Theorem efopnlem2 23681

Description: Lemma for efopn 23682. (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 23593	. . . . . . . 8 |- log : ( CC \ { 0 } ) -1-1-onto-> ran log
2		f1orn 5838	. . . . . . . . 9 |- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log <-> ( log Fn ( CC \ { 0 } ) /\ Fun `' log ) )
3	2	simprbi 471	. . . . . . . 8 |- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> Fun `' log )
4		funcnvres 5662	. . . . . . . 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 23585	. . . . . . . . . 10 |- log = `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
7	6	cnveqi 5014	. . . . . . . . 9 |- `' log = `' `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
8		relres 5138	. . . . . . . . . 10 |- Rel ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
9		dfrel2 5292	. . . . . . . . . 10 |- ( Rel ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) <-> `' `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) )
10	8, 9	mpbi 213	. . . . . . . . 9 |- `' `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
11	7, 10	eqtri 2493	. . . . . . . 8 |- `' log = ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
12	11	reseq1i 5107	. . . . . . 7 |- ( `' log |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) = ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( log " ( CC \ ( -oo (,] 0 ) ) ) )
13		imassrn 5185	. . . . . . . . 9 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) C_ ran log
14		logrn 23587	. . . . . . . . 9 |- ran log = ( `' Im " ( -u pi (,] pi ) )
15	13, 14	sseqtri 3450	. . . . . . . 8 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) C_ ( `' Im " ( -u pi (,] pi ) )
16		resabs1 5139	. . . . . . . 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 2497	. . . . . 6 |- `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) )
19	18	imaeq1i 5171	. . . . 5 |- ( `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) = ( ( exp |` ( log " ( CC \ ( -oo (,] 0 ) ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) )
20		cnxmet 21871	. . . . . . . . . . . . 13 |- ( abs o. - ) e. ( *Met ` CC )
21	20	a1i 11	. . . . . . . . . . . 12 |- ( ( R e. RR+ /\ R < pi ) -> ( abs o. - ) e. ( *Met ` CC ) )
22		0cnd 9654	. . . . . . . . . . . 12 |- ( ( R e. RR+ /\ R < pi ) -> 0 e. CC )
23		rpxr 11332	. . . . . . . . . . . . 13 |- ( R e. RR+ -> R e. RR* )
24	23	adantr 472	. . . . . . . . . . . 12 |- ( ( R e. RR+ /\ R < pi ) -> R e. RR* )
25		blssm 21511	. . . . . . . . . . . 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 1292	. . . . . . . . . . 11 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) C_ CC )
27	26	sselda 3418	. . . . . . . . . 10 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> x e. CC )
28	27	imcld 13335	. . . . . . . . . . 11 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( Im ` x ) e. RR )
29		efopnlem1 23680	. . . . . . . . . . . . 13 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( abs ` ( Im ` x ) ) < pi )
30		pire 23492	. . . . . . . . . . . . . 14 |- pi e. RR
31		abslt 13454	. . . . . . . . . . . . . 14 |- ( ( ( Im ` x ) e. RR /\ pi e. RR ) -> ( ( abs ` ( Im ` x ) ) < pi <-> ( -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) ) )
32	28, 30, 31	sylancl 675	. . . . . . . . . . . . 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 215	. . . . . . . . . . . 12 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) )
34	33	simpld 466	. . . . . . . . . . 11 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> -u pi < ( Im ` x ) )
35	33	simprd 470	. . . . . . . . . . 11 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( Im ` x ) < pi )
36	30	renegcli 9955	. . . . . . . . . . . . 13 |- -u pi e. RR
37	36	rexri 9711	. . . . . . . . . . . 12 |- -u pi e. RR*
38	30	rexri 9711	. . . . . . . . . . . 12 |- pi e. RR*
39		elioo2 11702	. . . . . . . . . . . 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 686	. . . . . . . . . . 11 |- ( ( Im ` x ) e. ( -u pi (,) pi ) <-> ( ( Im ` x ) e. RR /\ -u pi < ( Im ` x ) /\ ( Im ` x ) < pi ) )
41	28, 34, 35, 40	syl3anbrc 1214	. . . . . . . . . 10 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> ( Im ` x ) e. ( -u pi (,) pi ) )
42		imf 13253	. . . . . . . . . . 11 |- Im : CC --> RR
43		ffn 5739	. . . . . . . . . . 11 |- ( Im : CC --> RR -> Im Fn CC )
44		elpreima 6017	. . . . . . . . . . 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 677	. . . . . . . . 9 |- ( ( ( R e. RR+ /\ R < pi ) /\ x e. ( 0 ( ball ` ( abs o. - ) ) R ) ) -> x e. ( `' Im " ( -u pi (,) pi ) ) )
47	46	ex 441	. . . . . . . 8 |- ( ( R e. RR+ /\ R < pi ) -> ( x e. ( 0 ( ball ` ( abs o. - ) ) R ) -> x e. ( `' Im " ( -u pi (,) pi ) ) ) )
48	47	ssrdv 3424	. . . . . . 7 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) C_ ( `' Im " ( -u pi (,) pi ) ) )
49		df-ima 4852	. . . . . . . 8 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) = ran ( log |` ( CC \ ( -oo (,] 0 ) ) )
50		eqid 2471	. . . . . . . . . 10 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
51	50	logf1o2 23674	. . . . . . . . 9 |- ( log |` ( CC \ ( -oo (,] 0 ) ) ) : ( CC \ ( -oo (,] 0 ) ) -1-1-onto-> ( `' Im " ( -u pi (,) pi ) )
52		f1ofo 5835	. . . . . . . . 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 5809	. . . . . . . . 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 2493	. . . . . . 7 |- ( log " ( CC \ ( -oo (,] 0 ) ) ) = ( `' Im " ( -u pi (,) pi ) )
56	48, 55	syl6sseqr 3465	. . . . . 6 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) C_ ( log " ( CC \ ( -oo (,] 0 ) ) ) )
57		resima2 5144	. . . . . 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 2517	. . . 4 |- ( ( R e. RR+ /\ R < pi ) -> ( `' ( log |` ( CC \ ( -oo (,] 0 ) ) ) " ( 0 ( ball ` ( abs o. - ) ) R ) ) = ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) )
60	50	logcn 23671	. . . . . 6 |- ( log |` ( CC \ ( -oo (,] 0 ) ) ) e. ( ( CC \ ( -oo (,] 0 ) ) -cn-> CC )
61		difss 3549	. . . . . . 7 |- ( CC \ ( -oo (,] 0 ) ) C_ CC
62		ssid 3437	. . . . . . 7 |- CC C_ CC
63		efopn.j	. . . . . . . 8 |- J = ( TopOpen ` ℂfld )
64		eqid 2471	. . . . . . . 8 |- ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) = ( J ↾t ( CC \ ( -oo (,] 0 ) ) )
65	63	cnfldtop 21882	. . . . . . . . . 10 |- J e. Top
66	63	cnfldtopon 21881	. . . . . . . . . . . 12 |- J e. (TopOn ` CC )
67	66	toponunii 20024	. . . . . . . . . . 11 |- CC = U. J
68	67	restid 15410	. . . . . . . . . 10 |- ( J e. Top -> ( J ↾t CC ) = J )
69	65, 68	ax-mp 5	. . . . . . . . 9 |- ( J ↾t CC ) = J
70	69	eqcomi 2480	. . . . . . . 8 |- J = ( J ↾t CC )
71	63, 64, 70	cncfcn 22019	. . . . . . 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 686	. . . . . 6 |- ( ( CC \ ( -oo (,] 0 ) ) -cn-> CC ) = ( ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) Cn J )
73	60, 72	eleqtri 2547	. . . . 5 |- ( log |` ( CC \ ( -oo (,] 0 ) ) ) e. ( ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) Cn J )
74	63	cnfldtopn 21880	. . . . . . 7 |- J = ( MetOpen ` ( abs o. - ) )
75	74	blopn 21593	. . . . . 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 1292	. . . . 5 |- ( ( R e. RR+ /\ R < pi ) -> ( 0 ( ball ` ( abs o. - ) ) R ) e. J )
77		cnima 20358	. . . . 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 676	. . . 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 2550	. . 3 |- ( ( R e. RR+ /\ R < pi ) -> ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. ( J ↾t ( CC \ ( -oo (,] 0 ) ) ) )
80	50	logdmopn 23673	. . . . 5 |- ( CC \ ( -oo (,] 0 ) ) e. ( TopOpen ` ℂfld )
81	80, 63	eleqtrri 2548	. . . 4 |- ( CC \ ( -oo (,] 0 ) ) e. J
82		restopn2 20270	. . . 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 686	. . 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 201	. 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 466	1 |- ( ( R e. RR+ /\ R < pi ) -> ( exp " ( 0 ( ball ` ( abs o. - ) ) R ) ) e. J )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   \ cdif 3387   C_ wss 3390   {csn 3959   class class class wbr 4395   `'ccnv 4838   ran crn 4840   |` cres 4841   "cima 4842   o. ccom 4843   Rel wrel 4844   Fun wfun 5583   Fn wfn 5584   -->wf 5585   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   -oocmnf 9691   RR*cxr 9692   < clt 9693   - cmin 9880   -ucneg 9881   RR+crp 11325   (,)cioo 11660   (,]cioc 11661   Imcim 13238   abscabs 13374   expce 14191   picpi 14196   ↾_t crest 15397   TopOpenctopn 15398   *Metcxmt 19032   ballcbl 19034  ℂ_fldccnfld 19047   Topctop 19994   Cn ccn 20317   -cn->ccncf 21986   logclog 23583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585

This theorem is referenced by:  efopn  23682