MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvlog Structured version   Unicode version

Theorem dvlog 20544
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 2438 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtop 18820 . . . . . 6  |-  ( TopOpen ` fld )  e.  Top
31cnfldtopon 18819 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
43toponunii 16999 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
54restid 13663 . . . . . 6  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
62, 5ax-mp 8 . . . . 5  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
76eqcomi 2442 . . . 4  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8 cnex 9073 . . . . . 6  |-  CC  e.  _V
98prid2 3915 . . . . 5  |-  CC  e.  { RR ,  CC }
109a1i 11 . . . 4  |-  (  T. 
->  CC  e.  { RR ,  CC } )
11 logcn.d . . . . . 6  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
1211logdmopn 20542 . . . . 5  |-  D  e.  ( TopOpen ` fld )
1312a1i 11 . . . 4  |-  (  T. 
->  D  e.  ( TopOpen
` fld
) )
14 logf1o 20464 . . . . . . . . 9  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
15 f1of1 5675 . . . . . . . . 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 20535 . . . . . . . 8  |-  D  C_  ( CC  \  { 0 } )
18 f1ores 5691 . . . . . . . 8  |-  ( ( log : ( CC 
\  { 0 } ) -1-1-> ran  log  /\  D  C_  ( CC  \  { 0 } ) )  -> 
( log  |`  D ) : D -1-1-onto-> ( log " D
) )
1916, 17, 18mp2an 655 . . . . . . 7  |-  ( log  |`  D ) : D -1-1-onto-> ( log " D )
20 f1ocnv 5689 . . . . . . 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 20456 . . . . . . . . . . 11  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
2322reseq1i 5144 . . . . . . . . . 10  |-  ( log  |`  D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
2423cnveqi 5049 . . . . . . . . 9  |-  `' ( log  |`  D )  =  `' ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  |`  D )
25 eff 12686 . . . . . . . . . . 11  |-  exp : CC
--> CC
26 cnvimass 5226 . . . . . . . . . . . 12  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  dom  Im
27 imf 11920 . . . . . . . . . . . . 13  |-  Im : CC
--> RR
2827fdmi 5598 . . . . . . . . . . . 12  |-  dom  Im  =  CC
2926, 28sseqtri 3382 . . . . . . . . . . 11  |-  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC
30 fssres 5612 . . . . . . . . . . 11  |-  ( ( exp : CC --> CC  /\  ( `' Im " ( -u pi (,] pi ) ) 
C_  CC )  -> 
( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC )
3125, 29, 30mp2an 655 . . . . . . . . . 10  |-  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) ) : ( `' Im "
( -u pi (,] pi ) ) --> CC
32 ffun 5595 . . . . . . . . . 10  |-  ( ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC  ->  Fun  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) )
33 funcnvres2 5526 . . . . . . . . . 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 5226 . . . . . . . . . . 11  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  dom  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )
3631fdmi 5598 . . . . . . . . . . 11  |-  dom  ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )  =  ( `' Im "
( -u pi (,] pi ) )
3735, 36sseqtri 3382 . . . . . . . . . 10  |-  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) 
C_  ( `' Im " ( -u pi (,] pi ) )
38 resabs1 5177 . . . . . . . . . 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 2462 . . . . . . . 8  |-  `' ( log  |`  D )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
4122imaeq1i 5202 . . . . . . . . 9  |-  ( log " D )  =  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) ) " D )
4241reseq2i 5145 . . . . . . . 8  |-  ( exp  |`  ( log " D
) )  =  ( exp  |`  ( `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
" D ) )
4340, 42eqtr4i 2461 . . . . . . 7  |-  `' ( log  |`  D )  =  ( exp  |`  ( log " D ) )
44 f1oeq1 5667 . . . . . . 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 201 . . . . 5  |-  ( exp  |`  ( log " D
) ) : ( log " D ) -1-1-onto-> D
4746a1i 11 . . . 4  |-  (  T. 
->  ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D )
4843cnveqi 5049 . . . . . 6  |-  `' `' ( log  |`  D )  =  `' ( exp  |`  ( log " D ) )
49 relres 5176 . . . . . . 7  |-  Rel  ( log  |`  D )
50 dfrel2 5323 . . . . . . 7  |-  ( Rel  ( log  |`  D )  <->  `' `' ( log  |`  D )  =  ( log  |`  D ) )
5149, 50mpbi 201 . . . . . 6  |-  `' `' ( log  |`  D )  =  ( log  |`  D )
5248, 51eqtr3i 2460 . . . . 5  |-  `' ( exp  |`  ( log " D ) )  =  ( log  |`  D )
53 f1of 5676 . . . . . . 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 5218 . . . . . . . 8  |-  ( log " D )  C_  ran  log
56 logrncn 20462 . . . . . . . . 9  |-  ( x  e.  ran  log  ->  x  e.  CC )
5756ssriv 3354 . . . . . . . 8  |-  ran  log  C_  CC
5855, 57sstri 3359 . . . . . . 7  |-  ( log " D )  C_  CC
5911logcn 20540 . . . . . . 7  |-  ( log  |`  D )  e.  ( D -cn-> CC )
60 cncffvrn 18930 . . . . . . 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 655 . . . . . 6  |-  ( ( log  |`  D )  e.  ( D -cn-> ( log " D ) )  <->  ( log  |`  D ) : D --> ( log " D ) )
6254, 61sylibr 205 . . . . 5  |-  (  T. 
->  ( log  |`  D )  e.  ( D -cn-> ( log " D ) ) )
6352, 62syl5eqel 2522 . . . 4  |-  (  T. 
->  `' ( exp  |`  ( log " D ) )  e.  ( D -cn-> ( log " D ) ) )
64 ssid 3369 . . . . . . . . 9  |-  CC  C_  CC
651, 7dvres 19800 . . . . . . . . 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 656 . . . . . . . 8  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )
67 dvef 19866 . . . . . . . . 9  |-  ( CC 
_D  exp )  =  exp
6811dvloglem 20541 . . . . . . . . . 10  |-  ( log " D )  e.  (
TopOpen ` fld )
69 isopn3i 17148 . . . . . . . . . 10  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( log " D
)  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  ( log " D ) )  =  ( log " D
) )
702, 68, 69mp2an 655 . . . . . . . . 9  |-  ( ( int `  ( TopOpen ` fld )
) `  ( log " D ) )  =  ( log " D
)
7167, 70reseq12i 5146 . . . . . . . 8  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7266, 71eqtri 2458 . . . . . . 7  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  =  ( exp  |`  ( log " D
) )
7372dmeqi 5073 . . . . . 6  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  dom  ( exp  |`  ( log " D
) )
74 dmres 5169 . . . . . 6  |-  dom  ( exp  |`  ( log " D
) )  =  ( ( log " D
)  i^i  dom  exp )
7525fdmi 5598 . . . . . . . 8  |-  dom  exp  =  CC
7658, 75sseqtr4i 3383 . . . . . . 7  |-  ( log " D )  C_  dom  exp
77 df-ss 3336 . . . . . . 7  |-  ( ( log " D ) 
C_  dom  exp  <->  ( ( log " D )  i^i 
dom  exp )  =  ( log " D ) )
7876, 77mpbi 201 . . . . . 6  |-  ( ( log " D )  i^i  dom  exp )  =  ( log " D
)
7973, 74, 783eqtri 2462 . . . . 5  |-  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D )
8079a1i 11 . . . 4  |-  (  T. 
->  dom  ( CC  _D  ( exp  |`  ( log " D ) ) )  =  ( log " D
) )
81 neirr 2608 . . . . . 6  |-  -.  0  =/=  0
82 resss 5172 . . . . . . . . . . . . 13  |-  ( ( CC  _D  exp )  |`  ( ( int `  ( TopOpen
` fld
) ) `  ( log " D ) ) )  C_  ( CC  _D  exp )
8366, 82eqsstri 3380 . . . . . . . . . . . 12  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  ( CC  _D  exp )
8483, 67sseqtri 3382 . . . . . . . . . . 11  |-  ( CC 
_D  ( exp  |`  ( log " D ) ) )  C_  exp
85 rnss 5100 . . . . . . . . . . 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 12702 . . . . . . . . . . 11  |-  exp : CC
--> ( CC  \  {
0 } )
88 frn 5599 . . . . . . . . . . 11  |-  ( exp
: CC --> ( CC 
\  { 0 } )  ->  ran  exp  C_  ( CC  \  { 0 } ) )
8987, 88ax-mp 8 . . . . . . . . . 10  |-  ran  exp  C_  ( CC  \  {
0 } )
9086, 89sstri 3359 . . . . . . . . 9  |-  ran  ( CC  _D  ( exp  |`  ( log " D ) ) )  C_  ( CC  \  { 0 } )
9190sseli 3346 . . . . . . . 8  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  e.  ( CC  \  { 0 } ) )
92 eldifsn 3929 . . . . . . . 8  |-  ( 0  e.  ( CC  \  { 0 } )  <-> 
( 0  e.  CC  /\  0  =/=  0 ) )
9391, 92sylib 190 . . . . . . 7  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  ( 0  e.  CC  /\  0  =/=  0 ) )
9493simprd 451 . . . . . 6  |-  ( 0  e.  ran  ( CC 
_D  ( exp  |`  ( log " D ) ) )  ->  0  =/=  0 )
9581, 94mto 170 . . . . 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 19863 . . 3  |-  (  T. 
->  ( CC  _D  `' ( exp  |`  ( log " D ) ) )  =  ( x  e.  D  |->  ( 1  / 
( ( CC  _D  ( exp  |`  ( log " D ) ) ) `
 ( `' ( exp  |`  ( log " D ) ) `  x ) ) ) ) )
9897trud 1333 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( x  e.  D  |->  ( 1  /  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )
9952oveq2i 6094 . 2  |-  ( CC 
_D  `' ( exp  |`  ( log " D
) ) )  =  ( CC  _D  ( log  |`  D ) )
10072fveq1i 5731 . . . . 5  |-  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )
101 f1ocnvfv2 6017 . . . . . 6  |-  ( ( ( exp  |`  ( log " D ) ) : ( log " D
)
-1-1-onto-> D  /\  x  e.  D
)  ->  ( ( exp  |`  ( log " D
) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
10246, 101mpan 653 . . . . 5  |-  ( x  e.  D  ->  (
( exp  |`  ( log " D ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
103100, 102syl5eq 2482 . . . 4  |-  ( x  e.  D  ->  (
( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) )  =  x )
104103oveq2d 6099 . . 3  |-  ( x  e.  D  ->  (
1  /  ( ( CC  _D  ( exp  |`  ( log " D
) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) )  =  ( 1  /  x ) )
105104mpteq2ia 4293 . 2  |-  ( x  e.  D  |->  ( 1  /  ( ( CC 
_D  ( exp  |`  ( log " D ) ) ) `  ( `' ( exp  |`  ( log " D ) ) `
 x ) ) ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
10698, 99, 1053eqtr3i 2466 1  |-  ( CC 
_D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360    T. wtru 1326    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319    i^i cin 3321    C_ wss 3322   {csn 3816   {cpr 3817    e. cmpt 4268   `'ccnv 4879   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883   Rel wrel 4885   Fun wfun 5450   -->wf 5452   -1-1->wf1 5453   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    -oocmnf 9120   -ucneg 9294    / cdiv 9679   (,]cioc 10919   Imcim 11905   expce 12666   picpi 12671   ↾t crest 13650   TopOpenctopn 13651  ℂfldccnfld 16705   Topctop 16960   intcnt 17083   -cn->ccncf 18908    _D cdv 19752   logclog 20454
This theorem is referenced by:  dvlog2  20546  dvatan  20777  lgamgulmlem2  24816  dvreasin  26292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-tan 12676  df-pi 12677  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-cmp 17452  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456
  Copyright terms: Public domain W3C validator