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Theorem dvcjbr 19835
Description: The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19832 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 9047 . . . . 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 2436 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
65tgioo2 18834 . . . 4  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
72, 3, 4, 6, 5dvbssntr 19787 . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
8 dvcj.c . . 3  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
97, 8sseldd 3349 . 2  |-  ( ph  ->  C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X ) )
104, 1syl6ss 3360 . . . . . 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 19788 . . . . . . . 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 3349 . . . . . 6  |-  ( ph  ->  C  e.  X )
173, 10, 16dvlem 19783 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) )  e.  CC )
18 eqid 2436 . . . . 5  |-  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )
1917, 18fmptd 5893 . . . 4  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) : ( X  \  { C } ) --> CC )
20 ssid 3367 . . . . 5  |-  CC  C_  CC
2120a1i 11 . . . 4  |-  ( ph  ->  CC  C_  CC )
225cnfldtopon 18817 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
2322toponunii 16997 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
2423restid 13661 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2522, 24ax-mp 8 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2625eqcomi 2440 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
27 dvf 19794 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
28 ffun 5593 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
29 funfvbrb 5843 . . . . . . . 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 19785 . . . . . 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 18934 . . . . . 6  |-  *  e.  ( CC -cn-> CC )
365cncfcn1 18940 . . . . . 6  |-  ( CC
-cn-> CC )  =  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3735, 36eleqtri 2508 . . . . 5  |-  *  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3827ffvelrni 5869 . . . . . 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 17342 . . . . 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 19778 . . 3  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( *  o.  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) ) ) lim CC  C
) )
43 eqidd 2437 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) )
44 cjf 11909 . . . . . . . 8  |-  * : CC --> CC
4544a1i 11 . . . . . . 7  |-  ( ph  ->  * : CC --> CC )
4645feqmptd 5779 . . . . . 6  |-  ( ph  ->  *  =  ( y  e.  CC  |->  ( * `
 y ) ) )
47 fveq2 5728 . . . . . 6  |-  ( y  =  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) )  ->  (
* `  y )  =  ( * `  ( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) ) ) )
4817, 43, 46, 47fmptco 5901 . . . . 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 3469 . . . . . . . . . . 11  |-  ( x  e.  ( X  \  { C } )  ->  x  e.  X )
5150adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  X )
5249, 51ffvelrnd 5871 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  x
)  e.  CC )
533, 16ffvelrnd 5871 . . . . . . . . . 10  |-  ( ph  ->  ( F `  C
)  e.  CC )
5453adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  C
)  e.  CC )
5552, 54subcld 9411 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( F `  x )  -  ( F `  C )
)  e.  CC )
564sselda 3348 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  RR )
5750, 56sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  RR )
584, 16sseldd 3349 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
5958adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  RR )
6057, 59resubcld 9465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  RR )
6160recnd 9114 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  CC )
6257recnd 9114 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  CC )
6359recnd 9114 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  CC )
64 eldifsni 3928 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { C } )  ->  x  =/=  C )
6564adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  =/=  C )
6662, 63, 65subne0d 9420 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  =/=  0 )
6755, 61, 66cjdivd 12028 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( * `  ( ( F `  x )  -  ( F `  C ) ) )  /  ( * `  ( x  -  C
) ) ) )
68 cjsub 11954 . . . . . . . . . 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 5800 . . . . . . . . . . 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 5800 . . . . . . . . . . . 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 6099 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
7669, 75eqtr4d 2471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) ) )
7760cjred 12031 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
x  -  C ) )  =  ( x  -  C ) )
7876, 77oveq12d 6099 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( * `  ( ( F `  x )  -  ( F `  C )
) )  /  (
* `  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) )
7967, 78eqtrd 2468 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
8079mpteq2dva 4295 . . . . 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 2468 . . . 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 6096 . . 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 2512 . 2  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
84 eqid 2436 . . 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 5600 . . . 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 19785 . 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 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    C_ wss 3320   {csn 3814   class class class wbr 4212    e. cmpt 4266   dom cdm 4878   ran crn 4879    o. ccom 4882   Fun wfun 5448   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989    - cmin 9291    / cdiv 9677   (,)cioo 10916   *ccj 11901   ↾t crest 13648   TopOpenctopn 13649   topGenctg 13665  ℂfldccnfld 16703  TopOnctopon 16959   intcnt 17081    Cn ccn 17288    CnP ccnp 17289   -cn->ccncf 18906   lim CC climc 19749    _D cdv 19750
This theorem is referenced by:  dvcj  19836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-icc 10923  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-rest 13650  df-topn 13651  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-cncf 18908  df-limc 19753  df-dv 19754
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