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Theorem logcn 20538
Description: The logarithm function is continuous away from the branch cut at negative reals. (Contributed by Mario Carneiro, 25-Feb-2015.)
Hypothesis
Ref Expression
logcn.d  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
Assertion
Ref Expression
logcn  |-  ( log  |`  D )  e.  ( D -cn-> CC )

Proof of Theorem logcn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 logf1o 20462 . . . . . . 7  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1of 5674 . . . . . . 7  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  log : ( CC 
\  { 0 } ) --> ran  log )
31, 2ax-mp 8 . . . . . 6  |-  log :
( CC  \  {
0 } ) --> ran 
log
4 logcn.d . . . . . . 7  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
54logdmss 20533 . . . . . 6  |-  D  C_  ( CC  \  { 0 } )
6 fssres 5610 . . . . . 6  |-  ( ( log : ( CC 
\  { 0 } ) --> ran  log  /\  D  C_  ( CC  \  {
0 } ) )  ->  ( log  |`  D ) : D --> ran  log )
73, 5, 6mp2an 654 . . . . 5  |-  ( log  |`  D ) : D --> ran  log
8 ffn 5591 . . . . 5  |-  ( ( log  |`  D ) : D --> ran  log  ->  ( log  |`  D )  Fn  D )
97, 8ax-mp 8 . . . 4  |-  ( log  |`  D )  Fn  D
10 dffn5 5772 . . . 4  |-  ( ( log  |`  D )  Fn  D  <->  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `  x
) ) )
119, 10mpbi 200 . . 3  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `
 x ) )
12 fvres 5745 . . . . 5  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( log `  x
) )
134ellogdm 20530 . . . . . . . 8  |-  ( x  e.  D  <->  ( x  e.  CC  /\  ( x  e.  RR  ->  x  e.  RR+ ) ) )
1413simplbi 447 . . . . . . 7  |-  ( x  e.  D  ->  x  e.  CC )
154logdmn0 20531 . . . . . . 7  |-  ( x  e.  D  ->  x  =/=  0 )
1614, 15logcld 20468 . . . . . 6  |-  ( x  e.  D  ->  ( log `  x )  e.  CC )
1716replimd 12002 . . . . 5  |-  ( x  e.  D  ->  ( log `  x )  =  ( ( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
18 relog 20491 . . . . . . . 8  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
1914, 15, 18syl2anc 643 . . . . . . 7  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
2014, 15absrpcld 12250 . . . . . . . 8  |-  ( x  e.  D  ->  ( abs `  x )  e.  RR+ )
21 fvres 5745 . . . . . . . 8  |-  ( ( abs `  x )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( abs `  x ) )  =  ( log `  ( abs `  x
) ) )
2220, 21syl 16 . . . . . . 7  |-  ( x  e.  D  ->  (
( log  |`  RR+ ) `  ( abs `  x
) )  =  ( log `  ( abs `  x ) ) )
2319, 22eqtr4d 2471 . . . . . 6  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )
2423oveq1d 6096 . . . . 5  |-  ( x  e.  D  ->  (
( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
2512, 17, 243eqtrd 2472 . . . 4  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2625mpteq2ia 4291 . . 3  |-  ( x  e.  D  |->  ( ( log  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2711, 26eqtri 2456 . 2  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
28 eqid 2436 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2928addcn 18895 . . . . 5  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3029a1i 11 . . . 4  |-  (  T. 
->  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
3128cnfldtopon 18817 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3214ssriv 3352 . . . . . . . 8  |-  D  C_  CC
33 resttopon 17225 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
3431, 32, 33mp2an 654 . . . . . . 7  |-  ( (
TopOpen ` fld )t  D )  e.  (TopOn `  D )
3534a1i 11 . . . . . 6  |-  (  T. 
->  ( ( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
36 absf 12141 . . . . . . . . . . . 12  |-  abs : CC
--> RR
37 fssres 5610 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  D  C_  CC )  -> 
( abs  |`  D ) : D --> RR )
3836, 32, 37mp2an 654 . . . . . . . . . . 11  |-  ( abs  |`  D ) : D --> RR
3938a1i 11 . . . . . . . . . 10  |-  (  T. 
->  ( abs  |`  D ) : D --> RR )
4039feqmptd 5779 . . . . . . . . 9  |-  (  T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( ( abs  |`  D ) `  x
) ) )
41 fvres 5745 . . . . . . . . . 10  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  =  ( abs `  x
) )
4241mpteq2ia 4291 . . . . . . . . 9  |-  ( x  e.  D  |->  ( ( abs  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( abs `  x
) )
4340, 42syl6eq 2484 . . . . . . . 8  |-  (  T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( abs `  x
) ) )
44 ffn 5591 . . . . . . . . . . 11  |-  ( ( abs  |`  D ) : D --> RR  ->  ( abs  |`  D )  Fn  D )
4538, 44ax-mp 8 . . . . . . . . . 10  |-  ( abs  |`  D )  Fn  D
4641, 20eqeltrd 2510 . . . . . . . . . . 11  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  e.  RR+ )
4746rgen 2771 . . . . . . . . . 10  |-  A. x  e.  D  ( ( abs  |`  D ) `  x )  e.  RR+
48 ffnfv 5894 . . . . . . . . . 10  |-  ( ( abs  |`  D ) : D --> RR+  <->  ( ( abs  |`  D )  Fn  D  /\  A. x  e.  D  ( ( abs  |`  D ) `
 x )  e.  RR+ ) )
4945, 47, 48mpbir2an 887 . . . . . . . . 9  |-  ( abs  |`  D ) : D --> RR+
50 rpssre 10622 . . . . . . . . . . 11  |-  RR+  C_  RR
51 ax-resscn 9047 . . . . . . . . . . 11  |-  RR  C_  CC
5250, 51sstri 3357 . . . . . . . . . 10  |-  RR+  C_  CC
53 abscncf 18931 . . . . . . . . . . 11  |-  abs  e.  ( CC -cn-> RR )
54 rescncf 18927 . . . . . . . . . . 11  |-  ( D 
C_  CC  ->  ( abs 
e.  ( CC -cn-> RR )  ->  ( abs  |`  D )  e.  ( D -cn-> RR ) ) )
5532, 53, 54mp2 9 . . . . . . . . . 10  |-  ( abs  |`  D )  e.  ( D -cn-> RR )
56 cncffvrn 18928 . . . . . . . . . 10  |-  ( (
RR+  C_  CC  /\  ( abs  |`  D )  e.  ( D -cn-> RR ) )  ->  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ ) )
5752, 55, 56mp2an 654 . . . . . . . . 9  |-  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ )
5849, 57mpbir 201 . . . . . . . 8  |-  ( abs  |`  D )  e.  ( D -cn-> RR+ )
5943, 58syl6eqelr 2525 . . . . . . 7  |-  (  T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( D -cn-> RR+ ) )
60 eqid 2436 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
61 eqid 2436 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
6228, 60, 61cncfcn 18939 . . . . . . . 8  |-  ( ( D  C_  CC  /\  RR+  C_  CC )  ->  ( D -cn-> RR+ )  =  ( (
( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
6332, 52, 62mp2an 654 . . . . . . 7  |-  ( D
-cn->
RR+ )  =  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) )
6459, 63syl6eleq 2526 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
65 ssid 3367 . . . . . . . . . 10  |-  CC  C_  CC
66 cncfss 18929 . . . . . . . . . 10  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
6751, 65, 66mp2an 654 . . . . . . . . 9  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
68 relogcn 20529 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
6967, 68sselii 3345 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
7069a1i 11 . . . . . . 7  |-  (  T. 
->  ( log  |`  RR+ )  e.  ( RR+ -cn-> CC ) )
7128cnfldtop 18818 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
7231toponunii 16997 . . . . . . . . . . . 12  |-  CC  =  U. ( TopOpen ` fld )
7372restid 13661 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
7471, 73ax-mp 8 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
7574eqcomi 2440 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
7628, 61, 75cncfcn 18939 . . . . . . . 8  |-  ( (
RR+  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) ) )
7752, 65, 76mp2an 654 . . . . . . 7  |-  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) )
7870, 77syl6eleq 2526 . . . . . 6  |-  (  T. 
->  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen ` fld ) ) )
7935, 64, 78cnmpt11f 17696 . . . . 5  |-  (  T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( ( ( TopOpen ` fld )t  D
)  Cn  ( TopOpen ` fld )
) )
8028, 60, 75cncfcn 18939 . . . . . 6  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
8132, 65, 80mp2an 654 . . . . 5  |-  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) )
8279, 81syl6eleqr 2527 . . . 4  |-  (  T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( D -cn-> CC ) )
8316imcld 12000 . . . . . . . 8  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  RR )
8483recnd 9114 . . . . . . 7  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  CC )
8584adantl 453 . . . . . 6  |-  ( (  T.  /\  x  e.  D )  ->  (
Im `  ( log `  x ) )  e.  CC )
86 eqidd 2437 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  =  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) ) )
87 eqidd 2437 . . . . . 6  |-  (  T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  =  ( y  e.  CC  |->  ( _i  x.  y ) ) )
88 oveq2 6089 . . . . . 6  |-  ( y  =  ( Im `  ( log `  x ) )  ->  ( _i  x.  y )  =  ( _i  x.  ( Im
`  ( log `  x
) ) ) )
8985, 86, 87, 88fmptco 5901 . . . . 5  |-  (  T. 
->  ( ( y  e.  CC  |->  ( _i  x.  y ) )  o.  ( x  e.  D  |->  ( Im `  ( log `  x ) ) ) )  =  ( x  e.  D  |->  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
90 cncfss 18929 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> RR )  C_  ( D -cn-> CC ) )
9151, 65, 90mp2an 654 . . . . . . . 8  |-  ( D
-cn-> RR )  C_  ( D -cn-> CC )
924logcnlem5 20537 . . . . . . . 8  |-  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) )  e.  ( D -cn-> RR )
9391, 92sselii 3345 . . . . . . 7  |-  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) )  e.  ( D -cn-> CC )
9493a1i 11 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  e.  ( D
-cn-> CC ) )
95 ax-icn 9049 . . . . . . 7  |-  _i  e.  CC
96 eqid 2436 . . . . . . . 8  |-  ( y  e.  CC  |->  ( _i  x.  y ) )  =  ( y  e.  CC  |->  ( _i  x.  y ) )
9796mulc1cncf 18935 . . . . . . 7  |-  ( _i  e.  CC  ->  (
y  e.  CC  |->  ( _i  x.  y ) )  e.  ( CC
-cn-> CC ) )
9895, 97mp1i 12 . . . . . 6  |-  (  T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  e.  ( CC -cn-> CC ) )
9994, 98cncfco 18937 . . . . 5  |-  (  T. 
->  ( ( y  e.  CC  |->  ( _i  x.  y ) )  o.  ( x  e.  D  |->  ( Im `  ( log `  x ) ) ) )  e.  ( D -cn-> CC ) )
10089, 99eqeltrrd 2511 . . . 4  |-  (  T. 
->  ( x  e.  D  |->  ( _i  x.  (
Im `  ( log `  x ) ) ) )  e.  ( D
-cn-> CC ) )
10128, 30, 82, 100cncfmpt2f 18944 . . 3  |-  (  T. 
->  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )  e.  ( D -cn-> CC ) )
102101trud 1332 . 2  |-  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )  e.  ( D
-cn-> CC )
10327, 102eqeltri 2506 1  |-  ( log  |`  D )  e.  ( D -cn-> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    T. wtru 1325    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    \ cdif 3317    C_ wss 3320   {csn 3814    e. cmpt 4266   ran crn 4879    |` cres 4880    o. ccom 4882    Fn wfn 5449   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   _ici 8992    + caddc 8993    x. cmul 8995    -oocmnf 9118   RR+crp 10612   (,]cioc 10917   Recre 11902   Imcim 11903   abscabs 12039   ↾t crest 13648   TopOpenctopn 13649  ℂfldccnfld 16703   Topctop 16958  TopOnctopon 16959    Cn ccn 17288    tX ctx 17592   -cn->ccncf 18906   logclog 20452
This theorem is referenced by:  dvlog  20542  efopnlem2  20548  cxpcn  20629  lgamgulmlem2  24814  lgamcvg2  24839  areacirclem4  26295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-tan 12674  df-pi 12675  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-cmp 17450  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454
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