Theorem logcn 24393

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.

Step	Hyp	Ref	Expression
1		logf1o 24311	. . . . . . 7 |- log : ( CC \ { 0 } ) -1-1-onto-> ran log
2		f1of 6137	. . . . . . 7 |- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) --> ran log )
3	1, 2	ax-mp 5	. . . . . 6 |- log : ( CC \ { 0 } ) --> ran log
4		logcn.d	. . . . . . 7 |- D = ( CC \ ( -oo (,] 0 ) )
5	4	logdmss 24388	. . . . . 6 |- D C_ ( CC \ { 0 } )
6		fssres 6070	. . . . . 6 |- ( ( log : ( CC \ { 0 } ) --> ran log /\ D C_ ( CC \ { 0 } ) ) -> ( log |` D ) : D --> ran log )
7	3, 5, 6	mp2an 708	. . . . 5 |- ( log |` D ) : D --> ran log
8		ffn 6045	. . . . 5 |- ( ( log |` D ) : D --> ran log -> ( log |` D ) Fn D )
9	7, 8	ax-mp 5	. . . 4 |- ( log |` D ) Fn D
10		dffn5 6241	. . . 4 |- ( ( log |` D ) Fn D <-> ( log |` D ) = ( x e. D |-> ( ( log |` D ) ` x ) ) )
11	9, 10	mpbi 220	. . 3 |- ( log |` D ) = ( x e. D |-> ( ( log |` D ) ` x ) )
12		fvres 6207	. . . . 5 |- ( x e. D -> ( ( log |` D ) ` x ) = ( log ` x ) )
13	4	ellogdm 24385	. . . . . . . 8 |- ( x e. D <-> ( x e. CC /\ ( x e. RR -> x e. RR+ ) ) )
14	13	simplbi 476	. . . . . . 7 |- ( x e. D -> x e. CC )
15	4	logdmn0 24386	. . . . . . 7 |- ( x e. D -> x =/= 0 )
16	14, 15	logcld 24317	. . . . . 6 |- ( x e. D -> ( log ` x ) e. CC )
17	16	replimd 13937	. . . . 5 |- ( x e. D -> ( log ` x ) = ( ( Re ` ( log ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) )
18		relog 24343	. . . . . . . 8 |- ( ( x e. CC /\ x =/= 0 ) -> ( Re ` ( log ` x ) ) = ( log ` ( abs ` x ) ) )
19	14, 15, 18	syl2anc 693	. . . . . . 7 |- ( x e. D -> ( Re ` ( log ` x ) ) = ( log ` ( abs ` x ) ) )
20	14, 15	absrpcld 14187	. . . . . . . 8 |- ( x e. D -> ( abs ` x ) e. RR+ )
21		fvres 6207	. . . . . . . 8 |- ( ( abs ` x ) e. RR+ -> ( ( log |` RR+ ) ` ( abs ` x ) ) = ( log ` ( abs ` x ) ) )
22	20, 21	syl 17	. . . . . . 7 |- ( x e. D -> ( ( log |` RR+ ) ` ( abs ` x ) ) = ( log ` ( abs ` x ) ) )
23	19, 22	eqtr4d 2659	. . . . . 6 |- ( x e. D -> ( Re ` ( log ` x ) ) = ( ( log |` RR+ ) ` ( abs ` x ) ) )
24	23	oveq1d 6665	. . . . 5 |- ( x e. D -> ( ( Re ` ( log ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) = ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) )
25	12, 17, 24	3eqtrd 2660	. . . 4 |- ( x e. D -> ( ( log |` D ) ` x ) = ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) )
26	25	mpteq2ia 4740	. . 3 |- ( x e. D |-> ( ( log |` D ) ` x ) ) = ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) )
27	11, 26	eqtri 2644	. 2 |- ( log |` D ) = ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) )
28		eqid 2622	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
29	28	addcn 22668	. . . . 5 |- + e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
30	29	a1i 11	. . . 4 |- ( T. -> + e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
31	28	cnfldtopon 22586	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
32	14	ssriv 3607	. . . . . . . 8 |- D C_ CC
33		resttopon 20965	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ D C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t D ) e. (TopOn ` D ) )
34	31, 32, 33	mp2an 708	. . . . . . 7 |- ( ( TopOpen ` ℂfld ) ↾t D ) e. (TopOn ` D )
35	34	a1i 11	. . . . . 6 |- ( T. -> ( ( TopOpen ` ℂfld ) ↾t D ) e. (TopOn ` D ) )
36		absf 14077	. . . . . . . . . . . 12 |- abs : CC --> RR
37		fssres 6070	. . . . . . . . . . . 12 |- ( ( abs : CC --> RR /\ D C_ CC ) -> ( abs |` D ) : D --> RR )
38	36, 32, 37	mp2an 708	. . . . . . . . . . 11 |- ( abs |` D ) : D --> RR
39	38	a1i 11	. . . . . . . . . 10 |- ( T. -> ( abs |` D ) : D --> RR )
40	39	feqmptd 6249	. . . . . . . . 9 |- ( T. -> ( abs |` D ) = ( x e. D |-> ( ( abs |` D ) ` x ) ) )
41		fvres 6207	. . . . . . . . . 10 |- ( x e. D -> ( ( abs |` D ) ` x ) = ( abs ` x ) )
42	41	mpteq2ia 4740	. . . . . . . . 9 |- ( x e. D |-> ( ( abs |` D ) ` x ) ) = ( x e. D |-> ( abs ` x ) )
43	40, 42	syl6eq 2672	. . . . . . . 8 |- ( T. -> ( abs |` D ) = ( x e. D |-> ( abs ` x ) ) )
44		ffn 6045	. . . . . . . . . . 11 |- ( ( abs |` D ) : D --> RR -> ( abs |` D ) Fn D )
45	38, 44	ax-mp 5	. . . . . . . . . 10 |- ( abs |` D ) Fn D
46	41, 20	eqeltrd 2701	. . . . . . . . . . 11 |- ( x e. D -> ( ( abs |` D ) ` x ) e. RR+ )
47	46	rgen 2922	. . . . . . . . . 10 |- A. x e. D ( ( abs |` D ) ` x ) e. RR+
48		ffnfv 6388	. . . . . . . . . 10 |- ( ( abs |` D ) : D --> RR+ <-> ( ( abs |` D ) Fn D /\ A. x e. D ( ( abs |` D ) ` x ) e. RR+ ) )
49	45, 47, 48	mpbir2an 955	. . . . . . . . 9 |- ( abs |` D ) : D --> RR+
50		rpssre 11843	. . . . . . . . . . 11 |- RR+ C_ RR
51		ax-resscn 9993	. . . . . . . . . . 11 |- RR C_ CC
52	50, 51	sstri 3612	. . . . . . . . . 10 |- RR+ C_ CC
53		abscncf 22704	. . . . . . . . . . 11 |- abs e. ( CC -cn-> RR )
54		rescncf 22700	. . . . . . . . . . 11 |- ( D C_ CC -> ( abs e. ( CC -cn-> RR ) -> ( abs |` D ) e. ( D -cn-> RR ) ) )
55	32, 53, 54	mp2 9	. . . . . . . . . 10 |- ( abs |` D ) e. ( D -cn-> RR )
56		cncffvrn 22701	. . . . . . . . . 10 |- ( ( RR+ C_ CC /\ ( abs |` D ) e. ( D -cn-> RR ) ) -> ( ( abs |` D ) e. ( D -cn-> RR+ ) <-> ( abs |` D ) : D --> RR+ ) )
57	52, 55, 56	mp2an 708	. . . . . . . . 9 |- ( ( abs |` D ) e. ( D -cn-> RR+ ) <-> ( abs |` D ) : D --> RR+ )
58	49, 57	mpbir 221	. . . . . . . 8 |- ( abs |` D ) e. ( D -cn-> RR+ )
59	43, 58	syl6eqelr 2710	. . . . . . 7 |- ( T. -> ( x e. D |-> ( abs ` x ) ) e. ( D -cn-> RR+ ) )
60		eqid 2622	. . . . . . . . 9 |- ( ( TopOpen ` ℂfld ) ↾t D ) = ( ( TopOpen ` ℂfld ) ↾t D )
61		eqid 2622	. . . . . . . . 9 |- ( ( TopOpen ` ℂfld ) ↾t RR+ ) = ( ( TopOpen ` ℂfld ) ↾t RR+ )
62	28, 60, 61	cncfcn 22712	. . . . . . . 8 |- ( ( D C_ CC /\ RR+ C_ CC ) -> ( D -cn-> RR+ ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( ( TopOpen ` ℂfld ) ↾t RR+ ) ) )
63	32, 52, 62	mp2an 708	. . . . . . 7 |- ( D -cn-> RR+ ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( ( TopOpen ` ℂfld ) ↾t RR+ ) )
64	59, 63	syl6eleq 2711	. . . . . 6 |- ( T. -> ( x e. D |-> ( abs ` x ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( ( TopOpen ` ℂfld ) ↾t RR+ ) ) )
65		ssid 3624	. . . . . . . . . 10 |- CC C_ CC
66		cncfss 22702	. . . . . . . . . 10 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC ) )
67	51, 65, 66	mp2an 708	. . . . . . . . 9 |- ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC )
68		relogcn 24384	. . . . . . . . 9 |- ( log |` RR+ ) e. ( RR+ -cn-> RR )
69	67, 68	sselii 3600	. . . . . . . 8 |- ( log |` RR+ ) e. ( RR+ -cn-> CC )
70	69	a1i 11	. . . . . . 7 |- ( T. -> ( log |` RR+ ) e. ( RR+ -cn-> CC ) )
71	28	cnfldtop 22587	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) e. Top
72	31	toponunii 20721	. . . . . . . . . . . 12 |- CC = U. ( TopOpen ` ℂfld )
73	72	restid 16094	. . . . . . . . . . 11 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
74	71, 73	ax-mp 5	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
75	74	eqcomi 2631	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
76	28, 61, 75	cncfcn 22712	. . . . . . . 8 |- ( ( RR+ C_ CC /\ CC C_ CC ) -> ( RR+ -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t RR+ ) Cn ( TopOpen ` ℂfld ) ) )
77	52, 65, 76	mp2an 708	. . . . . . 7 |- ( RR+ -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t RR+ ) Cn ( TopOpen ` ℂfld ) )
78	70, 77	syl6eleq 2711	. . . . . 6 |- ( T. -> ( log |` RR+ ) e. ( ( ( TopOpen ` ℂfld ) ↾t RR+ ) Cn ( TopOpen ` ℂfld ) ) )
79	35, 64, 78	cnmpt11f 21467	. . . . 5 |- ( T. -> ( x e. D |-> ( ( log |` RR+ ) ` ( abs ` x ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) )
80	28, 60, 75	cncfcn 22712	. . . . . 6 |- ( ( D C_ CC /\ CC C_ CC ) -> ( D -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) )
81	32, 65, 80	mp2an 708	. . . . 5 |- ( D -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) )
82	79, 81	syl6eleqr 2712	. . . 4 |- ( T. -> ( x e. D |-> ( ( log |` RR+ ) ` ( abs ` x ) ) ) e. ( D -cn-> CC ) )
83	16	imcld 13935	. . . . . . . 8 |- ( x e. D -> ( Im ` ( log ` x ) ) e. RR )
84	83	recnd 10068	. . . . . . 7 |- ( x e. D -> ( Im ` ( log ` x ) ) e. CC )
85	84	adantl 482	. . . . . 6 |- ( ( T. /\ x e. D ) -> ( Im ` ( log ` x ) ) e. CC )
86		eqidd 2623	. . . . . 6 |- ( T. -> ( x e. D |-> ( Im ` ( log ` x ) ) ) = ( x e. D |-> ( Im ` ( log ` x ) ) ) )
87		eqidd 2623	. . . . . 6 |- ( T. -> ( y e. CC |-> ( _i x. y ) ) = ( y e. CC |-> ( _i x. y ) ) )
88		oveq2 6658	. . . . . 6 |- ( y = ( Im ` ( log ` x ) ) -> ( _i x. y ) = ( _i x. ( Im ` ( log ` x ) ) ) )
89	85, 86, 87, 88	fmptco 6396	. . . . 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 22702	. . . . . . . . 9 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( D -cn-> RR ) C_ ( D -cn-> CC ) )
91	51, 65, 90	mp2an 708	. . . . . . . 8 |- ( D -cn-> RR ) C_ ( D -cn-> CC )
92	4	logcnlem5 24392	. . . . . . . 8 |- ( x e. D |-> ( Im ` ( log ` x ) ) ) e. ( D -cn-> RR )
93	91, 92	sselii 3600	. . . . . . 7 |- ( x e. D |-> ( Im ` ( log ` x ) ) ) e. ( D -cn-> CC )
94	93	a1i 11	. . . . . 6 |- ( T. -> ( x e. D |-> ( Im ` ( log ` x ) ) ) e. ( D -cn-> CC ) )
95		ax-icn 9995	. . . . . . 7 |- _i e. CC
96		eqid 2622	. . . . . . . 8 |- ( y e. CC |-> ( _i x. y ) ) = ( y e. CC |-> ( _i x. y ) )
97	96	mulc1cncf 22708	. . . . . . 7 |- ( _i e. CC -> ( y e. CC |-> ( _i x. y ) ) e. ( CC -cn-> CC ) )
98	95, 97	mp1i 13	. . . . . 6 |- ( T. -> ( y e. CC |-> ( _i x. y ) ) e. ( CC -cn-> CC ) )
99	94, 98	cncfco 22710	. . . . 5 |- ( T. -> ( ( y e. CC |-> ( _i x. y ) ) o. ( x e. D |-> ( Im ` ( log ` x ) ) ) ) e. ( D -cn-> CC ) )
100	89, 99	eqeltrrd 2702	. . . 4 |- ( T. -> ( x e. D |-> ( _i x. ( Im ` ( log ` x ) ) ) ) e. ( D -cn-> CC ) )
101	28, 30, 82, 100	cncfmpt2f 22717	. . 3 |- ( T. -> ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) e. ( D -cn-> CC ) )
102	101	trud 1493	. 2 |- ( x e. D |-> ( ( ( log |` RR+ ) ` ( abs ` x ) ) + ( _i x. ( Im ` ( log ` x ) ) ) ) ) e. ( D -cn-> CC )
103	27, 102	eqeltri 2697	1 |- ( log |` D ) e. ( D -cn-> CC )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177   |-> cmpt 4729   ran crn 5115   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   _ici 9938   + caddc 9939   x. cmul 9941   -oocmnf 10072   RR+crp 11832   (,]cioc 12176   Recre 13837   Imcim 13838   abscabs 13974   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   -cn->ccncf 22679   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  dvlog  24397  efopnlem2  24403  dvcncxp1  24484  cxpcn  24486  lgamgulmlem2  24756  lgamcvg2  24781  areacirclem4  33503