ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvcjbr Unicode version

Theorem dvcjbr 12830
Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 12831. (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 7705 . . . . 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 2137 . . . . 5  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
65tgioo2cntop 12707 . . . 4  |-  ( topGen ` 
ran  (,) )  =  ( ( MetOpen `  ( abs  o. 
-  ) )t  RR )
72, 3, 4, 6, 5dvbssntrcntop 12811 . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
8 dvcj.c . . 3  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
97, 8sseldd 3093 . 2  |-  ( ph  ->  C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X ) )
104, 1sstrdi 3104 . . . . . 6  |-  ( ph  ->  X  C_  CC )
111a1i 9 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  RR  C_  CC )
12 simpl 108 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
13 simpr 109 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  X  C_  RR )
1411, 12, 13dvbss 12812 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  X )
153, 4, 14syl2anc 408 . . . . . . 7  |-  ( ph  ->  dom  ( RR  _D  F )  C_  X
)
1615, 8sseldd 3093 . . . . . 6  |-  ( ph  ->  C  e.  X )
173, 10, 16dvlemap 12807 . . . . 5  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) )  e.  CC )
1817fmpttd 5568 . . . 4  |-  ( ph  ->  ( x  e.  {
w  e.  X  |  w #  C }  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) : { w  e.  X  |  w #  C }
--> CC )
19 ssidd 3113 . . . 4  |-  ( ph  ->  CC  C_  CC )
205cntoptopon 12690 . . . . 5  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  (TopOn `  CC )
2120toponrestid 12177 . . . 4  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( (
MetOpen `  ( abs  o.  -  ) )t  CC )
223fdmd 5274 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  =  X )
2322feq2d 5255 . . . . . . . . . . . 12  |-  ( ph  ->  ( F : dom  F --> CC  <->  F : X --> CC ) )
243, 23mpbird 166 . . . . . . . . . . 11  |-  ( ph  ->  F : dom  F --> CC )
2522, 4eqsstrd 3128 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  C_  RR )
26 cnex 7737 . . . . . . . . . . . 12  |-  CC  e.  _V
27 reex 7747 . . . . . . . . . . . 12  |-  RR  e.  _V
2826, 27elpm2 6567 . . . . . . . . . . 11  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
2924, 25, 28sylanbrc 413 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
30 dvfpm 12816 . . . . . . . . . 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 5271 . . . . . . . 8  |-  ( ph  ->  Fun  ( RR  _D  F ) )
33 funfvbrb 5526 . . . . . . . 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 146 . . . . . 6  |-  ( ph  ->  C ( RR  _D  F ) ( ( RR  _D  F ) `
 C ) )
36 eqid 2137 . . . . . . 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 12809 . . . . . 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 146 . . . . 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 113 . . . 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 12733 . . . . . 6  |-  *  e.  ( CC -cn-> CC )
415cncfcn1cntop 12739 . . . . . 6  |-  ( CC
-cn-> CC )  =  ( ( MetOpen `  ( abs  o. 
-  ) )  Cn  ( MetOpen `  ( abs  o. 
-  ) ) )
4240, 41eleqtri 2212 . . . . 5  |-  *  e.  ( ( MetOpen `  ( abs  o.  -  ) )  Cn  ( MetOpen `  ( abs  o.  -  ) ) )
4331, 8ffvelrnd 5549 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  CC )
44 unicntopcntop 12694 . . . . . 6  |-  CC  =  U. ( MetOpen `  ( abs  o. 
-  ) )
4544cncnpi 12386 . . . . 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 410 . . . 4  |-  ( ph  ->  *  e.  ( ( ( MetOpen `  ( abs  o. 
-  ) )  CnP  ( MetOpen `  ( abs  o. 
-  ) ) ) `
 ( ( RR 
_D  F ) `  C ) ) )
4718, 19, 5, 21, 39, 46limccnpcntop 12802 . . 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 10612 . . . . . . 7  |-  * : CC --> CC
4948a1i 9 . . . . . 6  |-  ( ph  ->  * : CC --> CC )
5049, 17cofmpt 5582 . . . . 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 274 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  F : X --> CC )
52 elrabi 2832 . . . . . . . . . . 11  |-  ( x  e.  { w  e.  X  |  w #  C }  ->  x  e.  X
)
5352adantl 275 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x  e.  X )
5451, 53ffvelrnd 5549 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( F `  x
)  e.  CC )
553, 16ffvelrnd 5549 . . . . . . . . . 10  |-  ( ph  ->  ( F `  C
)  e.  CC )
5655adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( F `  C
)  e.  CC )
5754, 56subcld 8066 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( F `  x )  -  ( F `  C )
)  e.  CC )
584sselda 3092 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  RR )
5952, 58sylan2 284 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x  e.  RR )
604, 16sseldd 3093 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
6160adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  C  e.  RR )
6259, 61resubcld 8136 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( x  -  C
)  e.  RR )
6362recnd 7787 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( x  -  C
)  e.  CC )
6459recnd 7787 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x  e.  CC )
6561recnd 7787 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  C  e.  CC )
66 breq1 3927 . . . . . . . . . . . 12  |-  ( w  =  x  ->  (
w #  C  <->  x #  C
) )
6766elrab 2835 . . . . . . . . . . 11  |-  ( x  e.  { w  e.  X  |  w #  C } 
<->  ( x  e.  X  /\  x #  C )
)
6867simprbi 273 . . . . . . . . . 10  |-  ( x  e.  { w  e.  X  |  w #  C }  ->  x #  C )
6968adantl 275 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  ->  x #  C )
7064, 65, 69subap0d 8399 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( x  -  C
) #  0 )
7157, 63, 70cjdivapd 10733 . . . . . . 7  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( * `  ( ( F `  x )  -  ( F `  C ) ) )  /  ( * `  ( x  -  C
) ) ) )
72 cjsub 10657 . . . . . . . . . 10  |-  ( ( ( F `  x
)  e.  CC  /\  ( F `  C )  e.  CC )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
7354, 56, 72syl2anc 408 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
74 fvco3 5485 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
753, 52, 74syl2an 287 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
76 fvco3 5485 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
773, 16, 76syl2anc 408 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7877adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7975, 78oveq12d 5785 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
8073, 79eqtr4d 2173 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) ) )
8162cjred 10736 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
x  -  C ) )  =  ( x  -  C ) )
8280, 81oveq12d 5785 . . . . . . 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 2170 . . . . . 6  |-  ( (
ph  /\  x  e.  { w  e.  X  |  w #  C } )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
8483mpteq2dva 4013 . . . . 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 2170 . . . 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 5782 . . 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 2216 . 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 2137 . . 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 5283 . . . 4  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
9048, 3, 89sylancr 410 . . 3  |-  ( ph  ->  ( *  o.  F
) : X --> CC )
916, 5, 88, 2, 90, 4eldvap 12809 . 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 928 1  |-  ( ph  ->  C ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   {crab 2418    C_ wss 3066   class class class wbr 3924    |-> cmpt 3984   dom cdm 4534   ran crn 4535    o. ccom 4538   Fun wfun 5112   -->wf 5114   ` cfv 5118  (class class class)co 5767    ^pm cpm 6536   CCcc 7611   RRcr 7612    - cmin 7926   # cap 8336    / cdiv 8425   (,)cioo 9664   *ccj 10604   abscabs 10762   topGenctg 12124   MetOpencmopn 12143   intcnt 12251    Cn ccn 12343    CnP ccnp 12344   -cn->ccncf 12715   lim CC climc 12781    _D cdv 12782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-map 6537  df-pm 6538  df-sup 6864  df-inf 6865  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-xneg 9552  df-xadd 9553  df-ioo 9668  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-rest 12111  df-topgen 12130  df-psmet 12145  df-xmet 12146  df-met 12147  df-bl 12148  df-mopn 12149  df-top 12154  df-topon 12167  df-bases 12199  df-ntr 12254  df-cn 12346  df-cnp 12347  df-cncf 12716  df-limced 12783  df-dvap 12784
This theorem is referenced by:  dvcj  12831
  Copyright terms: Public domain W3C validator