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Theorem dvcjbr 19788
Description: The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19785 because  * is not a function on the reals, and even if we used complex derivatives, 
* is not complex-differentiable.) (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
Hypotheses
Ref Expression
dvcj.f  |-  ( ph  ->  F : X --> CC )
dvcj.x  |-  ( ph  ->  X  C_  RR )
dvcj.c  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
Assertion
Ref Expression
dvcjbr  |-  ( ph  ->  C ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  C ) ) )

Proof of Theorem dvcjbr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-resscn 9003 . . . . 5  |-  RR  C_  CC
21a1i 11 . . . 4  |-  ( ph  ->  RR  C_  CC )
3 dvcj.f . . . 4  |-  ( ph  ->  F : X --> CC )
4 dvcj.x . . . 4  |-  ( ph  ->  X  C_  RR )
5 eqid 2404 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
65tgioo2 18787 . . . 4  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
72, 3, 4, 6, 5dvbssntr 19740 . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
8 dvcj.c . . 3  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
97, 8sseldd 3309 . 2  |-  ( ph  ->  C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X ) )
104, 1syl6ss 3320 . . . . . 6  |-  ( ph  ->  X  C_  CC )
111a1i 11 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  RR  C_  CC )
12 simpl 444 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
13 simpr 448 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  X  C_  RR )
1411, 12, 13dvbss 19741 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  X )
153, 4, 14syl2anc 643 . . . . . . 7  |-  ( ph  ->  dom  ( RR  _D  F )  C_  X
)
1615, 8sseldd 3309 . . . . . 6  |-  ( ph  ->  C  e.  X )
173, 10, 16dvlem 19736 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) )  e.  CC )
18 eqid 2404 . . . . 5  |-  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )
1917, 18fmptd 5852 . . . 4  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) : ( X  \  { C } ) --> CC )
20 ssid 3327 . . . . 5  |-  CC  C_  CC
2120a1i 11 . . . 4  |-  ( ph  ->  CC  C_  CC )
225cnfldtopon 18770 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
2322toponunii 16952 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
2423restid 13616 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2522, 24ax-mp 8 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2625eqcomi 2408 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
27 dvf 19747 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
28 ffun 5552 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
29 funfvbrb 5802 . . . . . . . 8  |-  ( Fun  ( RR  _D  F
)  ->  ( C  e.  dom  ( RR  _D  F )  <->  C ( RR  _D  F ) ( ( RR  _D  F
) `  C )
) )
3027, 28, 29mp2b 10 . . . . . . 7  |-  ( C  e.  dom  ( RR 
_D  F )  <->  C ( RR  _D  F ) ( ( RR  _D  F
) `  C )
)
318, 30sylib 189 . . . . . 6  |-  ( ph  ->  C ( RR  _D  F ) ( ( RR  _D  F ) `
 C ) )
326, 5, 18, 2, 3, 4eldv 19738 . . . . . 6  |-  ( ph  ->  ( C ( RR 
_D  F ) ( ( RR  _D  F
) `  C )  <->  ( C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X )  /\  (
( RR  _D  F
) `  C )  e.  ( ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) lim
CC  C ) ) ) )
3331, 32mpbid 202 . . . . 5  |-  ( ph  ->  ( C  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  /\  ( ( RR  _D  F ) `  C
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) ) )
3433simprd 450 . . . 4  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
35 cjcncf 18887 . . . . . 6  |-  *  e.  ( CC -cn-> CC )
365cncfcn1 18893 . . . . . 6  |-  ( CC
-cn-> CC )  =  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3735, 36eleqtri 2476 . . . . 5  |-  *  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3827ffvelrni 5828 . . . . . 6  |-  ( C  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  C
)  e.  CC )
398, 38syl 16 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  CC )
4023cncnpi 17296 . . . . 5  |-  ( ( *  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
)  /\  ( ( RR  _D  F ) `  C )  e.  CC )  ->  *  e.  ( ( ( TopOpen ` fld )  CnP  ( TopOpen ` fld )
) `  ( ( RR  _D  F ) `  C ) ) )
4137, 39, 40sylancr 645 . . . 4  |-  ( ph  ->  *  e.  ( ( ( TopOpen ` fld )  CnP  ( TopOpen ` fld )
) `  ( ( RR  _D  F ) `  C ) ) )
4219, 21, 5, 26, 34, 41limccnp 19731 . . 3  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( *  o.  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) ) ) lim CC  C
) )
43 eqidd 2405 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) )
44 cjf 11864 . . . . . . . 8  |-  * : CC --> CC
4544a1i 11 . . . . . . 7  |-  ( ph  ->  * : CC --> CC )
4645feqmptd 5738 . . . . . 6  |-  ( ph  ->  *  =  ( y  e.  CC  |->  ( * `
 y ) ) )
47 fveq2 5687 . . . . . 6  |-  ( y  =  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) )  ->  (
* `  y )  =  ( * `  ( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) ) ) )
4817, 43, 46, 47fmptco 5860 . . . . 5  |-  ( ph  ->  ( *  o.  (
x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( * `
 ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) ) )
493adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  F : X --> CC )
50 eldifi 3429 . . . . . . . . . . 11  |-  ( x  e.  ( X  \  { C } )  ->  x  e.  X )
5150adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  X )
5249, 51ffvelrnd 5830 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  x
)  e.  CC )
533, 16ffvelrnd 5830 . . . . . . . . . 10  |-  ( ph  ->  ( F `  C
)  e.  CC )
5453adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  C
)  e.  CC )
5552, 54subcld 9367 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( F `  x )  -  ( F `  C )
)  e.  CC )
564sselda 3308 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  RR )
5750, 56sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  RR )
584, 16sseldd 3309 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
5958adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  RR )
6057, 59resubcld 9421 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  RR )
6160recnd 9070 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  CC )
6257recnd 9070 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  CC )
6359recnd 9070 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  CC )
64 eldifsni 3888 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { C } )  ->  x  =/=  C )
6564adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  =/=  C )
6662, 63, 65subne0d 9376 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  =/=  0 )
6755, 61, 66cjdivd 11983 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( * `  ( ( F `  x )  -  ( F `  C ) ) )  /  ( * `  ( x  -  C
) ) ) )
68 cjsub 11909 . . . . . . . . . 10  |-  ( ( ( F `  x
)  e.  CC  /\  ( F `  C )  e.  CC )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
6952, 54, 68syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
70 fvco3 5759 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
713, 50, 70syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
72 fvco3 5759 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
733, 16, 72syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7473adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7571, 74oveq12d 6058 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
7669, 75eqtr4d 2439 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) ) )
7760cjred 11986 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
x  -  C ) )  =  ( x  -  C ) )
7876, 77oveq12d 6058 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( * `  ( ( F `  x )  -  ( F `  C )
) )  /  (
* `  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) )
7967, 78eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
8079mpteq2dva 4255 . . . . 5  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( * `
 ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) ) )
8148, 80eqtrd 2436 . . . 4  |-  ( ph  ->  ( *  o.  (
x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) ) )
8281oveq1d 6055 . . 3  |-  ( ph  ->  ( ( *  o.  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) ) lim CC  C )  =  ( ( x  e.  ( X  \  { C } )  |->  ( ( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  /  ( x  -  C ) ) ) lim CC  C ) )
8342, 82eleqtrd 2480 . 2  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
84 eqid 2404 . . 3  |-  ( x  e.  ( X  \  { C } )  |->  ( ( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  /  ( x  -  C ) ) )
85 fco 5559 . . . 4  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
8644, 3, 85sylancr 645 . . 3  |-  ( ph  ->  ( *  o.  F
) : X --> CC )
876, 5, 84, 2, 86, 4eldv 19738 . 2  |-  ( ph  ->  ( C ( RR 
_D  ( *  o.  F ) ) ( * `  ( ( RR  _D  F ) `
 C ) )  <-> 
( C  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  /\  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) ) ) )
889, 83, 87mpbir2and 889 1  |-  ( ph  ->  C ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    C_ wss 3280   {csn 3774   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838    o. ccom 4841   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945    - cmin 9247    / cdiv 9633   (,)cioo 10872   *ccj 11856   ↾t crest 13603   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658  TopOnctopon 16914   intcnt 17036    Cn ccn 17242    CnP ccnp 17243   -cn->ccncf 18859   lim CC climc 19702    _D cdv 19703
This theorem is referenced by:  dvcj  19789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-cncf 18861  df-limc 19706  df-dv 19707
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