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Theorem dvcjbr 21428
Description: The derivative of the conjugate of a function. (This doesn't follow from dvcobr 21425 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 9344 . . . . 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 2443 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
65tgioo2 20385 . . . 4  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
72, 3, 4, 6, 5dvbssntr 21380 . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
8 dvcj.c . . 3  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
97, 8sseldd 3362 . 2  |-  ( ph  ->  C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X ) )
104, 1syl6ss 3373 . . . . . 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 21381 . . . . . . . 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 3362 . . . . . 6  |-  ( ph  ->  C  e.  X )
173, 10, 16dvlem 21376 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) )  e.  CC )
18 eqid 2443 . . . . 5  |-  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )
1917, 18fmptd 5872 . . . 4  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) : ( X  \  { C } ) --> CC )
20 ssid 3380 . . . . 5  |-  CC  C_  CC
2120a1i 11 . . . 4  |-  ( ph  ->  CC  C_  CC )
225cnfldtopon 20367 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
2322toponunii 18542 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
2423restid 14377 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2522, 24ax-mp 5 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2625eqcomi 2447 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
27 dvf 21387 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
28 ffun 5566 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
29 funfvbrb 5821 . . . . . . . 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 21378 . . . . . 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 20485 . . . . . 6  |-  *  e.  ( CC -cn-> CC )
365cncfcn1 20491 . . . . . 6  |-  ( CC
-cn-> CC )  =  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3735, 36eleqtri 2515 . . . . 5  |-  *  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3827ffvelrni 5847 . . . . . 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 18887 . . . . 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 21371 . . 3  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( *  o.  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) ) ) lim CC  C
) )
43 eqidd 2444 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) )
44 cjf 12598 . . . . . . . 8  |-  * : CC --> CC
4544a1i 11 . . . . . . 7  |-  ( ph  ->  * : CC --> CC )
4645feqmptd 5749 . . . . . 6  |-  ( ph  ->  *  =  ( y  e.  CC  |->  ( * `
 y ) ) )
47 fveq2 5696 . . . . . 6  |-  ( y  =  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) )  ->  (
* `  y )  =  ( * `  ( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) ) ) )
4817, 43, 46, 47fmptco 5881 . . . . 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 3483 . . . . . . . . . . 11  |-  ( x  e.  ( X  \  { C } )  ->  x  e.  X )
5150adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  X )
5249, 51ffvelrnd 5849 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  x
)  e.  CC )
533, 16ffvelrnd 5849 . . . . . . . . . 10  |-  ( ph  ->  ( F `  C
)  e.  CC )
5453adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  C
)  e.  CC )
5552, 54subcld 9724 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( F `  x )  -  ( F `  C )
)  e.  CC )
564sselda 3361 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  RR )
5750, 56sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  RR )
584, 16sseldd 3362 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
5958adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  RR )
6057, 59resubcld 9781 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  RR )
6160recnd 9417 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  CC )
6257recnd 9417 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  CC )
6359recnd 9417 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  CC )
64 eldifsni 4006 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { C } )  ->  x  =/=  C )
6564adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  =/=  C )
6662, 63, 65subne0d 9733 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  =/=  0 )
6755, 61, 66cjdivd 12717 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( * `  ( ( F `  x )  -  ( F `  C ) ) )  /  ( * `  ( x  -  C
) ) ) )
68 cjsub 12643 . . . . . . . . . 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 5773 . . . . . . . . . . 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 5773 . . . . . . . . . . . 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 6114 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
7669, 75eqtr4d 2478 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) ) )
7760cjred 12720 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
x  -  C ) )  =  ( x  -  C ) )
7876, 77oveq12d 6114 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( * `  ( ( F `  x )  -  ( F `  C )
) )  /  (
* `  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) )
7967, 78eqtrd 2475 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
8079mpteq2dva 4383 . . . . 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 2475 . . . 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 6111 . . 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 2519 . 2  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
84 eqid 2443 . . 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 5573 . . . 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 21378 . 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 913 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 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330    C_ wss 3333   {csn 3882   class class class wbr 4297    e. cmpt 4355   dom cdm 4845   ran crn 4846    o. ccom 4849   Fun wfun 5417   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286    - cmin 9600    / cdiv 9998   (,)cioo 11305   *ccj 12590   ↾t crest 14364   TopOpenctopn 14365   topGenctg 14381  ℂfldccnfld 17823  TopOnctopon 18504   intcnt 18626    Cn ccn 18833    CnP ccnp 18834   -cn->ccncf 20457   lim CC climc 21342    _D cdv 21343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fi 7666  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-icc 11312  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-rest 14366  df-topn 14367  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-cncf 20459  df-limc 21346  df-dv 21347
This theorem is referenced by:  dvcj  21429
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