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Theorem logcn 19994
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 19922 . . . . . . 7  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1of 5472 . . . . . . 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 19989 . . . . . 6  |-  D  C_  ( CC  \  { 0 } )
6 fssres 5408 . . . . . 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 5389 . . . . 5  |-  ( ( log  |`  D ) : D --> ran  log  ->  ( log  |`  D )  Fn  D )
97, 8ax-mp 8 . . . 4  |-  ( log  |`  D )  Fn  D
10 dffn5 5568 . . . 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 5542 . . . . 5  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( log `  x
) )
134ellogdm 19986 . . . . . . . 8  |-  ( x  e.  D  <->  ( x  e.  CC  /\  ( x  e.  RR  ->  x  e.  RR+ ) ) )
1413simplbi 446 . . . . . . 7  |-  ( x  e.  D  ->  x  e.  CC )
154logdmn0 19987 . . . . . . 7  |-  ( x  e.  D  ->  x  =/=  0 )
16 logcl 19926 . . . . . . 7  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( log `  x
)  e.  CC )
1714, 15, 16syl2anc 642 . . . . . 6  |-  ( x  e.  D  ->  ( log `  x )  e.  CC )
1817replimd 11682 . . . . 5  |-  ( x  e.  D  ->  ( log `  x )  =  ( ( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
19 relog 19950 . . . . . . . 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 11930 . . . . . . . 8  |-  ( x  e.  D  ->  ( abs `  x )  e.  RR+ )
22 fvres 5542 . . . . . . . 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 2318 . . . . . 6  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )
2524oveq1d 5873 . . . . 5  |-  ( x  e.  D  ->  (
( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
2612, 18, 253eqtrd 2319 . . . 4  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2726mpteq2ia 4102 . . 3  |-  ( x  e.  D  |->  ( ( log  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2811, 27eqtri 2303 . 2  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
29 eqid 2283 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3029addcn 18369 . . . . 5  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3130a1i 10 . . . 4  |-  (  T. 
->  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
3229cnfldtopon 18292 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3314ssriv 3184 . . . . . . . 8  |-  D  C_  CC
34 resttopon 16892 . . . . . . . 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 11821 . . . . . . . . . . . 12  |-  abs : CC
--> RR
38 fssres 5408 . . . . . . . . . . . 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 5575 . . . . . . . . 9  |-  (  T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( ( abs  |`  D ) `  x
) ) )
42 fvres 5542 . . . . . . . . . 10  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  =  ( abs `  x
) )
4342mpteq2ia 4102 . . . . . . . . 9  |-  ( x  e.  D  |->  ( ( abs  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( abs `  x
) )
4441, 43syl6eq 2331 . . . . . . . 8  |-  (  T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( abs `  x
) ) )
45 ffn 5389 . . . . . . . . . . 11  |-  ( ( abs  |`  D ) : D --> RR  ->  ( abs  |`  D )  Fn  D )
4639, 45ax-mp 8 . . . . . . . . . 10  |-  ( abs  |`  D )  Fn  D
4742, 21eqeltrd 2357 . . . . . . . . . . 11  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  e.  RR+ )
4847rgen 2608 . . . . . . . . . 10  |-  A. x  e.  D  ( ( abs  |`  D ) `  x )  e.  RR+
49 ffnfv 5685 . . . . . . . . . 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 10364 . . . . . . . . . . 11  |-  RR+  C_  RR
52 ax-resscn 8794 . . . . . . . . . . 11  |-  RR  C_  CC
5351, 52sstri 3188 . . . . . . . . . 10  |-  RR+  C_  CC
54 abscncf 18405 . . . . . . . . . . 11  |-  abs  e.  ( CC -cn-> RR )
55 rescncf 18401 . . . . . . . . . . 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 18402 . . . . . . . . . 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 2372 . . . . . . 7  |-  (  T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( D -cn-> RR+ ) )
61 eqid 2283 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
62 eqid 2283 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
6329, 61, 62cncfcn 18413 . . . . . . . 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 2373 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
66 ssid 3197 . . . . . . . . . 10  |-  CC  C_  CC
67 cncfss 18403 . . . . . . . . . 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 19985 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
7068, 69sselii 3177 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
7170a1i 10 . . . . . . 7  |-  (  T. 
->  ( log  |`  RR+ )  e.  ( RR+ -cn-> CC ) )
7229cnfldtop 18293 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
7332toponunii 16670 . . . . . . . . . . . 12  |-  CC  =  U. ( TopOpen ` fld )
7473restid 13338 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
7572, 74ax-mp 8 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
7675eqcomi 2287 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
7729, 62, 76cncfcn 18413 . . . . . . . 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 2373 . . . . . 6  |-  (  T. 
->  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen ` fld ) ) )
8036, 65, 79cnmpt11f 17358 . . . . 5  |-  (  T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( ( ( TopOpen ` fld )t  D
)  Cn  ( TopOpen ` fld )
) )
8129, 61, 76cncfcn 18413 . . . . . 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 2374 . . . 4  |-  (  T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( D -cn-> CC ) )
8417imcld 11680 . . . . . . . 8  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  RR )
8584recnd 8861 . . . . . . 7  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  CC )
8685adantl 452 . . . . . 6  |-  ( (  T.  /\  x  e.  D )  ->  (
Im `  ( log `  x ) )  e.  CC )
87 eqidd 2284 . . . . . 6  |-  (  T. 
->  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  =  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) ) )
88 eqidd 2284 . . . . . 6  |-  (  T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  =  ( y  e.  CC  |->  ( _i  x.  y ) ) )
89 oveq2 5866 . . . . . 6  |-  ( y  =  ( Im `  ( log `  x ) )  ->  ( _i  x.  y )  =  ( _i  x.  ( Im
`  ( log `  x
) ) ) )
9086, 87, 88, 89fmptco 5691 . . . . 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 18403 . . . . . . . . 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 19993 . . . . . . . 8  |-  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) )  e.  ( D -cn-> RR )
9492, 93sselii 3177 . . . . . . 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 8796 . . . . . . 7  |-  _i  e.  CC
97 eqid 2283 . . . . . . . 8  |-  ( y  e.  CC  |->  ( _i  x.  y ) )  =  ( y  e.  CC  |->  ( _i  x.  y ) )
9897mulc1cncf 18409 . . . . . . 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 18411 . . . . 5  |-  (  T. 
->  ( ( y  e.  CC  |->  ( _i  x.  y ) )  o.  ( x  e.  D  |->  ( Im `  ( log `  x ) ) ) )  e.  ( D -cn-> CC ) )
10190, 100eqeltrrd 2358 . . . 4  |-  (  T. 
->  ( x  e.  D  |->  ( _i  x.  (
Im `  ( log `  x ) ) ) )  e.  ( D
-cn-> CC ) )
10229, 31, 83, 101cncfmpt2f 18418 . . 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 2353 1  |-  ( log  |`  D )  e.  ( D -cn-> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    T. wtru 1307    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077   ran crn 4690    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   _ici 8739    + caddc 8740    x. cmul 8742    -oocmnf 8865   RR+crp 10354   (,]cioc 10657   Recre 11582   Imcim 11583   abscabs 11719   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255   -cn->ccncf 18380   logclog 19912
This theorem is referenced by:  dvlog  19998  efopnlem2  20004  cxpcn  20085  areacirclem5  24929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-tan 12353  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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