 38, 39mulassd 9608 . . . . . . . . . . . . 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 2515 . . . . . . . . . . . 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 2504 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
4321, 25negsubdi2d 9935 . . . . . . . . . . . 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 6291 . . . . . . . . . . 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 2504 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
46 mulneg12 9984 . . . . . . . . . . 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 2504 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  (
-u _i  x.  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
4948fveq2d 5861 . . . . . . . 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 9566 . . . . . . . . . 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 13007 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Re
`  ( 2  x.  (arctan `  A )
) )  =  ( 2  x.  (arctan `  A ) ) )
5328, 49, 523eqtr2rd 2508 . . . . . . 7  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( Im `  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
54 rpgt0 11220 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  < 
A )
551rered 13007 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( Re
`  A )  =  A )
5654, 55breqtrrd 4466 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
( Re `  A
) )
57 atanlogsublem 22967 . . . . . . . 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 2548 . . . . . 6  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  (
-u pi (,) pi ) )
60 eliooord 11573 . . . . . 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 22578 . . . . . . 7  |-  pi  e.  RR
6463renegcli 9869 . . . . . 6  |-  -u pi  e.  RR
6564a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  -u pi  e.  RR )
66 2pos 10616 . . . . . 6  |-  0  <  2
6766a1i 11 . . . . 5  |-  ( A  e.  RR+  ->  0  <  2 )
68 ltdivmul 10406 . . . . 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 1227 . . . 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 4473 . 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 10405 . . . 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 1227 . . 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 22583 . . . . 5  |-  ( pi 
/  2 )  e.  RR
7877renegcli 9869 . . . 4  |-  -u (
pi  /  2 )  e.  RR
7978rexri 9635 . . 3  |-  -u (
pi  /  2 )  e.  RR*
8077rexri 9635 . . 3  |-  ( pi 
/  2 )  e. 
RR*
81 elioo2 11559 . . 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 1175 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   dom cdm 4992   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486   RR*cxr 9616    < clt 9617    - cmin 9794   -ucneg 9795    / cdiv 10195   2c2 10574   RR+crp 11209   (,)cioo 11518   Recre 12880   Imcim 12881   picpi 13653   logclog 22663  arctancatan 22916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-ef 13654  df-sin 13656  df-cos 13657  df-tan 13658  df-pi 13659  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999  df-log 22665  df-atan 22919
This theorem is referenced by:  atanbnd  22978
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