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Theorem logcn 20010
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 19938 . . . . . . 7  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1of 5488 . . . . . . 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 20005 . . . . . 6  |-  D  C_  ( CC  \  { 0 } )
6 fssres 5424 . . . . . 6  |-  ( ( log : ( CC 
\  { 0 } ) --> ran  log  /\  D  C_  ( CC  \  {
0 } ) )  ->  ( log  |`  D ) : D --> ran  log )
73, 5, 6mp2an 653 . . . . 5  |-  ( log  |`  D ) : D --> ran  log
8 ffn 5405 . . . . 5  |-  ( ( log  |`  D ) : D --> ran  log  ->  ( log  |`  D )  Fn  D )
97, 8ax-mp 8 . . . 4  |-  ( log  |`  D )  Fn  D
10 dffn5 5584 . . . 4  |-  ( ( log  |`  D )  Fn  D  <->  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `  x
) ) )
119, 10mpbi 199 . . 3  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `
 x ) )
12 fvres 5558 . . . . 5  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( log `  x
) )
134ellogdm 20002 . . . . . . . 8  |-  ( x  e.  D  <->  ( x  e.  CC  /\  ( x  e.  RR  ->  x  e.  RR+ ) ) )
1413simplbi 446 . . . . . . 7  |-  ( x  e.  D  ->  x  e.  CC )
154logdmn0 20003 . . . . . . 7  |-  ( x  e.  D  ->  x  =/=  0 )
16 logcl 19942 . . . . . . 7  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( log `  x
)  e.  CC )
1714, 15, 16syl2anc 642 . . . . . 6  |-  ( x  e.  D  ->  ( log `  x )  e.  CC )
1817replimd 11698 . . . . 5  |-  ( x  e.  D  ->  ( log `  x )  =  ( ( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
19 relog 19966 . . . . . . . 8  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
2014, 15, 19syl2anc 642 . . . . . . 7  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
2114, 15absrpcld 11946 . . . . . . . 8  |-  ( x  e.  D  ->  ( abs `  x )  e.  RR+ )
22 fvres 5558 . . . . . . . 8  |-  ( ( abs `  x )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( abs `  x ) )  =  ( log `  ( abs `  x
) ) )
2321, 22syl 15 . . . . . . 7  |-  ( x  e.  D  ->  (
( log  |`  RR+ ) `  ( abs `  x
) )  =  ( log `  ( abs `  x ) ) )
2420, 23eqtr4d 2331 . . . . . 6  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )
2524oveq1d 5889 . . . . 5  |-  ( x  e.  D  ->  (
( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
2612, 18, 253eqtrd 2332 . . . 4  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2726mpteq2ia 4118 . . 3  |-  ( x  e.  D  |->  ( ( log  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2811, 27eqtri 2316 . 2  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
29 eqid 2296 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3029addcn 18385 . . . . 5  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3130a1i 10 . . . 4  |-  (  T. 
->  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
3229cnfldtopon 18308 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3314ssriv 3197 . . . . . . . 8  |-  D  C_  CC
34 resttopon 16908 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
3532, 33, 34mp2an 653 . . . . . . 7  |-  ( (
TopOpen ` fld )t  D )  e.  (TopOn `  D )
3635a1i 10 . . . . . 6  |-  (  T. 
->  ( ( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
37 absf 11837 . . . . . . . . . . . 12  |-  abs : CC
--> RR
38 fssres 5424 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  D  C_  CC )  -> 
( abs  |`  D ) : D --> RR )
3937, 33, 38mp2an 653 . . . . . . . . . . 11  |-  ( abs  |`  D ) : D --> RR
4039a1i 10 . . . . . . . . . 10  |-  (  T. 
->  ( abs  |`  D ) : D --> RR )
4140feqmptd 5591 . . . . . . . . 9  |-  (  T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( ( abs  |`  D ) `  x
) ) )
42 fvres 5558 . . . . . . . . . 10  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  =  ( abs `  x
) )
4342mpteq2ia 4118 . . . . . . . . 9  |-  ( x  e.  D  |->  ( ( abs  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( abs `  x
) )
4441, 43syl6eq 2344 . . . . . . . 8  |-  (  T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( abs `  x
) ) )
45 ffn 5405 . . . . . . . . . . 11  |-  ( ( abs  |`  D ) : D --> RR  ->  ( abs  |`  D )  Fn  D )
4639, 45ax-mp 8 . . . . . . . . . 10  |-  ( abs  |`  D )  Fn  D
4742, 21eqeltrd 2370 . . . . . . . . . . 11  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  e.  RR+ )
4847rgen 2621 . . . . . . . . . 10  |-  A. x  e.  D  ( ( abs  |`  D ) `  x )  e.  RR+
49 ffnfv 5701 . . . . . . . . . 10  |-  ( ( abs  |`  D ) : D --> RR+  <->  ( ( abs  |`  D )  Fn  D  /\  A. x  e.  D  ( ( abs  |`  D ) `
 x )  e.  RR+ ) )
5046, 48, 49mpbir2an 886 . . . . . . . . 9  |-  ( abs  |`  D ) : D --> RR+
51 rpssre 10380 . . . . . . . . . . 11  |-  RR+  C_  RR
52 ax-resscn 8810 . . . . . . . . . . 11  |-  RR  C_  CC
5351, 52sstri 3201 . . . . . . . . . 10  |-  RR+  C_  CC
54 abscncf 18421 . . . . . . . . . . 11  |-  abs  e.  ( CC -cn-> RR )
55 rescncf 18417 . . . . . . . . . . 11  |-  ( D 
C_  CC  ->  ( abs 
e.  ( CC -cn-> RR )  ->  ( abs  |`  D )  e.  ( D -cn-> RR ) ) )
5633, 54, 55mp2 17 . . . . . . . . . 10  |-  ( abs  |`  D )  e.  ( D -cn-> RR )
57 cncffvrn 18418 . . . . . . . . . 10  |-  ( (
RR+  C_  CC  /\  ( abs  |`  D )  e.  ( D -cn-> RR ) )  ->  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ ) )
5853, 56, 57mp2an 653 . . . . . . . . 9  |-  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ )
5950, 58mpbir 200 . . . . . . . 8  |-  ( abs  |`  D )  e.  ( D -cn-> RR+ )
6044, 59syl6eqelr 2385 . . . . . . 7  |-  (  T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( D -cn-> RR+ ) )
61 eqid 2296 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
62 eqid 2296 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
6329, 61, 62cncfcn 18429 . . . . . . . 8  |-  ( ( D  C_  CC  /\  RR+  C_  CC )  ->  ( D -cn-> RR+ )  =  ( (
( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
6433, 53, 63mp2an 653 . . . . . . 7  |-  ( D
-cn->
RR+ )  =  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) )
6560, 64syl6eleq 2386 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
66 ssid 3210 . . . . . . . . . 10  |-  CC  C_  CC
67 cncfss 18419 . . . . . . . . . 10  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
6852, 66, 67mp2an 653 . . . . . . . . 9  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
69 relogcn 20001 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
7068, 69sselii 3190 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
7170a1i 10 . . . . . . 7  |-  (  T. 
->  ( log  |`  RR+ )  e.  ( RR+ -cn-> CC ) )
7229cnfldtop 18309 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
7332toponunii 16686 . . . . . . . . . . . 12  |-  CC  =  U. ( TopOpen ` fld )
7473restid 13354 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
7572, 74ax-mp 8 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
7675eqcomi 2300 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
7729, 62, 76cncfcn 18429 . . . . . . . 8  |-  ( (
RR+  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) ) )
7853, 66, 77mp2an 653 . . . . . . 7  |-  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) )
7971, 78syl6eleq 2386 . . . . . 6  |-  (  T. 
->  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen ` fld ) ) )
8036, 65, 79cnmpt11f 17374 . . . . 5  |-  (  T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( ( ( TopOpen ` fld )t  D
)  Cn  ( TopOpen ` fld )
) )
8129, 61, 76cncfcn 18429 . . . . . 6  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
8233, 66, 81mp2an 653 . . . . 5  |-  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) )
8380, 82syl6eleqr 2387 . . . 4  |-  (  T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( D -cn-> CC ) )
8417imcld 11696 . . . . . . . 8  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  RR )
8584recnd 8877 . . . . . . 7  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  CC )
8685adantl 452 . . . . . 6  |-  ( (  T.  /\  x  e.  D )  ->  (
Im `  ( log `  x ) )  e.  CC )
87 eqidd 2297 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  =  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) ) )
88 eqidd 2297 . . . . . 6  |-  (  T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  =  ( y  e.  CC  |->  ( _i  x.  y ) ) )
89 oveq2 5882 . . . . . 6  |-  ( y  =  ( Im `  ( log `  x ) )  ->  ( _i  x.  y )  =  ( _i  x.  ( Im
`  ( log `  x
) ) ) )
9086, 87, 88, 89fmptco 5707 . . . . 5  |-  (  T. 
->  ( ( y  e.  CC  |->  ( _i  x.  y ) )  o.  ( x  e.  D  |->  ( Im `  ( log `  x ) ) ) )  =  ( x  e.  D  |->  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
91 cncfss 18419 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> RR )  C_  ( D -cn-> CC ) )
9252, 66, 91mp2an 653 . . . . . . . 8  |-  ( D
-cn-> RR )  C_  ( D -cn-> CC )
934logcnlem5 20009 . . . . . . . 8  |-  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) )  e.  ( D -cn-> RR )
9492, 93sselii 3190 . . . . . . 7  |-  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) )  e.  ( D -cn-> CC )
9594a1i 10 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  e.  ( D
-cn-> CC ) )
96 ax-icn 8812 . . . . . . 7  |-  _i  e.  CC
97 eqid 2296 . . . . . . . 8  |-  ( y  e.  CC  |->  ( _i  x.  y ) )  =  ( y  e.  CC  |->  ( _i  x.  y ) )
9897mulc1cncf 18425 . . . . . . 7  |-  ( _i  e.  CC  ->  (
y  e.  CC  |->  ( _i  x.  y ) )  e.  ( CC
-cn-> CC ) )
9996, 98mp1i 11 . . . . . 6  |-  (  T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  e.  ( CC -cn-> CC ) )
10095, 99cncfco 18427 . . . . 5  |-  (  T. 
->  ( ( y  e.  CC  |->  ( _i  x.  y ) )  o.  ( x  e.  D  |->  ( Im `  ( log `  x ) ) ) )  e.  ( D -cn-> CC ) )
10190, 100eqeltrrd 2371 . . . 4  |-  (  T. 
->  ( x  e.  D  |->  ( _i  x.  (
Im `  ( log `  x ) ) ) )  e.  ( D
-cn-> CC ) )
10229, 31, 83, 101cncfmpt2f 18434 . . 3  |-  (  T. 
->  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )  e.  ( D -cn-> CC ) )
103102trud 1314 . 2  |-  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )  e.  ( D
-cn-> CC )
10428, 103eqeltri 2366 1  |-  ( log  |`  D )  e.  ( D -cn-> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    T. wtru 1307    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    C_ wss 3165   {csn 3653    e. cmpt 4093   ran crn 4706    |` cres 4707    o. ccom 4709    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   _ici 8755    + caddc 8756    x. cmul 8758    -oocmnf 8881   RR+crp 10370   (,]cioc 10673   Recre 11598   Imcim 11599   abscabs 11735   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393   Topctop 16647  TopOnctopon 16648    Cn ccn 16970    tX ctx 17271   -cn->ccncf 18396   logclog 19928
This theorem is referenced by:  dvlog  20014  efopnlem2  20020  cxpcn  20101  areacirclem5  25032
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-inf2 7358  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  ax-addf 8832  ax-mulf 8833
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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  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-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-tan 12369  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  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-cmp 17130  df-tx 17273  df-hmeo 17462  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-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930
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