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Theorem dvcjbr 14675
Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 14676. (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  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-resscn 7938 . . . . 5  |-  RR  C_  CC
21a1i 9 . . . 4  |-  ( ph  ->  RR  C_  CC )
3 dvcj.f . . . 4  |-  ( ph  ->  F : X --> CC )
4 dvcj.x . . . 4  |-  ( ph  ->  X  C_  RR )
5 eqid 2189 . . . . 5  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
65tgioo2cntop 14535 . . . 4  |-  ( topGen ` 
ran  (,) )  =  ( ( MetOpen `  ( abs  o. 
-  ) )t  RR )
72, 3, 4, 6, 5dvbssntrcntop 14656 . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
8 dvcj.c . . 3  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
97, 8sseldd 3171 . 2  |-  ( ph  ->  C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X ) )
104, 1sstrdi 3182 . . . . . 6  |-  ( ph  ->  X  C_  CC )
111a1i 9 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  RR  C_  CC )
12 simpl 109 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
13 simpr 110 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  X  C_  RR )
1411, 12, 13dvbss 14657 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  X )
153, 4, 14syl2anc 411 . . . . . . 7  |-  ( ph  ->  dom  ( RR  _D  F )  C_  X
)
1615, 8sseldd 3171 . . . . . 6  |-  ( ph  ->  C  e.  X )
173, 10, 16dvlemap 14652 . . . . 5  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) )  e.  CC )
1817fmpttd 5695 . . . 4  |-  ( ph  ->  ( x  e.  {
w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) : { w  e.  X  |  w #  C }
--> CC )
19 ssidd 3191 . . . 4  |-  ( ph  ->  CC  C_  CC )
205cntoptopon 14518 . . . . 5  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  (TopOn `  CC )
2120toponrestid 14007 . . . 4  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( (
MetOpen `  ( abs  o.  -  ) )t  CC )
223fdmd 5394 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  =  X )
2322feq2d 5375 . . . . . . . . . . . 12  |-  ( ph  ->  ( F : dom  F --> CC  <->  F : X --> CC ) )
243, 23mpbird 167 . . . . . . . . . . 11  |-  ( ph  ->  F : dom  F --> CC )
2522, 4eqsstrd 3206 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  C_  RR )
26 cnex 7970 . . . . . . . . . . . 12  |-  CC  e.  _V
27 reex 7980 . . . . . . . . . . . 12  |-  RR  e.  _V
2826, 27elpm2 6710 . . . . . . . . . . 11  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
2924, 25, 28sylanbrc 417 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
30 dvfpm 14661 . . . . . . . . . 10  |-  ( F  e.  ( CC  ^pm  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC )
3129, 30syl 14 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> CC )
3231ffund 5391 . . . . . . . 8  |-  ( ph  ->  Fun  ( RR  _D  F ) )
33 funfvbrb 5653 . . . . . . . 8  |-  ( Fun  ( RR  _D  F
)  ->  ( C  e.  dom  ( RR  _D  F )  <->  C ( RR  _D  F ) ( ( RR  _D  F
) `  C )
) )
3432, 33syl 14 . . . . . . 7  |-  ( ph  ->  ( C  e.  dom  ( RR  _D  F
)  <->  C ( RR  _D  F ) ( ( RR  _D  F ) `
 C ) ) )
358, 34mpbid 147 . . . . . 6  |-  ( ph  ->  C ( RR  _D  F ) ( ( RR  _D  F ) `
 C ) )
36 eqid 2189 . . . . . . 7  |-  ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) )  =  ( x  e.  {
w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) )
376, 5, 36, 2, 3, 4eldvap 14654 . . . . . 6  |-  ( ph  ->  ( C ( RR 
_D  F ) ( ( RR  _D  F
) `  C )  <->  ( C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X )  /\  (
( RR  _D  F
) `  C )  e.  ( ( x  e. 
{ w  e.  X  |  w #  C }  |->  ( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) ) ) lim CC  C ) ) ) )
3835, 37mpbid 147 . . . . 5  |-  ( ph  ->  ( C  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  /\  ( ( RR  _D  F ) `  C
)  e.  ( ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) lim
CC  C ) ) )
3938simprd 114 . . . 4  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  ( ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) lim
CC  C ) )
40 cjcncf 14561 . . . . . 6  |-  *  e.  ( CC -cn-> CC )
415cncfcn1cntop 14567 . . . . . 6  |-  ( CC
-cn-> CC )  =  ( ( MetOpen `  ( abs  o. 
-  ) )  Cn  ( MetOpen `  ( abs  o. 
-  ) ) )
4240, 41eleqtri 2264 . . . . 5  |-  *  e.  ( ( MetOpen `  ( abs  o.  -  ) )  Cn  ( MetOpen `  ( abs  o.  -  ) ) )
4331, 8ffvelcdmd 5676 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  CC )
44 unicntopcntop 14522 . . . . . 6  |-  CC  =  U. ( MetOpen `  ( abs  o. 
-  ) )
4544cncnpi 14214 . . . . 5  |-  ( ( *  e.  ( (
MetOpen `  ( abs  o.  -  ) )  Cn  ( MetOpen `  ( abs  o. 
-  ) ) )  /\  ( ( RR 
_D  F ) `  C )  e.  CC )  ->  *  e.  ( ( ( MetOpen `  ( abs  o.  -  ) )  CnP  ( MetOpen `  ( abs  o.  -  ) ) ) `  ( ( RR  _D  F ) `
 C ) ) )
4642, 43, 45sylancr 414 . . . 4  |-  ( ph  ->  *  e.  ( ( ( MetOpen `  ( abs  o. 
-  ) )  CnP  ( MetOpen `  ( abs  o. 
-  ) ) ) `
 ( ( RR 
_D  F ) `  C ) ) )
4718, 19, 5, 21, 39, 46limccnpcntop 14647 . . 3  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( *  o.  ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) ) lim
CC  C ) )
48 cjf 10898 . . . . . . 7  |-  * : CC --> CC
4948a1i 9 . . . . . 6  |-  ( ph  ->  * : CC --> CC )
5049, 17cofmpt 5709 . . . . 5  |-  ( ph  ->  ( *  o.  (
x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) )  =  ( x  e.  { w  e.  X  |  w #  C }  |->  ( * `  ( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) ) ) ) )
513adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  F : X --> CC )
52 elrabi 2905 . . . . . . . . . . 11  |-  ( x  e.  { w  e.  X  |  w #  C }  ->  x  e.  X
)
5352adantl 277 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x  e.  X )
5451, 53ffvelcdmd 5676 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( F `  x
)  e.  CC )
553, 16ffvelcdmd 5676 . . . . . . . . . 10  |-  ( ph  ->  ( F `  C
)  e.  CC )
5655adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( F `  C
)  e.  CC )
5754, 56subcld 8304 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( F `  x )  -  ( F `  C )
)  e.  CC )
584sselda 3170 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  RR )
5952, 58sylan2 286 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x  e.  RR )
604, 16sseldd 3171 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
6160adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  C  e.  RR )
6259, 61resubcld 8374 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( x  -  C
)  e.  RR )
6362recnd 8022 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( x  -  C
)  e.  CC )
6459recnd 8022 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x  e.  CC )
6561recnd 8022 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  C  e.  CC )
66 breq1 4024 . . . . . . . . . . . 12  |-  ( w  =  x  ->  (
w #  C  <->  x #  C
) )
6766elrab 2908 . . . . . . . . . . 11  |-  ( x  e.  { w  e.  X  |  w #  C } 
<->  ( x  e.  X  /\  x #  C )
)
6867simprbi 275 . . . . . . . . . 10  |-  ( x  e.  { w  e.  X  |  w #  C }  ->  x #  C )
6968adantl 277 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x #  C )
7064, 65, 69subap0d 8637 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( x  -  C
) #  0 )
7157, 63, 70cjdivapd 11019 . . . . . . 7  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( * `  ( ( F `  x )  -  ( F `  C ) ) )  /  ( * `  ( x  -  C
) ) ) )
72 cjsub 10943 . . . . . . . . . 10  |-  ( ( ( F `  x
)  e.  CC  /\  ( F `  C )  e.  CC )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
7354, 56, 72syl2anc 411 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
74 fvco3 5611 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
753, 52, 74syl2an 289 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
76 fvco3 5611 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
773, 16, 76syl2anc 411 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7877adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7975, 78oveq12d 5918 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
8073, 79eqtr4d 2225 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) ) )
8162cjred 11022 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
x  -  C ) )  =  ( x  -  C ) )
8280, 81oveq12d 5918 . . . . . . 7  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( * `  ( ( F `  x )  -  ( F `  C )
) )  /  (
* `  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) )
8371, 82eqtrd 2222 . . . . . 6  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
8483mpteq2dva 4111 . . . . 5  |-  ( ph  ->  ( x  e.  {
w  e.  X  |  w #  C }  |->  ( * `
 ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) )  =  ( x  e. 
{ w  e.  X  |  w #  C }  |->  ( ( ( ( *  o.  F ) `
 x )  -  ( ( *  o.  F ) `  C
) )  /  (
x  -  C ) ) ) )
8550, 84eqtrd 2222 . . . 4  |-  ( ph  ->  ( *  o.  (
x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) )  =  ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) )
8685oveq1d 5915 . . 3  |-  ( ph  ->  ( ( *  o.  ( x  e.  {
w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) ) lim CC  C )  =  ( ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
8747, 86eleqtrd 2268 . 2  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) ) lim
CC  C ) )
88 eqid 2189 . . 3  |-  ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) )  =  ( x  e.  {
w  e.  X  |  w #  C }  |->  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
89 fco 5403 . . . 4  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
9048, 3, 89sylancr 414 . . 3  |-  ( ph  ->  ( *  o.  F
) : X --> CC )
916, 5, 88, 2, 90, 4eldvap 14654 . 2  |-  ( ph  ->  ( C ( RR 
_D  ( *  o.  F ) ) ( * `  ( ( RR  _D  F ) `
 C ) )  <-> 
( C  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  /\  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  { w  e.  X  |  w #  C }  |->  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) ) lim
CC  C ) ) ) )
929, 87, 91mpbir2and 946 1  |-  ( ph  ->  C ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   {crab 2472    C_ wss 3144   class class class wbr 4021    |-> cmpt 4082   dom cdm 4647   ran crn 4648    o. ccom 4651   Fun wfun 5232   -->wf 5234   ` cfv 5238  (class class class)co 5900    ^pm cpm 6679   CCcc 7844   RRcr 7845    - cmin 8164   # cap 8574    / cdiv 8665   (,)cioo 9925   *ccj 10890   abscabs 11048   topGenctg 12771   MetOpencmopn 13880   intcnt 14079    Cn ccn 14171    CnP ccnp 14172   -cn->ccncf 14543   lim CC climc 14626    _D cdv 14627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964  ax-arch 7965  ax-caucvg 7966
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-isom 5247  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-map 6680  df-pm 6681  df-sup 7017  df-inf 7018  df-pnf 8030  df-mnf 8031  df-xr 8032  df-ltxr 8033  df-le 8034  df-sub 8166  df-neg 8167  df-reap 8568  df-ap 8575  df-div 8666  df-inn 8956  df-2 9014  df-3 9015  df-4 9016  df-n0 9213  df-z 9290  df-uz 9565  df-q 9657  df-rp 9691  df-xneg 9809  df-xadd 9810  df-ioo 9929  df-seqfrec 10486  df-exp 10561  df-cj 10893  df-re 10894  df-im 10895  df-rsqrt 11049  df-abs 11050  df-rest 12758  df-topgen 12777  df-psmet 13882  df-xmet 13883  df-met 13884  df-bl 13885  df-mopn 13886  df-top 13984  df-topon 13997  df-bases 14029  df-ntr 14082  df-cn 14174  df-cnp 14175  df-cncf 14544  df-limced 14628  df-dvap 14629
This theorem is referenced by:  dvcj  14676
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