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Theorem dvcjbr 19314
Description: The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19311 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 8810 . . . . 5  |-  RR  C_  CC
21a1i 10 . . . 4  |-  ( ph  ->  RR  C_  CC )
3 dvcj.f . . . 4  |-  ( ph  ->  F : X --> CC )
4 dvcj.x . . . 4  |-  ( ph  ->  X  C_  RR )
5 eqid 2296 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
65tgioo2 18325 . . . 4  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
72, 3, 4, 6, 5dvbssntr 19266 . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
8 dvcj.c . . 3  |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )
97, 8sseldd 3194 . 2  |-  ( ph  ->  C  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X ) )
104, 1syl6ss 3204 . . . . . 6  |-  ( ph  ->  X  C_  CC )
111a1i 10 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  RR  C_  CC )
12 simpl 443 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
13 simpr 447 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  X  C_  RR )
1411, 12, 13dvbss 19267 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  X )
153, 4, 14syl2anc 642 . . . . . . 7  |-  ( ph  ->  dom  ( RR  _D  F )  C_  X
)
1615, 8sseldd 3194 . . . . . 6  |-  ( ph  ->  C  e.  X )
173, 10, 16dvlem 19262 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) )  e.  CC )
18 eqid 2296 . . . . 5  |-  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )
1917, 18fmptd 5700 . . . 4  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) : ( X  \  { C } ) --> CC )
20 ssid 3210 . . . . 5  |-  CC  C_  CC
2120a1i 10 . . . 4  |-  ( ph  ->  CC  C_  CC )
225cnfldtopon 18308 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
2322toponunii 16686 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
2423restid 13354 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2522, 24ax-mp 8 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2625eqcomi 2300 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
27 dvf 19273 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
28 ffun 5407 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
29 funfvbrb 5654 . . . . . . . 8  |-  ( Fun  ( RR  _D  F
)  ->  ( C  e.  dom  ( RR  _D  F )  <->  C ( RR  _D  F ) ( ( RR  _D  F
) `  C )
) )
3027, 28, 29mp2b 9 . . . . . . 7  |-  ( C  e.  dom  ( RR 
_D  F )  <->  C ( RR  _D  F ) ( ( RR  _D  F
) `  C )
)
318, 30sylib 188 . . . . . 6  |-  ( ph  ->  C ( RR  _D  F ) ( ( RR  _D  F ) `
 C ) )
326, 5, 18, 2, 3, 4eldv 19264 . . . . . 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 201 . . . . 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 449 . . . 4  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
35 cjcncf 18424 . . . . . 6  |-  *  e.  ( CC -cn-> CC )
365cncfcn1 18430 . . . . . 6  |-  ( CC
-cn-> CC )  =  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3735, 36eleqtri 2368 . . . . 5  |-  *  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )
3827ffvelrni 5680 . . . . . 6  |-  ( C  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  C
)  e.  CC )
398, 38syl 15 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  C
)  e.  CC )
4023cncnpi 17023 . . . . 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 644 . . . 4  |-  ( ph  ->  *  e.  ( ( ( TopOpen ` fld )  CnP  ( TopOpen ` fld )
) `  ( ( RR  _D  F ) `  C ) ) )
4219, 21, 5, 26, 34, 41limccnp 19257 . . 3  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( *  o.  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) ) ) lim CC  C
) )
43 eqidd 2297 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( F `  x
)  -  ( F `
 C ) )  /  ( x  -  C ) ) ) )
44 cjf 11605 . . . . . . . 8  |-  * : CC --> CC
4544a1i 10 . . . . . . 7  |-  ( ph  ->  * : CC --> CC )
4645feqmptd 5591 . . . . . 6  |-  ( ph  ->  *  =  ( y  e.  CC  |->  ( * `
 y ) ) )
47 fveq2 5541 . . . . . 6  |-  ( y  =  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) )  ->  (
* `  y )  =  ( * `  ( ( ( F `
 x )  -  ( F `  C ) )  /  ( x  -  C ) ) ) )
4817, 43, 46, 47fmptco 5707 . . . . 5  |-  ( ph  ->  ( *  o.  (
x  e.  ( X 
\  { C }
)  |->  ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( * `
 ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) ) )
493adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  F : X --> CC )
50 eldifi 3311 . . . . . . . . . . 11  |-  ( x  e.  ( X  \  { C } )  ->  x  e.  X )
5150adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  X )
52 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( F `  x
)  e.  CC )
5349, 51, 52syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  x
)  e.  CC )
54 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( F `  C
)  e.  CC )
553, 16, 54syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( F `  C
)  e.  CC )
5655adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( F `  C
)  e.  CC )
5753, 56subcld 9173 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( F `  x )  -  ( F `  C )
)  e.  CC )
584sselda 3193 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  RR )
5950, 58sylan2 460 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  RR )
604, 16sseldd 3194 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
6160adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  RR )
6259, 61resubcld 9227 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  RR )
6362recnd 8877 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  e.  CC )
64 eldifsni 3763 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { C } )  ->  x  =/=  C )
6564adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  =/=  C )
6659recnd 8877 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  x  e.  CC )
6761recnd 8877 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  ->  C  e.  CC )
68 subeq0 9089 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  C  e.  CC )  ->  ( ( x  -  C )  =  0  <-> 
x  =  C ) )
6966, 67, 68syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( x  -  C )  =  0  <-> 
x  =  C ) )
7069necon3bid 2494 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( x  -  C )  =/=  0  <->  x  =/=  C ) )
7165, 70mpbird 223 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( x  -  C
)  =/=  0 )
7257, 63, 71cjdivd 11724 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( * `  ( ( F `  x )  -  ( F `  C ) ) )  /  ( * `  ( x  -  C
) ) ) )
73 cjsub 11650 . . . . . . . . . 10  |-  ( ( ( F `  x
)  e.  CC  /\  ( F `  C )  e.  CC )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
7453, 56, 73syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
75 fvco3 5612 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
763, 50, 75syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( *  o.  F ) `  x
)  =  ( * `
 ( F `  x ) ) )
77 fvco3 5612 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
783, 16, 77syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
7978adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( *  o.  F ) `  C
)  =  ( * `
 ( F `  C ) ) )
8076, 79oveq12d 5892 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( ( *  o.  F ) `  x )  -  (
( *  o.  F
) `  C )
)  =  ( ( * `  ( F `
 x ) )  -  ( * `  ( F `  C ) ) ) )
8174, 80eqtr4d 2331 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( F `  x
)  -  ( F `
 C ) ) )  =  ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) ) )
8262cjred 11727 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
x  -  C ) )  =  ( x  -  C ) )
8381, 82oveq12d 5892 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( ( * `  ( ( F `  x )  -  ( F `  C )
) )  /  (
* `  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) )
8472, 83eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X  \  { C } ) )  -> 
( * `  (
( ( F `  x )  -  ( F `  C )
)  /  ( x  -  C ) ) )  =  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) )
8584mpteq2dva 4122 . . . . 5  |-  ( ph  ->  ( x  e.  ( X  \  { C } )  |->  ( * `
 ( ( ( F `  x )  -  ( F `  C ) )  / 
( x  -  C
) ) ) )  =  ( x  e.  ( X  \  { C } )  |->  ( ( ( ( *  o.  F ) `  x
)  -  ( ( *  o.  F ) `
 C ) )  /  ( x  -  C ) ) ) )
8648, 85eqtrd 2328 . . . 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 ) ) ) )
8786oveq1d 5889 . . 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 ) )
8842, 87eleqtrd 2372 . 2  |-  ( ph  ->  ( * `  (
( RR  _D  F
) `  C )
)  e.  ( ( x  e.  ( X 
\  { C }
)  |->  ( ( ( ( *  o.  F
) `  x )  -  ( ( *  o.  F ) `  C ) )  / 
( x  -  C
) ) ) lim CC  C ) )
89 eqid 2296 . . 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 ) ) )
90 fco 5414 . . . 4  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
9144, 3, 90sylancr 644 . . 3  |-  ( ph  ->  ( *  o.  F
) : X --> CC )
926, 5, 89, 2, 91, 4eldv 19264 . 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 ) ) ) )
939, 88, 92mpbir2and 888 1  |-  ( ph  ->  C ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    o. ccom 4709   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    - cmin 9053    / cdiv 9439   (,)cioo 10672   *ccj 11597   ↾t crest 13341   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393  TopOnctopon 16648   intcnt 16770    Cn ccn 16970    CnP ccnp 16971   -cn->ccncf 18396   lim CC climc 19228    _D cdv 19229
This theorem is referenced by:  dvcj  19315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-cncf 18398  df-limc 19232  df-dv 19233
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