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Theorem dvlog 20495
Description: The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)
Hypothesis
Ref Expression
logcn.d  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
Assertion
Ref Expression
dvlog  |-  ( CC 
_D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
Distinct variable group:    x, D

Proof of Theorem dvlog
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtop 18771 . . . . . 6  |-  ( TopOpen ` fld )  e.  Top
31cnfldtopon 18770 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
43toponunii 16952 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
54restid 13616 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
62, 5ax-mp 8 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
76eqcomi 2408 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8 cnex 9027 . . . . . 6  |-  CC  e.  _V
98prid2 3873 . . . . 5  |-  CC  e.  { RR ,  CC }
109a1i 11 . . . 4  |-  (  T. 
->  CC  e.  { RR ,  CC } )
11 logcn.d . . . . . 6  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
1211logdmopn 20493 . . . . 5  |-  D  e.  ( TopOpen ` fld )
1312a1i 11 . . . 4  |-  (  T. 
->  D  e.  ( TopOpen
` fld
) )
14 logf1o 20415 . . . . . . . . 9  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
15 f1of1 5632 . . . . . . . . 9  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  log : ( CC 
\  { 0 } ) -1-1-> ran  log )
1614, 15ax-mp 8 . . . . . . . 8  |-  log :
( CC  \  {
0 } ) -1-1-> ran  log
1711logdmss 20486 . . . . . . . 8  |-  D  C_  ( CC  \  { 0 } )
18 f1ores 5648 . . . . . . . 8  |-  ( ( log : ( CC 
\  { 0 } ) -1-1-> ran  log  /\  D  C_  ( CC  \  { 0 } ) )  -> 
( log  |`  D ) : D -1-1-onto-> ( log " D
) )
1916, 17, 18mp2an 654 . . . . . . 7  |-  ( log  |`  D ) : D -1-1-onto-> ( log " D )
20 f1ocnv 5646 . . . . . . 7  |-  ( ( log  |`  D ) : D -1-1-onto-> ( log " D
)  ->  `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D )
2119, 20ax-mp 8 . . . . . 6  |-  `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D
22 df-log 20407 . . . . . . . . . . 11  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
2322reseq1i 5101 . . . . . . . . . 10  |-  ( log  |`  D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
2423cnveqi 5006 . . . . . . . . 9  |-  `' ( log  |`  D )  =  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
25 eff 12639 . . . . . . . . . . 11  |-  exp : CC
--> CC
26 cnvimass 5183 . . . . . . . . . . . 12  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  dom  Im
27 imf 11873 . . . . . . . . . . . . 13  |-  Im : CC
--> RR
2827fdmi 5555 . . . . . . . . . . . 12  |-  dom  Im  =  CC
2926, 28sseqtri 3340 . . . . . . . . . . 11  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC
30 fssres 5569 . . . . . . . . . . 11  |-  ( ( exp : CC --> CC  /\  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC )  -> 
( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC )
3125, 29, 30mp2an 654 . . . . . . . . . 10  |-  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) : ( `' Im "
( -u pi (,] pi ) ) --> CC
32 ffun 5552 . . . . . . . . . 10  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC  ->  Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) )
33 funcnvres2 5483 . . . . . . . . . 10  |-  ( Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  ->  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )  =  ( ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) )
3431, 32, 33mp2b 10 . . . . . . . . 9  |-  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )  =  ( ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
35 cnvimass 5183 . . . . . . . . . . 11  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  dom  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )
3631fdmi 5555 . . . . . . . . . . 11  |-  dom  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( `' Im "
( -u pi (,] pi ) )
3735, 36sseqtri 3340 . . . . . . . . . 10  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )
38 resabs1 5134 . . . . . . . . . 10  |-  ( ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )  ->  (
( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) )
3937, 38ax-mp 8 . . . . . . . . 9  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  ( `' ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) " D ) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
4024, 34, 393eqtri 2428 . . . . . . . 8  |-  `' ( log  |`  D )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
4122imaeq1i 5159 . . . . . . . . 9  |-  ( log " D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D )
4241reseq2i 5102 . . . . . . . 8  |-  ( exp  |`  ( log " D
) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )
4340, 42eqtr4i 2427 . . . . . . 7  |-  `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )
44 f1oeq1 5624 . . . . . . 7  |-  ( `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )  ->  ( `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D 
<->  ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D ) )
4543, 44ax-mp 8 . . . . . 6  |-  ( `' ( log  |`  D ) : ( log " D
)
-1-1-onto-> D 
<->  ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D )
4621, 45mpbi 200 . . . . 5  |-  ( exp  |`  ( log " D
) ) : ( log " D ) -1-1-onto-> D
4746a1i 11 . . . 4  |-  (  T. 
->  ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D )
4843cnveqi 5006 . . . . . 6  |-  `' `' ( log  |`  D )  =  `' ( exp  |`  ( log " D ) )
49 relres 5133 . . . . . . 7  |-  Rel  ( log  |`  D )
50 dfrel2 5280 . . . . . . 7  |-  ( Rel  ( log  |`  D )  <->  `' `' ( log  |`  D )  =  ( log  |`  D ) )
5149, 50mpbi 200 . . . . . 6  |-  `' `' ( log  |`  D )  =  ( log  |`  D )
5248, 51eqtr3i 2426 . . . . 5  |-  `' ( exp  |`  ( log " D ) )  =  ( log  |`  D )
53 f1of 5633 . . . . . . 7  |-  ( ( log  |`  D ) : D -1-1-onto-> ( log " D
)  ->  ( log  |`  D ) : D --> ( log " D ) )
5419, 53mp1i 12 . . . . . 6  |-  (  T. 
->  ( log  |`  D ) : D --> ( log " D ) )
55 imassrn 5175 . . . . . . . 8  |-  ( log " D )  C_  ran  log
56 logrncn 20413 . . . . . . . . 9  |-  ( x  e.  ran  log  ->  x  e.  CC )
5756ssriv 3312 . . . . . . . 8  |-  ran  log  C_  CC
5855, 57sstri 3317 . . . . . . 7  |-  ( log " D )  C_  CC
5911logcn 20491 . . . . . . 7  |-  ( log  |`  D )  e.  ( D -cn-> CC )
60 cncffvrn 18881 . . . . . . 7  |-  ( ( ( log " D
)  C_  CC  /\  ( log  |`  D )  e.  ( D -cn-> CC ) )  ->  ( ( log  |`  D )  e.  ( D -cn-> ( log " D ) )  <->  ( log  |`  D ) : D --> ( log " D ) ) )
6158, 59, 60mp2an 654 . . . . . 6  |-  ( ( log  |`  D )  e.  ( D -cn-> ( log " D ) )  <->  ( log  |`  D ) : D --> ( log " D ) )
6254, 61sylibr 204 . . . . 5  |-  (  T. 
->  ( log  |`  D )  e.  ( D -cn-> ( log " D ) ) )
6352, 62syl5eqel 2488 . . . 4  |-  (  T. 
->  `' ( exp  |`  ( log " D ) )  e.  ( D -cn-> ( log " D ) ) )
64 ssid 3327 . . . . . . . . 9  |-  CC  C_  CC
651, 7dvres 19751 . . . . . . . . 9  |-  ( ( ( CC  C_  CC  /\ 
exp : CC --> CC )  /\  ( CC  C_  CC  /\  ( log " D
)  C_  CC )
)  ->  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) ) )
6664, 25, 64, 58, 65mp4an 655 . . . . . . . 8  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )
67 dvef 19817 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
6811dvloglem 20492 . . . . . . . . . 10  |-  ( log " D )  e.  (
TopOpen ` fld )
69 isopn3i 17101 . . . . . . . . . 10  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( log " D
)  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  ( log " D ) )  =  ( log " D
) )
702, 68, 69mp2an 654 . . . . . . . . 9  |-  ( ( int `  ( TopOpen ` fld )
) `  ( log " D ) )  =  ( log " D
)
7167, 70reseq12i 5103 . . . . . . . 8  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7266, 71eqtri 2424 . . . . . . 7  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7372dmeqi 5030 . . . . . 6  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  dom  ( exp  |`  ( log " D
) )
74 dmres 5126 . . . . . 6  |-  dom  ( exp  |`  ( log " D
) )  =  ( ( log " D
)  i^i  dom  exp )
7525fdmi 5555 . . . . . . . 8  |-  dom  exp  =  CC
7658, 75sseqtr4i 3341 . . . . . . 7  |-  ( log " D )  C_  dom  exp
77 df-ss 3294 . . . . . . 7  |-  ( ( log " D ) 
C_  dom  exp  <->  ( ( log " D )  i^i 
dom  exp )  =  ( log " D ) )
7876, 77mpbi 200 . . . . . 6  |-  ( ( log " D )  i^i  dom  exp )  =  ( log " D
)
7973, 74, 783eqtri 2428 . . . . 5  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D )
8079a1i 11 . . . 4  |-  (  T. 
->  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D
) )
81 neirr 2572 . . . . . 6  |-  -.  0  =/=  0
82 resss 5129 . . . . . . . . . . . . 13  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  C_  ( CC  _D  exp )
8366, 82eqsstri 3338 . . . . . . . . . . . 12  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  ( CC  _D  exp )
8483, 67sseqtri 3340 . . . . . . . . . . 11  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  exp
85 rnss 5057 . . . . . . . . . . 11  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) )  C_  exp  ->  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ran  exp )
8684, 85ax-mp 8 . . . . . . . . . 10  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ran  exp
87 eff2 12655 . . . . . . . . . . 11  |-  exp : CC
--> ( CC  \  {
0 } )
88 frn 5556 . . . . . . . . . . 11  |-  ( exp
: CC --> ( CC 
\  { 0 } )  ->  ran  exp  C_  ( CC  \  { 0 } ) )
8987, 88ax-mp 8 . . . . . . . . . 10  |-  ran  exp  C_  ( CC  \  {
0 } )
9086, 89sstri 3317 . . . . . . . . 9  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ( CC  \  { 0 } )
9190sseli 3304 . . . . . . . 8  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  e.  ( CC  \  { 0 } ) )
92 eldifsn 3887 . . . . . . . 8  |-  ( 0  e.  ( CC  \  { 0 } )  <-> 
( 0  e.  CC  /\  0  =/=  0 ) )
9391, 92sylib 189 . . . . . . 7  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  ( 0  e.  CC  /\  0  =/=  0 ) )
9493simprd 450 . . . . . 6  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  =/=  0 )
9581, 94mto 169 . . . . 5  |-  -.  0  e.  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )
9695a1i 11 . . . 4  |-  (  T. 
->  -.  0  e.  ran  ( CC  _D  ( exp  |`  ( log " D
) ) ) )
971, 7, 10, 13, 47, 63, 80, 96dvcnv 19814 . . 3  |-  (  T. 
->  ( CC  _D  `' ( exp  |`  ( log " D ) ) )  =  ( x  e.  D  |->  ( 1  / 
( ( CC  _D  ( exp  |`  ( log " D ) ) ) `
 ( `' ( exp  |`  ( log " D ) ) `  x ) ) ) ) )
9897trud 1329 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( x  e.  D  |->  ( 1  /  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )
9952oveq2i 6051 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( CC  _D  ( log  |`  D ) )
10072fveq1i 5688 . . . . 5  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )
101 f1ocnvfv2 5974 . . . . . 6  |-  ( ( ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D  /\  x  e.  D
)  ->  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
10246, 101mpan 652 . . . . 5  |-  ( x  e.  D  ->  (
( exp  |`  ( log " D ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
103100, 102syl5eq 2448 . . . 4  |-  ( x  e.  D  ->  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
104103oveq2d 6056 . . 3  |-  ( x  e.  D  ->  (
1  /  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) )  =  ( 1  /  x ) )
105104mpteq2ia 4251 . 2  |-  ( x  e.  D  |->  ( 1  /  ( ( CC 
_D  ( exp  |`  ( log " D ) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
10698, 99, 1053eqtr3i 2432 1  |-  ( CC 
_D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    i^i cin 3279    C_ wss 3280   {csn 3774   {cpr 3775    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Rel wrel 4842   Fun wfun 5407   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    -oocmnf 9074   -ucneg 9248    / cdiv 9633   (,]cioc 10873   Imcim 11858   expce 12619   picpi 12624   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658   Topctop 16913   intcnt 17036   -cn->ccncf 18859    _D cdv 19703   logclog 20405
This theorem is referenced by:  dvlog2  20497  dvatan  20728  lgamgulmlem2  24767  dvreasin  26179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-tan 12629  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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