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Theorem dvcjbr 22220
Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 22221. (This doesn't follow from dvcobr 22217 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 9561 . . . . 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 2467 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
65tgioo2 21176 . . . 4  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
72, 3, 4, 6, 5dvbssntr 22172 . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
8 dvcj.c . . 3  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
97, 8sseldd 3510 . 2  |-  ( ph  ->  C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X ) )
104, 1syl6ss 3521 . . . . . 6  |-  ( ph  ->  X  C_  CC )
111a1i 11 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  RR  C_  CC )
12 simpl 457 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
13 simpr 461 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  X  C_  RR )
1411, 12, 13dvbss 22173 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  X )
153, 4, 14syl2anc 661 . . . . . . 7  |-  ( ph  ->  dom  ( RR  _D  F )  C_  X
)
1615, 8sseldd 3510 . . . . . 6  |-  ( ph  ->  C  e.  X )
173, 10, 16dvlem 22168 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) )  e.  CC )
18 eqid 2467 . . . . 5  |-  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )
1917, 18fmptd 6056 . . . 4  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) : ( X  \  { C } ) --> CC )
20 ssid 3528 . . . . 5  |-  CC  C_  CC
2120a1i 11 . . . 4  |-  ( ph  ->  CC  C_  CC )
225cnfldtopon 21158 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
2322toponunii 19302 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
2423restid 14706 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2522, 24ax-mp 5 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2625eqcomi 2480 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
27 dvf 22179 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
28 ffun 5739 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
29 funfvbrb 6001 . . . . . . . 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 196 . . . . . 6  |-  ( ph  ->  C ( RR  _D  F ) ( ( RR  _D  F ) `
 C ) )
326, 5, 18, 2, 3, 4eldv 22170 . . . . . 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 210 . . . . 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 463 . . . 4  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
35 cjcncf 21276 . . . . . 6  |-  *  e.  ( CC -cn-> CC )
365cncfcn1 21282 . . . . . 6  |-  ( CC
-cn-> CC )  =  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3735, 36eleqtri 2553 . . . . 5  |-  *  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3827ffvelrni 6031 . . . . . 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 19647 . . . . 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 663 . . . 4  |-  ( ph  ->  *  e.  ( ( ( TopOpen ` fld )  CnP  ( TopOpen ` fld )
) `  ( ( RR  _D  F ) `  C ) ) )
4219, 21, 5, 26, 34, 41limccnp 22163 . . 3  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( *  o.  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) ) ) lim CC  C
) )
43 eqidd 2468 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) )
44 cjf 12917 . . . . . . . 8  |-  * : CC --> CC
4544a1i 11 . . . . . . 7  |-  ( ph  ->  * : CC --> CC )
4645feqmptd 5927 . . . . . 6  |-  ( ph  ->  *  =  ( y  e.  CC  |->  ( * `
 y ) ) )
47 fveq2 5872 . . . . . 6  |-  ( y  =  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) )  ->  (
* `  y )  =  ( * `  ( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) ) ) )
4817, 43, 46, 47fmptco 6065 . . . . 5  |-  ( ph  ->  ( *  o.  (
x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( * `
 ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) ) )
493adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  F : X --> CC )
50 eldifi 3631 . . . . . . . . . . 11  |-  ( x  e.  ( X  \  { C } )  ->  x  e.  X )
5150adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  X )
5249, 51ffvelrnd 6033 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  x
)  e.  CC )
533, 16ffvelrnd 6033 . . . . . . . . . 10  |-  ( ph  ->  ( F `  C
)  e.  CC )
5453adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  C
)  e.  CC )
5552, 54subcld 9942 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( F `  x )  -  ( F `  C )
)  e.  CC )
564sselda 3509 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  RR )
5750, 56sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  RR )
584, 16sseldd 3510 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
5958adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  RR )
6057, 59resubcld 9999 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  RR )
6160recnd 9634 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  CC )
6257recnd 9634 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  CC )
6359recnd 9634 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  CC )
64 eldifsni 4159 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { C } )  ->  x  =/=  C )
6564adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  =/=  C )
6662, 63, 65subne0d 9951 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  =/=  0 )
6755, 61, 66cjdivd 13036 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( * `  ( ( F `  x )  -  ( F `  C ) ) )  /  ( * `  ( x  -  C
) ) ) )
68 cjsub 12962 . . . . . . . . . 10  |-  ( ( ( F `  x
)  e.  CC  /\  ( F `  C )  e.  CC )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
6952, 54, 68syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
70 fvco3 5951 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
713, 50, 70syl2an 477 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
72 fvco3 5951 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
733, 16, 72syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7473adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7571, 74oveq12d 6313 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
7669, 75eqtr4d 2511 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) ) )
7760cjred 13039 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
x  -  C ) )  =  ( x  -  C ) )
7876, 77oveq12d 6313 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( * `  ( ( F `  x )  -  ( F `  C )
) )  /  (
* `  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) )
7967, 78eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
8079mpteq2dva 4539 . . . . 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 2508 . . . 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 6310 . . 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 2557 . 2  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
84 eqid 2467 . . 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 5747 . . . 4  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
8644, 3, 85sylancr 663 . . 3  |-  ( ph  ->  ( *  o.  F
) : X --> CC )
876, 5, 84, 2, 86, 4eldv 22170 . 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 920 1  |-  ( ph  ->  C ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478    C_ wss 3481   {csn 4033   class class class wbr 4453    |-> cmpt 4511   dom cdm 5005   ran crn 5006    o. ccom 5009   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503    - cmin 9817    / cdiv 10218   (,)cioo 11541   *ccj 12909   ↾t crest 14693   TopOpenctopn 14694   topGenctg 14710  ℂfldccnfld 18290  TopOnctopon 19264   intcnt 19386    Cn ccn 19593    CnP ccnp 19594   -cn->ccncf 21248   lim CC climc 22134    _D cdv 22135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-cncf 21250  df-limc 22138  df-dv 22139
This theorem is referenced by:  dvcj  22221
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