Theorem xrge0iifiso 29981

Description: The defined bijection from the closed unit interval and the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)

Hypothesis

Ref	Expression
xrge0iifhmeo.1	|- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) )

Assertion

Ref	Expression
xrge0iifiso	|- F Isom < , `' < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) )

Distinct variable group:   x, F

Proof of Theorem xrge0iifiso

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 12256	. . 3 |- ( 0 [,] 1 ) C_ RR*
2		xrltso 11974	. . 3 |- < Or RR*
3		soss 5053	. . 3 |- ( ( 0 [,] 1 ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] 1 ) ) )
4	1, 2, 3	mp2 9	. 2 |- < Or ( 0 [,] 1 )
5		iccssxr 12256	. . 3 |- ( 0 [,] +oo ) C_ RR*
6		cnvso 5674	. . . . 5 |- ( < Or RR* <-> `' < Or RR* )
7	2, 6	mpbi 220	. . . 4 |- `' < Or RR*
8		sopo 5052	. . . 4 |- ( `' < Or RR* -> `' < Po RR* )
9	7, 8	ax-mp 5	. . 3 |- `' < Po RR*
10		poss 5037	. . 3 |- ( ( 0 [,] +oo ) C_ RR* -> ( `' < Po RR* -> `' < Po ( 0 [,] +oo ) ) )
11	5, 9, 10	mp2 9	. 2 |- `' < Po ( 0 [,] +oo )
12		xrge0iifhmeo.1	. . . . 5 |- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) )
13	12	xrge0iifcnv 29979	. . . 4 |- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( z e. ( 0 [,] +oo ) |-> if ( z = +oo , 0 , ( exp ` -u z ) ) ) )
14	13	simpli 474	. . 3 |- F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )
15		f1ofo 6144	. . 3 |- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) -> F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo ) )
16	14, 15	ax-mp 5	. 2 |- F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo )
17		0xr 10086	. . . . . . . 8 |- 0 e. RR*
18		1re 10039	. . . . . . . . 9 |- 1 e. RR
19	18	rexri 10097	. . . . . . . 8 |- 1 e. RR*
20		0le1 10551	. . . . . . . 8 |- 0 <_ 1
21		snunioc 12300	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR* /\ 0 <_ 1 ) -> ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 ) )
22	17, 19, 20, 21	mp3an 1424	. . . . . . 7 |- ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 )
23	22	eleq2i 2693	. . . . . 6 |- ( w e. ( { 0 } u. ( 0 (,] 1 ) ) <-> w e. ( 0 [,] 1 ) )
24		elun 3753	. . . . . 6 |- ( w e. ( { 0 } u. ( 0 (,] 1 ) ) <-> ( w e. { 0 } \/ w e. ( 0 (,] 1 ) ) )
25	23, 24	bitr3i 266	. . . . 5 |- ( w e. ( 0 [,] 1 ) <-> ( w e. { 0 } \/ w e. ( 0 (,] 1 ) ) )
26		velsn 4193	. . . . . . 7 |- ( w e. { 0 } <-> w = 0 )
27		elunitrn 29943	. . . . . . . . . . . 12 |- ( z e. ( 0 [,] 1 ) -> z e. RR )
28	27	adantr 481	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ 0 < z ) -> z e. RR )
29		simpr 477	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ 0 < z ) -> 0 < z )
30		0re 10040	. . . . . . . . . . . . . 14 |- 0 e. RR
31	30, 18	elicc2i 12239	. . . . . . . . . . . . 13 |- ( z e. ( 0 [,] 1 ) <-> ( z e. RR /\ 0 <_ z /\ z <_ 1 ) )
32	31	simp3bi 1078	. . . . . . . . . . . 12 |- ( z e. ( 0 [,] 1 ) -> z <_ 1 )
33	32	adantr 481	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ 0 < z ) -> z <_ 1 )
34		elioc2 12236	. . . . . . . . . . . 12 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( z e. ( 0 (,] 1 ) <-> ( z e. RR /\ 0 < z /\ z <_ 1 ) ) )
35	17, 18, 34	mp2an 708	. . . . . . . . . . 11 |- ( z e. ( 0 (,] 1 ) <-> ( z e. RR /\ 0 < z /\ z <_ 1 ) )
36	28, 29, 33, 35	syl3anbrc 1246	. . . . . . . . . 10 |- ( ( z e. ( 0 [,] 1 ) /\ 0 < z ) -> z e. ( 0 (,] 1 ) )
37		pnfxr 10092	. . . . . . . . . . . . . . 15 |- +oo e. RR*
38		0le0 11110	. . . . . . . . . . . . . . 15 |- 0 <_ 0
39		ltpnf 11954	. . . . . . . . . . . . . . . 16 |- ( 1 e. RR -> 1 < +oo )
40	18, 39	ax-mp 5	. . . . . . . . . . . . . . 15 |- 1 < +oo
41		iocssioo 12263	. . . . . . . . . . . . . . 15 |- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 <_ 0 /\ 1 < +oo ) ) -> ( 0 (,] 1 ) C_ ( 0 (,) +oo ) )
42	17, 37, 38, 40, 41	mp4an 709	. . . . . . . . . . . . . 14 |- ( 0 (,] 1 ) C_ ( 0 (,) +oo )
43		ioorp 12251	. . . . . . . . . . . . . 14 |- ( 0 (,) +oo ) = RR+
44	42, 43	sseqtri 3637	. . . . . . . . . . . . 13 |- ( 0 (,] 1 ) C_ RR+
45	44	sseli 3599	. . . . . . . . . . . 12 |- ( z e. ( 0 (,] 1 ) -> z e. RR+ )
46		relogcl 24322	. . . . . . . . . . . . . . 15 |- ( z e. RR+ -> ( log ` z ) e. RR )
47	46	renegcld 10457	. . . . . . . . . . . . . 14 |- ( z e. RR+ -> -u ( log ` z ) e. RR )
48		ltpnf 11954	. . . . . . . . . . . . . 14 |- ( -u ( log ` z ) e. RR -> -u ( log ` z ) < +oo )
49	47, 48	syl 17	. . . . . . . . . . . . 13 |- ( z e. RR+ -> -u ( log ` z ) < +oo )
50		brcnvg 5303	. . . . . . . . . . . . . 14 |- ( ( +oo e. RR* /\ -u ( log ` z ) e. RR ) -> ( +oo `' < -u ( log ` z ) <-> -u ( log ` z ) < +oo ) )
51	37, 47, 50	sylancr 695	. . . . . . . . . . . . 13 |- ( z e. RR+ -> ( +oo `' < -u ( log ` z ) <-> -u ( log ` z ) < +oo ) )
52	49, 51	mpbird 247	. . . . . . . . . . . 12 |- ( z e. RR+ -> +oo `' < -u ( log ` z ) )
53	45, 52	syl 17	. . . . . . . . . . 11 |- ( z e. ( 0 (,] 1 ) -> +oo `' < -u ( log ` z ) )
54	12	xrge0iifcv 29980	. . . . . . . . . . 11 |- ( z e. ( 0 (,] 1 ) -> ( F ` z ) = -u ( log ` z ) )
55	53, 54	breqtrrd 4681	. . . . . . . . . 10 |- ( z e. ( 0 (,] 1 ) -> +oo `' < ( F ` z ) )
56	36, 55	syl 17	. . . . . . . . 9 |- ( ( z e. ( 0 [,] 1 ) /\ 0 < z ) -> +oo `' < ( F ` z ) )
57	56	ex 450	. . . . . . . 8 |- ( z e. ( 0 [,] 1 ) -> ( 0 < z -> +oo `' < ( F ` z ) ) )
58		breq1 4656	. . . . . . . . 9 |- ( w = 0 -> ( w < z <-> 0 < z ) )
59		fveq2 6191	. . . . . . . . . . 11 |- ( w = 0 -> ( F ` w ) = ( F ` 0 ) )
60		0elunit 12290	. . . . . . . . . . . 12 |- 0 e. ( 0 [,] 1 )
61		iftrue 4092	. . . . . . . . . . . . 13 |- ( x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) = +oo )
62		pnfex 10093	. . . . . . . . . . . . 13 |- +oo e. _V
63	61, 12, 62	fvmpt 6282	. . . . . . . . . . . 12 |- ( 0 e. ( 0 [,] 1 ) -> ( F ` 0 ) = +oo )
64	60, 63	ax-mp 5	. . . . . . . . . . 11 |- ( F ` 0 ) = +oo
65	59, 64	syl6eq 2672	. . . . . . . . . 10 |- ( w = 0 -> ( F ` w ) = +oo )
66	65	breq1d 4663	. . . . . . . . 9 |- ( w = 0 -> ( ( F ` w ) `' < ( F ` z ) <-> +oo `' < ( F ` z ) ) )
67	58, 66	imbi12d 334	. . . . . . . 8 |- ( w = 0 -> ( ( w < z -> ( F ` w ) `' < ( F ` z ) ) <-> ( 0 < z -> +oo `' < ( F ` z ) ) ) )
68	57, 67	syl5ibr 236	. . . . . . 7 |- ( w = 0 -> ( z e. ( 0 [,] 1 ) -> ( w < z -> ( F ` w ) `' < ( F ` z ) ) ) )
69	26, 68	sylbi 207	. . . . . 6 |- ( w e. { 0 } -> ( z e. ( 0 [,] 1 ) -> ( w < z -> ( F ` w ) `' < ( F ` z ) ) ) )
70		simpll 790	. . . . . . . . 9 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> w e. ( 0 (,] 1 ) )
71	27	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> z e. RR )
72	30	a1i 11	. . . . . . . . . . 11 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> 0 e. RR )
73	44	sseli 3599	. . . . . . . . . . . . 13 |- ( w e. ( 0 (,] 1 ) -> w e. RR+ )
74	73	rpred 11872	. . . . . . . . . . . 12 |- ( w e. ( 0 (,] 1 ) -> w e. RR )
75	74	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> w e. RR )
76		elioc2 12236	. . . . . . . . . . . . . 14 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( w e. ( 0 (,] 1 ) <-> ( w e. RR /\ 0 < w /\ w <_ 1 ) ) )
77	17, 18, 76	mp2an 708	. . . . . . . . . . . . 13 |- ( w e. ( 0 (,] 1 ) <-> ( w e. RR /\ 0 < w /\ w <_ 1 ) )
78	77	simp2bi 1077	. . . . . . . . . . . 12 |- ( w e. ( 0 (,] 1 ) -> 0 < w )
79	78	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> 0 < w )
80		simpr 477	. . . . . . . . . . 11 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> w < z )
81	72, 75, 71, 79, 80	lttrd 10198	. . . . . . . . . 10 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> 0 < z )
82	32	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> z <_ 1 )
83	71, 81, 82, 35	syl3anbrc 1246	. . . . . . . . 9 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> z e. ( 0 (,] 1 ) )
84	70, 83	jca 554	. . . . . . . 8 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) )
85	73	adantr 481	. . . . . . . . . . . . 13 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> w e. RR+ )
86	85	relogcld 24369	. . . . . . . . . . . 12 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( log ` w ) e. RR )
87	45	adantl 482	. . . . . . . . . . . . 13 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> z e. RR+ )
88	87	relogcld 24369	. . . . . . . . . . . 12 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( log ` z ) e. RR )
89	86, 88	ltnegd 10605	. . . . . . . . . . 11 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( ( log ` w ) < ( log ` z ) <-> -u ( log ` z ) < -u ( log ` w ) ) )
90		logltb 24346	. . . . . . . . . . . 12 |- ( ( w e. RR+ /\ z e. RR+ ) -> ( w < z <-> ( log ` w ) < ( log ` z ) ) )
91	73, 45, 90	syl2an 494	. . . . . . . . . . 11 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( w < z <-> ( log ` w ) < ( log ` z ) ) )
92		negex 10279	. . . . . . . . . . . . 13 |- -u ( log ` w ) e. _V
93		negex 10279	. . . . . . . . . . . . 13 |- -u ( log ` z ) e. _V
94	92, 93	brcnv 5305	. . . . . . . . . . . 12 |- ( -u ( log ` w ) `' < -u ( log ` z ) <-> -u ( log ` z ) < -u ( log ` w ) )
95	94	a1i 11	. . . . . . . . . . 11 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( -u ( log ` w ) `' < -u ( log ` z ) <-> -u ( log ` z ) < -u ( log ` w ) ) )
96	89, 91, 95	3bitr4d 300	. . . . . . . . . 10 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( w < z <-> -u ( log ` w ) `' < -u ( log ` z ) ) )
97	96	biimpd 219	. . . . . . . . 9 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( w < z -> -u ( log ` w ) `' < -u ( log ` z ) ) )
98	12	xrge0iifcv 29980	. . . . . . . . . 10 |- ( w e. ( 0 (,] 1 ) -> ( F ` w ) = -u ( log ` w ) )
99	98, 54	breqan12d 4669	. . . . . . . . 9 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( ( F ` w ) `' < ( F ` z ) <-> -u ( log ` w ) `' < -u ( log ` z ) ) )
100	97, 99	sylibrd 249	. . . . . . . 8 |- ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 (,] 1 ) ) -> ( w < z -> ( F ` w ) `' < ( F ` z ) ) )
101	84, 80, 100	sylc 65	. . . . . . 7 |- ( ( ( w e. ( 0 (,] 1 ) /\ z e. ( 0 [,] 1 ) ) /\ w < z ) -> ( F ` w ) `' < ( F ` z ) )
102	101	exp31 630	. . . . . 6 |- ( w e. ( 0 (,] 1 ) -> ( z e. ( 0 [,] 1 ) -> ( w < z -> ( F ` w ) `' < ( F ` z ) ) ) )
103	69, 102	jaoi 394	. . . . 5 |- ( ( w e. { 0 } \/ w e. ( 0 (,] 1 ) ) -> ( z e. ( 0 [,] 1 ) -> ( w < z -> ( F ` w ) `' < ( F ` z ) ) ) )
104	25, 103	sylbi 207	. . . 4 |- ( w e. ( 0 [,] 1 ) -> ( z e. ( 0 [,] 1 ) -> ( w < z -> ( F ` w ) `' < ( F ` z ) ) ) )
105	104	imp 445	. . 3 |- ( ( w e. ( 0 [,] 1 ) /\ z e. ( 0 [,] 1 ) ) -> ( w < z -> ( F ` w ) `' < ( F ` z ) ) )
106	105	rgen2a 2977	. 2 |- A. w e. ( 0 [,] 1 ) A. z e. ( 0 [,] 1 ) ( w < z -> ( F ` w ) `' < ( F ` z ) )
107		soisoi 6578	. 2 |- ( ( ( < Or ( 0 [,] 1 ) /\ `' < Po ( 0 [,] +oo ) ) /\ ( F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo ) /\ A. w e. ( 0 [,] 1 ) A. z e. ( 0 [,] 1 ) ( w < z -> ( F ` w ) `' < ( F ` z ) ) ) ) -> F Isom < , `' < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) )
108	4, 11, 16, 106, 107	mp4an 709	1 |- F Isom < , `' < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   Po wpo 5033   Or wor 5034   `'ccnv 5113   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   RR+crp 11832   (,)cioo 12175   (,]cioc 12176   [,]cicc 12178   expce 14792   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-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-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:  xrge0iifhmeo  29982