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Theorem logcn 23457
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 23379 . . . . . . 7  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1of 5831 . . . . . . 7  |-  ( log
: ( CC  \  { 0 } ) -1-1-onto-> ran 
log  ->  log : ( CC 
\  { 0 } ) --> ran  log )
31, 2ax-mp 5 . . . . . 6  |-  log :
( CC  \  {
0 } ) --> ran 
log
4 logcn.d . . . . . . 7  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
54logdmss 23452 . . . . . 6  |-  D  C_  ( CC  \  { 0 } )
6 fssres 5766 . . . . . 6  |-  ( ( log : ( CC 
\  { 0 } ) --> ran  log  /\  D  C_  ( CC  \  {
0 } ) )  ->  ( log  |`  D ) : D --> ran  log )
73, 5, 6mp2an 676 . . . . 5  |-  ( log  |`  D ) : D --> ran  log
8 ffn 5746 . . . . 5  |-  ( ( log  |`  D ) : D --> ran  log  ->  ( log  |`  D )  Fn  D )
97, 8ax-mp 5 . . . 4  |-  ( log  |`  D )  Fn  D
10 dffn5 5926 . . . 4  |-  ( ( log  |`  D )  Fn  D  <->  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `  x
) ) )
119, 10mpbi 211 . . 3  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `
 x ) )
12 fvres 5895 . . . . 5  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( log `  x
) )
134ellogdm 23449 . . . . . . . 8  |-  ( x  e.  D  <->  ( x  e.  CC  /\  ( x  e.  RR  ->  x  e.  RR+ ) ) )
1413simplbi 461 . . . . . . 7  |-  ( x  e.  D  ->  x  e.  CC )
154logdmn0 23450 . . . . . . 7  |-  ( x  e.  D  ->  x  =/=  0 )
1614, 15logcld 23385 . . . . . 6  |-  ( x  e.  D  ->  ( log `  x )  e.  CC )
1716replimd 13239 . . . . 5  |-  ( x  e.  D  ->  ( log `  x )  =  ( ( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
18 relog 23411 . . . . . . . 8  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
1914, 15, 18syl2anc 665 . . . . . . 7  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
2014, 15absrpcld 13488 . . . . . . . 8  |-  ( x  e.  D  ->  ( abs `  x )  e.  RR+ )
21 fvres 5895 . . . . . . . 8  |-  ( ( abs `  x )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( abs `  x ) )  =  ( log `  ( abs `  x
) ) )
2220, 21syl 17 . . . . . . 7  |-  ( x  e.  D  ->  (
( log  |`  RR+ ) `  ( abs `  x
) )  =  ( log `  ( abs `  x ) ) )
2319, 22eqtr4d 2473 . . . . . 6  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )
2423oveq1d 6320 . . . . 5  |-  ( x  e.  D  ->  (
( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
2512, 17, 243eqtrd 2474 . . . 4  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2625mpteq2ia 4508 . . 3  |-  ( x  e.  D  |->  ( ( log  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2711, 26eqtri 2458 . 2  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
28 eqid 2429 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2928addcn 21793 . . . . 5  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3029a1i 11 . . . 4  |-  ( T. 
->  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
3128cnfldtopon 21714 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3214ssriv 3474 . . . . . . . 8  |-  D  C_  CC
33 resttopon 20108 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
3431, 32, 33mp2an 676 . . . . . . 7  |-  ( (
TopOpen ` fld )t  D )  e.  (TopOn `  D )
3534a1i 11 . . . . . 6  |-  ( T. 
->  ( ( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
36 absf 13379 . . . . . . . . . . . 12  |-  abs : CC
--> RR
37 fssres 5766 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  D  C_  CC )  -> 
( abs  |`  D ) : D --> RR )
3836, 32, 37mp2an 676 . . . . . . . . . . 11  |-  ( abs  |`  D ) : D --> RR
3938a1i 11 . . . . . . . . . 10  |-  ( T. 
->  ( abs  |`  D ) : D --> RR )
4039feqmptd 5934 . . . . . . . . 9  |-  ( T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( ( abs  |`  D ) `  x
) ) )
41 fvres 5895 . . . . . . . . . 10  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  =  ( abs `  x
) )
4241mpteq2ia 4508 . . . . . . . . 9  |-  ( x  e.  D  |->  ( ( abs  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( abs `  x
) )
4340, 42syl6eq 2486 . . . . . . . 8  |-  ( T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( abs `  x
) ) )
44 ffn 5746 . . . . . . . . . . 11  |-  ( ( abs  |`  D ) : D --> RR  ->  ( abs  |`  D )  Fn  D )
4538, 44ax-mp 5 . . . . . . . . . 10  |-  ( abs  |`  D )  Fn  D
4641, 20eqeltrd 2517 . . . . . . . . . . 11  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  e.  RR+ )
4746rgen 2792 . . . . . . . . . 10  |-  A. x  e.  D  ( ( abs  |`  D ) `  x )  e.  RR+
48 ffnfv 6064 . . . . . . . . . 10  |-  ( ( abs  |`  D ) : D --> RR+  <->  ( ( abs  |`  D )  Fn  D  /\  A. x  e.  D  ( ( abs  |`  D ) `
 x )  e.  RR+ ) )
4945, 47, 48mpbir2an 928 . . . . . . . . 9  |-  ( abs  |`  D ) : D --> RR+
50 rpssre 11312 . . . . . . . . . . 11  |-  RR+  C_  RR
51 ax-resscn 9595 . . . . . . . . . . 11  |-  RR  C_  CC
5250, 51sstri 3479 . . . . . . . . . 10  |-  RR+  C_  CC
53 abscncf 21829 . . . . . . . . . . 11  |-  abs  e.  ( CC -cn-> RR )
54 rescncf 21825 . . . . . . . . . . 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 21826 . . . . . . . . . 10  |-  ( (
RR+  C_  CC  /\  ( abs  |`  D )  e.  ( D -cn-> RR ) )  ->  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ ) )
5752, 55, 56mp2an 676 . . . . . . . . 9  |-  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ )
5849, 57mpbir 212 . . . . . . . 8  |-  ( abs  |`  D )  e.  ( D -cn-> RR+ )
5943, 58syl6eqelr 2526 . . . . . . 7  |-  ( T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( D -cn-> RR+ ) )
60 eqid 2429 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
61 eqid 2429 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
6228, 60, 61cncfcn 21837 . . . . . . . 8  |-  ( ( D  C_  CC  /\  RR+  C_  CC )  ->  ( D -cn-> RR+ )  =  ( (
( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
6332, 52, 62mp2an 676 . . . . . . 7  |-  ( D
-cn->
RR+ )  =  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) )
6459, 63syl6eleq 2527 . . . . . 6  |-  ( T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
65 ssid 3489 . . . . . . . . . 10  |-  CC  C_  CC
66 cncfss 21827 . . . . . . . . . 10  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
6751, 65, 66mp2an 676 . . . . . . . . 9  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
68 relogcn 23448 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
6967, 68sselii 3467 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
7069a1i 11 . . . . . . 7  |-  ( T. 
->  ( log  |`  RR+ )  e.  ( RR+ -cn-> CC ) )
7128cnfldtop 21715 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
7231toponunii 19878 . . . . . . . . . . . 12  |-  CC  =  U. ( TopOpen ` fld )
7372restid 15291 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
7471, 73ax-mp 5 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
7574eqcomi 2442 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
7628, 61, 75cncfcn 21837 . . . . . . . 8  |-  ( (
RR+  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) ) )
7752, 65, 76mp2an 676 . . . . . . 7  |-  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) )
7870, 77syl6eleq 2527 . . . . . 6  |-  ( T. 
->  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen ` fld ) ) )
7935, 64, 78cnmpt11f 20610 . . . . 5  |-  ( T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( ( ( TopOpen ` fld )t  D
)  Cn  ( TopOpen ` fld )
) )
8028, 60, 75cncfcn 21837 . . . . . 6  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
8132, 65, 80mp2an 676 . . . . 5  |-  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) )
8279, 81syl6eleqr 2528 . . . 4  |-  ( T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( D -cn-> CC ) )
8316imcld 13237 . . . . . . . 8  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  RR )
8483recnd 9668 . . . . . . 7  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  CC )
8584adantl 467 . . . . . 6  |-  ( ( T.  /\  x  e.  D )  ->  (
Im `  ( log `  x ) )  e.  CC )
86 eqidd 2430 . . . . . 6  |-  ( T. 
->  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  =  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) ) )
87 eqidd 2430 . . . . . 6  |-  ( T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  =  ( y  e.  CC  |->  ( _i  x.  y ) ) )
88 oveq2 6313 . . . . . 6  |-  ( y  =  ( Im `  ( log `  x ) )  ->  ( _i  x.  y )  =  ( _i  x.  ( Im
`  ( log `  x
) ) ) )
8985, 86, 87, 88fmptco 6071 . . . . 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 21827 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> RR )  C_  ( D -cn-> CC ) )
9151, 65, 90mp2an 676 . . . . . . . 8  |-  ( D
-cn-> RR )  C_  ( D -cn-> CC )
924logcnlem5 23456 . . . . . . . 8  |-  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) )  e.  ( D -cn-> RR )
9391, 92sselii 3467 . . . . . . 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 9597 . . . . . . 7  |-  _i  e.  CC
96 eqid 2429 . . . . . . . 8  |-  ( y  e.  CC  |->  ( _i  x.  y ) )  =  ( y  e.  CC  |->  ( _i  x.  y ) )
9796mulc1cncf 21833 . . . . . . 7  |-  ( _i  e.  CC  ->  (
y  e.  CC  |->  ( _i  x.  y ) )  e.  ( CC
-cn-> CC ) )
9895, 97mp1i 13 . . . . . 6  |-  ( T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  e.  ( CC -cn-> CC ) )
9994, 98cncfco 21835 . . . . 5  |-  ( T. 
->  ( ( y  e.  CC  |->  ( _i  x.  y ) )  o.  ( x  e.  D  |->  ( Im `  ( log `  x ) ) ) )  e.  ( D -cn-> CC ) )
10089, 99eqeltrrd 2518 . . . 4  |-  ( T. 
->  ( x  e.  D  |->  ( _i  x.  (
Im `  ( log `  x ) ) ) )  e.  ( D
-cn-> CC ) )
10128, 30, 82, 100cncfmpt2f 21842 . . 3  |-  ( T. 
->  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )  e.  ( D -cn-> CC ) )
102101trud 1446 . 2  |-  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )  e.  ( D
-cn-> CC )
10327, 102eqeltri 2513 1  |-  ( log  |`  D )  e.  ( D -cn-> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   T. wtru 1438    e. wcel 1870    =/= wne 2625   A.wral 2782    \ cdif 3439    C_ wss 3442   {csn 4002    |-> cmpt 4484   ran crn 4855    |` cres 4856    o. ccom 4858    Fn wfn 5596   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   _ici 9540    + caddc 9541    x. cmul 9543   -oocmnf 9672   RR+crp 11302   (,]cioc 11636   Recre 13139   Imcim 13140   abscabs 13276   ↾t crest 15278   TopOpenctopn 15279  ℂfldccnfld 18905   Topctop 19848  TopOnctopon 19849    Cn ccn 20171    tX ctx 20506   -cn->ccncf 21804   logclog 23369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-tan 14103  df-pi 14104  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-cmp 20333  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699  df-log 23371
This theorem is referenced by:  dvlog  23461  efopnlem2  23467  dvcncxp1  23548  cxpcn  23550  lgamgulmlem2  23820  lgamcvg2  23845  areacirclem4  31739
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