Theorem efopnlem2 24403

Description: Lemma for efopn 24404. (Contributed by Mario Carneiro, 2-May-2015.)

Hypothesis

Ref	Expression
efopn.j	|- J = ( TopOpen ` ℂfld )

Assertion

Ref	Expression
efopnlem2	|- ( ( R e. RR+ /\ R < pi ) -> ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. J )

Proof of Theorem efopnlem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		logf1o 24311	. . . . . . . 8 |- log : ( CC \ { 0 } ) -1-1-onto-> ran log
2		f1orn 6147	. . . . . . . . 9 |- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log <-> ( log Fn ( CC \ { 0 } ) /\ Fun `' log ) )
3	2	simprbi 480	. . . . . . . 8 |- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> Fun `' log )
4		funcnvres 5967	. . . . . . . 8 |- ( Fun `' log -> `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( `' log |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) )
5	1, 3, 4	mp2b 10	. . . . . . 7 |- `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( `' log |` ( log " ( CC \ ( -oo (,] 0 ) ) ) )
6		df-log 24303	. . . . . . . . . 10 |- log = `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
7	6	cnveqi 5297	. . . . . . . . 9 |- `' log = `' `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
8		relres 5426	. . . . . . . . . 10 |- Rel ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
9		dfrel2 5583	. . . . . . . . . 10 |- ( Rel ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) <-> `' `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) )
10	8, 9	mpbi 220	. . . . . . . . 9 |- `' `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
11	7, 10	eqtri 2644	. . . . . . . 8 |- `' log = ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
12	11	reseq1i 5392	. . . . . . 7 |- ( `' log |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) = ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( log " ( CC \ ( -oo (,] 0 ) ) ) )
13		imassrn 5477	. . . . . . . . 9 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) C_ ran log
14		logrn 24305	. . . . . . . . 9 |- ran log = ( `' Im " ( -u pi (,] pi ) )
15	13, 14	sseqtri 3637	. . . . . . . 8 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) C_ ( `' Im " ( -u pi (,] pi ) )
16		resabs1 5427	. . . . . . . 8 |- ( ( log " ( CC \ ( -oo (,] 0 ) ) ) C_ ( `' Im " ( -u pi (,] pi ) ) -> ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) = ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) )
17	15, 16	ax-mp 5	. . . . . . 7 |- ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) = ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) )
18	5, 12, 17	3eqtri 2648	. . . . . 6 |- `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) )
19	18	imaeq1i 5463	. . . . 5 |- ( `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) = ( ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) )
20		cnxmet 22576	. . . . . . . . . . . . 13 |- ( abs o. - ) e. ( *Met ` CC )
21	20	a1i 11	. . . . . . . . . . . 12 |- ( ( R e. RR+ /\ R < pi ) -> ( abs o. - ) e. ( *Met ` CC ) )
22		0cnd 10033	. . . . . . . . . . . 12 |- ( ( R e. RR+ /\ R < pi ) -> 0 e. CC )
23		rpxr 11840	. . . . . . . . . . . . 13 |- ( R e. RR+ -> R e. RR* )
24	23	adantr 481	. . . . . . . . . . . 12 |- ( ( R e. RR+ /\ R < pi ) -> R e. RR* )
25		blssm 22223	. . . . . . . . . . . 12 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ R e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) R ) C_ CC )
26	21, 22, 24, 25	syl3anc 1326	. . . . . . . . . . 11 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) C_ CC )
27	26	sselda 3603	. . . . . . . . . 10 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> x e. CC )
28	27	imcld 13935	. . . . . . . . . . 11 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( Im ` x ) e. RR )
29		efopnlem1 24402	. . . . . . . . . . . . 13 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( abs ` ( Im ` x ) ) < pi )
30		pire 24210	. . . . . . . . . . . . . 14 |- pi e. RR
31		abslt 14054	. . . . . . . . . . . . . 14 |- ( ( ( Im ` x ) e. RR /\ pi e. RR ) -> ( ( abs ` ( Im ` x ) ) < pi <-> ( -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) ) )
32	28, 30, 31	sylancl 694	. . . . . . . . . . . . 13 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( ( abs ` ( Im ` x ) ) < pi <-> ( -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) ) )
33	29, 32	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) )
34	33	simpld 475	. . . . . . . . . . 11 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> -u pi < ( Im ` x ) )
35	33	simprd 479	. . . . . . . . . . 11 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( Im ` x ) < pi )
36	30	renegcli 10342	. . . . . . . . . . . . 13 |- -u pi e. RR
37	36	rexri 10097	. . . . . . . . . . . 12 |- -u pi e. RR*
38	30	rexri 10097	. . . . . . . . . . . 12 |- pi e. RR*
39		elioo2 12216	. . . . . . . . . . . 12 |- ( ( -u pi e. RR* /\ pi e. RR* ) -> ( ( Im ` x ) e. ( -u pi (,) pi ) <-> ( ( Im ` x ) e. RR /\ -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) ) )
40	37, 38, 39	mp2an 708	. . . . . . . . . . 11 |- ( ( Im ` x ) e. ( -u pi (,) pi ) <-> ( ( Im ` x ) e. RR /\ -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) )
41	28, 34, 35, 40	syl3anbrc 1246	. . . . . . . . . 10 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( Im ` x ) e. ( -u pi (,) pi ) )
42		imf 13853	. . . . . . . . . . 11 |- Im : CC --> RR
43		ffn 6045	. . . . . . . . . . 11 |- ( Im : CC --> RR -> Im Fn CC )
44		elpreima 6337	. . . . . . . . . . 11 |- ( Im Fn CC -> ( x e. ( `' Im " ( -u pi (,) pi ) ) <-> ( x e. CC /\ ( Im ` x ) e. ( -u pi (,) pi ) ) ) )
45	42, 43, 44	mp2b 10	. . . . . . . . . 10 |- ( x e. ( `' Im " ( -u pi (,) pi ) ) <-> ( x e. CC /\ ( Im ` x ) e. ( -u pi (,) pi ) ) )
46	27, 41, 45	sylanbrc 698	. . . . . . . . 9 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> x e. ( `' Im " ( -u pi (,) pi ) ) )
47	46	ex 450	. . . . . . . 8 |- ( ( R e. RR+ /\ R < pi ) -> ( x e. ( 0 ( ball ` ( abs o. - ) ) R ) -> x e. ( `' Im " ( -u pi (,) pi ) ) ) )
48	47	ssrdv 3609	. . . . . . 7 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) C_ ( `' Im " ( -u pi (,) pi ) ) )
49		df-ima 5127	. . . . . . . 8 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) = ran ( log |` ( CC \ ( -oo (,] 0 ) ) )
50		eqid 2622	. . . . . . . . . 10 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
51	50	logf1o2 24396	. . . . . . . . 9 |- ( log |` ( CC \ ( -oo (,] 0 ) ) ) : ( CC \ ( -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
52		f1ofo 6144	. . . . . . . . 9 |- ( ( log |` ( CC \ ( -oo (,] 0 ) ) ) : ( CC \ ( -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) ) -> ( log |` ( CC \ ( -oo (,] 0 ) ) ) : ( CC \ ( -oo (,] 0 ) ) -onto-> ( `' Im " ( -u pi (,) pi ) ) )
53		forn 6118	. . . . . . . . 9 |- ( ( log |` ( CC \ ( -oo (,] 0 ) ) ) : ( CC \ ( -oo (,] 0 ) ) -onto-> ( `' Im " ( -u pi (,) pi ) ) -> ran ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( `' Im " ( -u pi (,) pi ) ) )
54	51, 52, 53	mp2b 10	. . . . . . . 8 |- ran ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( `' Im " ( -u pi (,) pi ) )
55	49, 54	eqtri 2644	. . . . . . 7 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) = ( `' Im " ( -u pi (,) pi ) )
56	48, 55	syl6sseqr 3652	. . . . . 6 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) C_ ( log " ( CC \ ( -oo (,] 0 ) ) ) )
57		resima2 5432	. . . . . 6 |- ( ( 0 ( ball ` ( abs o. - ) ) R ) C_ ( log " ( CC \ ( -oo (,] 0 ) ) ) -> ( ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) = ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) )
58	56, 57	syl 17	. . . . 5 |- ( ( R e. RR+ /\ R < pi ) -> ( ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) = ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) )
59	19, 58	syl5eq 2668	. . . 4 |- ( ( R e. RR+ /\ R < pi ) -> ( `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) = ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) )
60	50	logcn 24393	. . . . . 6 |- ( log |` ( CC \ ( -oo (,] 0 ) ) ) e. ( ( CC \ ( -oo (,] 0 ) ) -cn-> CC )
61		difss 3737	. . . . . . 7 |- ( CC \ ( -oo (,] 0 ) ) C_ CC
62		ssid 3624	. . . . . . 7 |- CC C_ CC
63		efopn.j	. . . . . . . 8 |- J = ( TopOpen ` ℂfld )
64		eqid 2622	. . . . . . . 8 |- ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) = ( J ↾t ( CC \ ( -oo (,] 0 ) ) )
65	63	cnfldtop 22587	. . . . . . . . . 10 |- J e. Top
66	63	cnfldtopon 22586	. . . . . . . . . . . 12 |- J e. (TopOn ` CC )
67	66	toponunii 20721	. . . . . . . . . . 11 |- CC = U. J
68	67	restid 16094	. . . . . . . . . 10 |- ( J e. Top -> ( J ↾t CC ) = J )
69	65, 68	ax-mp 5	. . . . . . . . 9 |- ( J ↾t CC ) = J
70	69	eqcomi 2631	. . . . . . . 8 |- J = ( J ↾t CC )
71	63, 64, 70	cncfcn 22712	. . . . . . 7 |- ( ( ( CC \ ( -oo (,] 0 ) ) C_ CC /\ CC C_ CC ) -> ( ( CC \ ( -oo (,] 0 ) ) -cn-> CC ) = ( ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) Cn J ) )
72	61, 62, 71	mp2an 708	. . . . . 6 |- ( ( CC \ ( -oo (,] 0 ) ) -cn-> CC ) = ( ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) Cn J )
73	60, 72	eleqtri 2699	. . . . 5 |- ( log |` ( CC \ ( -oo (,] 0 ) ) ) e. ( ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) Cn J )
74	63	cnfldtopn 22585	. . . . . . 7 |- J = ( MetOpen ` ( abs o. - ) )
75	74	blopn 22305	. . . . . 6 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ R e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) R ) e. J )
76	21, 22, 24, 75	syl3anc 1326	. . . . 5 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) e. J )
77		cnima 21069	. . . . 5 |- ( ( ( log |` ( CC \ ( -oo (,] 0 ) ) ) e. ( ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) Cn J ) /\ ( 0 ( ball ` ( abs o. - ) ) R ) e. J ) -> ( `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) )
78	73, 76, 77	sylancr 695	. . . 4 |- ( ( R e. RR+ /\ R < pi ) -> ( `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) )
79	59, 78	eqeltrrd 2702	. . 3 |- ( ( R e. RR+ /\ R < pi ) -> ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) )
80	50	logdmopn 24395	. . . . 5 |- ( CC \ ( -oo (,] 0 ) ) e. ( TopOpen ` ℂfld )
81	80, 63	eleqtrri 2700	. . . 4 |- ( CC \ ( -oo (,] 0 ) ) e. J
82		restopn2 20981	. . . 4 |- ( ( J e. Top /\ ( CC \ ( -oo (,] 0 ) ) e. J ) -> ( ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) <-> ( ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. J /\ ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) C_ ( CC \ ( -oo (,] 0 ) ) ) ) )
83	65, 81, 82	mp2an 708	. . 3 |- ( ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) <-> ( ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. J /\ ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) C_ ( CC \ ( -oo (,] 0 ) ) ) )
84	79, 83	sylib 208	. 2 |- ( ( R e. RR+ /\ R < pi ) -> ( ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. J /\ ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) C_ ( CC \ ( -oo (,] 0 ) ) ) )
85	84	simpld 475	1 |- ( ( R e. RR+ /\ R < pi ) -> ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. J )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   -oocmnf 10072   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   RR+crp 11832   (,)cioo 12175   (,]cioc 12176   Imcim 13838   abscabs 13974   expce 14792   picpi 14797   ↾_t crest 16081   TopOpenctopn 16082   *Metcxmt 19731   ballcbl 19733  ℂ_fldccnfld 19746   Topctop 20698   Cn ccn 21028   -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:  efopn  24404