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Theorem logcn 22220
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 22144 . . . . . . 7  |-  log :
( CC  \  {
0 } ) -1-1-onto-> ran  log
2 f1of 5744 . . . . . . 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 22215 . . . . . 6  |-  D  C_  ( CC  \  { 0 } )
6 fssres 5681 . . . . . 6  |-  ( ( log : ( CC 
\  { 0 } ) --> ran  log  /\  D  C_  ( CC  \  {
0 } ) )  ->  ( log  |`  D ) : D --> ran  log )
73, 5, 6mp2an 672 . . . . 5  |-  ( log  |`  D ) : D --> ran  log
8 ffn 5662 . . . . 5  |-  ( ( log  |`  D ) : D --> ran  log  ->  ( log  |`  D )  Fn  D )
97, 8ax-mp 5 . . . 4  |-  ( log  |`  D )  Fn  D
10 dffn5 5841 . . . 4  |-  ( ( log  |`  D )  Fn  D  <->  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `  x
) ) )
119, 10mpbi 208 . . 3  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `
 x ) )
12 fvres 5808 . . . . 5  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( log `  x
) )
134ellogdm 22212 . . . . . . . 8  |-  ( x  e.  D  <->  ( x  e.  CC  /\  ( x  e.  RR  ->  x  e.  RR+ ) ) )
1413simplbi 460 . . . . . . 7  |-  ( x  e.  D  ->  x  e.  CC )
154logdmn0 22213 . . . . . . 7  |-  ( x  e.  D  ->  x  =/=  0 )
1614, 15logcld 22150 . . . . . 6  |-  ( x  e.  D  ->  ( log `  x )  e.  CC )
1716replimd 12799 . . . . 5  |-  ( x  e.  D  ->  ( log `  x )  =  ( ( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
18 relog 22173 . . . . . . . 8  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
1914, 15, 18syl2anc 661 . . . . . . 7  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( log `  ( abs `  x ) ) )
2014, 15absrpcld 13047 . . . . . . . 8  |-  ( x  e.  D  ->  ( abs `  x )  e.  RR+ )
21 fvres 5808 . . . . . . . 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 2496 . . . . . 6  |-  ( x  e.  D  ->  (
Re `  ( log `  x ) )  =  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )
2423oveq1d 6210 . . . . 5  |-  ( x  e.  D  ->  (
( Re `  ( log `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
2512, 17, 243eqtrd 2497 . . . 4  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2625mpteq2ia 4477 . . 3  |-  ( x  e.  D  |->  ( ( log  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )
2711, 26eqtri 2481 . 2  |-  ( log  |`  D )  =  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )
28 eqid 2452 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2928addcn 20568 . . . . 5  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3029a1i 11 . . . 4  |-  ( T. 
->  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
3128cnfldtopon 20489 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3214ssriv 3463 . . . . . . . 8  |-  D  C_  CC
33 resttopon 18892 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
3431, 32, 33mp2an 672 . . . . . . 7  |-  ( (
TopOpen ` fld )t  D )  e.  (TopOn `  D )
3534a1i 11 . . . . . 6  |-  ( T. 
->  ( ( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
36 absf 12938 . . . . . . . . . . . 12  |-  abs : CC
--> RR
37 fssres 5681 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  D  C_  CC )  -> 
( abs  |`  D ) : D --> RR )
3836, 32, 37mp2an 672 . . . . . . . . . . 11  |-  ( abs  |`  D ) : D --> RR
3938a1i 11 . . . . . . . . . 10  |-  ( T. 
->  ( abs  |`  D ) : D --> RR )
4039feqmptd 5848 . . . . . . . . 9  |-  ( T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( ( abs  |`  D ) `  x
) ) )
41 fvres 5808 . . . . . . . . . 10  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  =  ( abs `  x
) )
4241mpteq2ia 4477 . . . . . . . . 9  |-  ( x  e.  D  |->  ( ( abs  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( abs `  x
) )
4340, 42syl6eq 2509 . . . . . . . 8  |-  ( T. 
->  ( abs  |`  D )  =  ( x  e.  D  |->  ( abs `  x
) ) )
44 ffn 5662 . . . . . . . . . . 11  |-  ( ( abs  |`  D ) : D --> RR  ->  ( abs  |`  D )  Fn  D )
4538, 44ax-mp 5 . . . . . . . . . 10  |-  ( abs  |`  D )  Fn  D
4641, 20eqeltrd 2540 . . . . . . . . . . 11  |-  ( x  e.  D  ->  (
( abs  |`  D ) `
 x )  e.  RR+ )
4746rgen 2893 . . . . . . . . . 10  |-  A. x  e.  D  ( ( abs  |`  D ) `  x )  e.  RR+
48 ffnfv 5973 . . . . . . . . . 10  |-  ( ( abs  |`  D ) : D --> RR+  <->  ( ( abs  |`  D )  Fn  D  /\  A. x  e.  D  ( ( abs  |`  D ) `
 x )  e.  RR+ ) )
4945, 47, 48mpbir2an 911 . . . . . . . . 9  |-  ( abs  |`  D ) : D --> RR+
50 rpssre 11107 . . . . . . . . . . 11  |-  RR+  C_  RR
51 ax-resscn 9445 . . . . . . . . . . 11  |-  RR  C_  CC
5250, 51sstri 3468 . . . . . . . . . 10  |-  RR+  C_  CC
53 abscncf 20604 . . . . . . . . . . 11  |-  abs  e.  ( CC -cn-> RR )
54 rescncf 20600 . . . . . . . . . . 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 20601 . . . . . . . . . 10  |-  ( (
RR+  C_  CC  /\  ( abs  |`  D )  e.  ( D -cn-> RR ) )  ->  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ ) )
5752, 55, 56mp2an 672 . . . . . . . . 9  |-  ( ( abs  |`  D )  e.  ( D -cn-> RR+ )  <->  ( abs  |`  D ) : D --> RR+ )
5849, 57mpbir 209 . . . . . . . 8  |-  ( abs  |`  D )  e.  ( D -cn-> RR+ )
5943, 58syl6eqelr 2549 . . . . . . 7  |-  ( T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( D -cn-> RR+ ) )
60 eqid 2452 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
61 eqid 2452 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
6228, 60, 61cncfcn 20612 . . . . . . . 8  |-  ( ( D  C_  CC  /\  RR+  C_  CC )  ->  ( D -cn-> RR+ )  =  ( (
( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
6332, 52, 62mp2an 672 . . . . . . 7  |-  ( D
-cn->
RR+ )  =  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) )
6459, 63syl6eleq 2550 . . . . . 6  |-  ( T. 
->  ( x  e.  D  |->  ( abs `  x
) )  e.  ( ( ( TopOpen ` fld )t  D )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
65 ssid 3478 . . . . . . . . . 10  |-  CC  C_  CC
66 cncfss 20602 . . . . . . . . . 10  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
6751, 65, 66mp2an 672 . . . . . . . . 9  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
68 relogcn 22211 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
6967, 68sselii 3456 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
7069a1i 11 . . . . . . 7  |-  ( T. 
->  ( log  |`  RR+ )  e.  ( RR+ -cn-> CC ) )
7128cnfldtop 20490 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
7231toponunii 18664 . . . . . . . . . . . 12  |-  CC  =  U. ( TopOpen ` fld )
7372restid 14486 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
7471, 73ax-mp 5 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
7574eqcomi 2465 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
7628, 61, 75cncfcn 20612 . . . . . . . 8  |-  ( (
RR+  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) ) )
7752, 65, 76mp2an 672 . . . . . . 7  |-  ( RR+ -cn-> CC )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen
` fld
) )
7870, 77syl6eleq 2550 . . . . . 6  |-  ( T. 
->  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( TopOpen ` fld ) ) )
7935, 64, 78cnmpt11f 19364 . . . . 5  |-  ( T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( ( ( TopOpen ` fld )t  D
)  Cn  ( TopOpen ` fld )
) )
8028, 60, 75cncfcn 20612 . . . . . 6  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
8132, 65, 80mp2an 672 . . . . 5  |-  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) )
8279, 81syl6eleqr 2551 . . . 4  |-  ( T. 
->  ( x  e.  D  |->  ( ( log  |`  RR+ ) `  ( abs `  x
) ) )  e.  ( D -cn-> CC ) )
8316imcld 12797 . . . . . . . 8  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  RR )
8483recnd 9518 . . . . . . 7  |-  ( x  e.  D  ->  (
Im `  ( log `  x ) )  e.  CC )
8584adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  D )  ->  (
Im `  ( log `  x ) )  e.  CC )
86 eqidd 2453 . . . . . 6  |-  ( T. 
->  ( x  e.  D  |->  ( Im `  ( log `  x ) ) )  =  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) ) )
87 eqidd 2453 . . . . . 6  |-  ( T. 
->  ( y  e.  CC  |->  ( _i  x.  y
) )  =  ( y  e.  CC  |->  ( _i  x.  y ) ) )
88 oveq2 6203 . . . . . 6  |-  ( y  =  ( Im `  ( log `  x ) )  ->  ( _i  x.  y )  =  ( _i  x.  ( Im
`  ( log `  x
) ) ) )
8985, 86, 87, 88fmptco 5980 . . . . 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 20602 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> RR )  C_  ( D -cn-> CC ) )
9151, 65, 90mp2an 672 . . . . . . . 8  |-  ( D
-cn-> RR )  C_  ( D -cn-> CC )
924logcnlem5 22219 . . . . . . . 8  |-  ( x  e.  D  |->  ( Im
`  ( log `  x
) ) )  e.  ( D -cn-> RR )
9391, 92sselii 3456 . . . . . . 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 9447 . . . . . . 7  |-  _i  e.  CC
96 eqid 2452 . . . . . . . 8  |-  ( y  e.  CC  |->  ( _i  x.  y ) )  =  ( y  e.  CC  |->  ( _i  x.  y ) )
9796mulc1cncf 20608 . . . . . . 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 20610 . . . . 5  |-  ( T. 
->  ( ( y  e.  CC  |->  ( _i  x.  y ) )  o.  ( x  e.  D  |->  ( Im `  ( log `  x ) ) ) )  e.  ( D -cn-> CC ) )
10089, 99eqeltrrd 2541 . . . 4  |-  ( T. 
->  ( x  e.  D  |->  ( _i  x.  (
Im `  ( log `  x ) ) ) )  e.  ( D
-cn-> CC ) )
10128, 30, 82, 100cncfmpt2f 20617 . . 3  |-  ( T. 
->  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x ) )  +  ( _i  x.  ( Im `  ( log `  x ) ) ) ) )  e.  ( D -cn-> CC ) )
102101trud 1379 . 2  |-  ( x  e.  D  |->  ( ( ( log  |`  RR+ ) `  ( abs `  x
) )  +  ( _i  x.  ( Im
`  ( log `  x
) ) ) ) )  e.  ( D
-cn-> CC )
10327, 102eqeltri 2536 1  |-  ( log  |`  D )  e.  ( D -cn-> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   T. wtru 1371    e. wcel 1758    =/= wne 2645   A.wral 2796    \ cdif 3428    C_ wss 3431   {csn 3980    |-> cmpt 4453   ran crn 4944    |` cres 4945    o. ccom 4947    Fn wfn 5516   -->wf 5517   -1-1-onto->wf1o 5520   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   _ici 9390    + caddc 9391    x. cmul 9393   -oocmnf 9522   RR+crp 11097   (,]cioc 11407   Recre 12699   Imcim 12700   abscabs 12836   ↾t crest 14473   TopOpenctopn 14474  ℂfldccnfld 17938   Topctop 18625  TopOnctopon 18626    Cn ccn 18955    tX ctx 19260   -cn->ccncf 20579   logclog 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-fi 7767  df-sup 7797  df-oi 7830  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-ioc 11411  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-fac 12164  df-bc 12191  df-hash 12216  df-shft 12669  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-limsup 13062  df-clim 13079  df-rlim 13080  df-sum 13277  df-ef 13466  df-sin 13468  df-cos 13469  df-tan 13470  df-pi 13471  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-hom 14376  df-cco 14377  df-rest 14475  df-topn 14476  df-0g 14494  df-gsum 14495  df-topgen 14496  df-pt 14497  df-prds 14500  df-xrs 14554  df-qtop 14559  df-imas 14560  df-xps 14562  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-submnd 15579  df-mulg 15662  df-cntz 15949  df-cmn 16395  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-fbas 17934  df-fg 17935  df-cnfld 17939  df-top 18630  df-bases 18632  df-topon 18633  df-topsp 18634  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-lp 18867  df-perf 18868  df-cn 18958  df-cnp 18959  df-haus 19046  df-cmp 19117  df-tx 19262  df-hmeo 19455  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-xms 20022  df-ms 20023  df-tms 20024  df-cncf 20581  df-limc 21469  df-dv 21470  df-log 22136
This theorem is referenced by:  dvlog  22224  efopnlem2  22230  cxpcn  22311  lgamgulmlem2  27155  lgamcvg2  27180  dvcncxp1  28620  areacirclem4  28630
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