Theorem icopnfhmeo 22742

Description: The defined bijection from [ 0 , 1 ) to [ 0 , +oo ) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypotheses

Ref	Expression
icopnfhmeo.f	|- F = ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) )
icopnfhmeo.j	|- J = ( TopOpen ` ℂfld )

Assertion

Ref	Expression
icopnfhmeo	|- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) /\ F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) ) )

Distinct variable group:   x, J

Allowed substitution hint:   F( x)

Proof of Theorem icopnfhmeo

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

Step	Hyp	Ref	Expression
1		icopnfhmeo.f	. . . . 5 |- F = ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) )
2	1	icopnfcnv 22741	. . . 4 |- ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) /\ `' F = ( y e. ( 0 [,) +oo ) |-> ( y / ( 1 + y ) ) ) )
3	2	simpli 474	. . 3 |- F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )
4		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
5		1re 10039	. . . . . . . . . . . 12 |- 1 e. RR
6	5	rexri 10097	. . . . . . . . . . 11 |- 1 e. RR*
7		elico2 12237	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) ) )
8	4, 6, 7	mp2an 708	. . . . . . . . . 10 |- ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) )
9	8	simp1bi 1076	. . . . . . . . 9 |- ( x e. ( 0 [,) 1 ) -> x e. RR )
10	9	ssriv 3607	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR
11	10	sseli 3599	. . . . . . 7 |- ( z e. ( 0 [,) 1 ) -> z e. RR )
12	11	adantr 481	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. RR )
13		elico2 12237	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) ) )
14	4, 6, 13	mp2an 708	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) )
15	14	simp3bi 1078	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> w < 1 )
16	10	sseli 3599	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) -> w e. RR )
17		difrp 11868	. . . . . . . . . 10 |- ( ( w e. RR /\ 1 e. RR ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
18	16, 5, 17	sylancl 694	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
19	15, 18	mpbid 222	. . . . . . . 8 |- ( w e. ( 0 [,) 1 ) -> ( 1 - w ) e. RR+ )
20	19	rpregt0d 11878	. . . . . . 7 |- ( w e. ( 0 [,) 1 ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
21	20	adantl 482	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
22	16	adantl 482	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. RR )
23		elico2 12237	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) ) )
24	4, 6, 23	mp2an 708	. . . . . . . . . 10 |- ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) )
25	24	simp3bi 1078	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> z < 1 )
26		difrp 11868	. . . . . . . . . 10 |- ( ( z e. RR /\ 1 e. RR ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
27	11, 5, 26	sylancl 694	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
28	25, 27	mpbid 222	. . . . . . . 8 |- ( z e. ( 0 [,) 1 ) -> ( 1 - z ) e. RR+ )
29	28	adantr 481	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( 1 - z ) e. RR+ )
30	29	rpregt0d 11878	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - z ) e. RR /\ 0 < ( 1 - z ) ) )
31		lt2mul2div 10901	. . . . . 6 |- ( ( ( z e. RR /\ ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) ) /\ ( w e. RR /\ ( ( 1 - z ) e. RR /\ 0 < ( 1 - z ) ) ) ) -> ( ( z x. ( 1 - w ) ) < ( w x. ( 1 - z ) ) <-> ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) ) )
32	12, 21, 22, 30, 31	syl22anc 1327	. . . . 5 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( z x. ( 1 - w ) ) < ( w x. ( 1 - z ) ) <-> ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) ) )
33	12, 22	remulcld 10070	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. w ) e. RR )
34	12, 22, 33	ltsub1d 10636	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( z - ( z x. w ) ) < ( w - ( z x. w ) ) ) )
35	12	recnd 10068	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. CC )
36		1cnd 10056	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> 1 e. CC )
37	22	recnd 10068	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. CC )
38	35, 36, 37	subdid 10486	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( ( z x. 1 ) - ( z x. w ) ) )
39	35	mulid1d 10057	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. 1 ) = z )
40	39	oveq1d 6665	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( z x. 1 ) - ( z x. w ) ) = ( z - ( z x. w ) ) )
41	38, 40	eqtrd 2656	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( z - ( z x. w ) ) )
42	37, 36, 35	subdid 10486	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( ( w x. 1 ) - ( w x. z ) ) )
43	37	mulid1d 10057	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. 1 ) = w )
44	37, 35	mulcomd 10061	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. z ) = ( z x. w ) )
45	43, 44	oveq12d 6668	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( w x. 1 ) - ( w x. z ) ) = ( w - ( z x. w ) ) )
46	42, 45	eqtrd 2656	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( w - ( z x. w ) ) )
47	41, 46	breq12d 4666	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( z x. ( 1 - w ) ) < ( w x. ( 1 - z ) ) <-> ( z - ( z x. w ) ) < ( w - ( z x. w ) ) ) )
48	34, 47	bitr4d 271	. . . . 5 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( z x. ( 1 - w ) ) < ( w x. ( 1 - z ) ) ) )
49		id 22	. . . . . . . 8 |- ( x = z -> x = z )
50		oveq2 6658	. . . . . . . 8 |- ( x = z -> ( 1 - x ) = ( 1 - z ) )
51	49, 50	oveq12d 6668	. . . . . . 7 |- ( x = z -> ( x / ( 1 - x ) ) = ( z / ( 1 - z ) ) )
52		ovex 6678	. . . . . . 7 |- ( z / ( 1 - z ) ) e. _V
53	51, 1, 52	fvmpt 6282	. . . . . 6 |- ( z e. ( 0 [,) 1 ) -> ( F ` z ) = ( z / ( 1 - z ) ) )
54		id 22	. . . . . . . 8 |- ( x = w -> x = w )
55		oveq2 6658	. . . . . . . 8 |- ( x = w -> ( 1 - x ) = ( 1 - w ) )
56	54, 55	oveq12d 6668	. . . . . . 7 |- ( x = w -> ( x / ( 1 - x ) ) = ( w / ( 1 - w ) ) )
57		ovex 6678	. . . . . . 7 |- ( w / ( 1 - w ) ) e. _V
58	56, 1, 57	fvmpt 6282	. . . . . 6 |- ( w e. ( 0 [,) 1 ) -> ( F ` w ) = ( w / ( 1 - w ) ) )
59	53, 58	breqan12d 4669	. . . . 5 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( F ` z ) < ( F ` w ) <-> ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) ) )
60	32, 48, 59	3bitr4d 300	. . . 4 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( F ` z ) < ( F ` w ) ) )
61	60	rgen2a 2977	. . 3 |- A. z e. ( 0 [,) 1 ) A. w e. ( 0 [,) 1 ) ( z < w <-> ( F ` z ) < ( F ` w ) )
62		df-isom 5897	. . 3 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) /\ A. z e. ( 0 [,) 1 ) A. w e. ( 0 [,) 1 ) ( z < w <-> ( F ` z ) < ( F ` w ) ) ) )
63	3, 61, 62	mpbir2an 955	. 2 |- F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
64		letsr 17227	. . . . . 6 |- <_ e. TosetRel
65	64	elexi 3213	. . . . 5 |- <_ e. _V
66	65	inex1 4799	. . . 4 |- ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) e. _V
67	65	inex1 4799	. . . 4 |- ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) e. _V
68		icossxr 12258	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR*
69		icossxr 12258	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR*
70		leiso 13243	. . . . . . . 8 |- ( ( ( 0 [,) 1 ) C_ RR* /\ ( 0 [,) +oo ) C_ RR* ) -> ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) ) )
71	68, 69, 70	mp2an 708	. . . . . . 7 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
72	63, 71	mpbi 220	. . . . . 6 |- F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
73		isores1 6584	. . . . . 6 |- ( F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
74	72, 73	mpbi 220	. . . . 5 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
75		isores2 6583	. . . . 5 |- ( F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
76	74, 75	mpbi 220	. . . 4 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
77		tsrps 17221	. . . . . . . 8 |- ( <_ e. TosetRel -> <_ e. PosetRel )
78	64, 77	ax-mp 5	. . . . . . 7 |- <_ e. PosetRel
79		ledm 17224	. . . . . . . 8 |- RR* = dom <_
80	79	psssdm 17216	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) 1 ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 ) )
81	78, 68, 80	mp2an 708	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 )
82	81	eqcomi 2631	. . . . 5 |- ( 0 [,) 1 ) = dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) )
83	79	psssdm 17216	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) +oo ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo ) )
84	78, 69, 83	mp2an 708	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
85	84	eqcomi 2631	. . . . 5 |- ( 0 [,) +oo ) = dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) )
86	82, 85	ordthmeo 21605	. . . 4 |- ( ( ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) e. _V /\ ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) e. _V /\ F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) ) -> F e. ( (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) ) Homeo (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ) ) )
87	66, 67, 76, 86	mp3an 1424	. . 3 |- F e. ( (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) ) Homeo (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ) )
88		icopnfhmeo.j	. . . . . . 7 |- J = ( TopOpen ` ℂfld )
89		eqid 2622	. . . . . . 7 |- (ordTop ` <_ ) = (ordTop ` <_ )
90	88, 89	xrrest2 22611	. . . . . 6 |- ( ( 0 [,) 1 ) C_ RR -> ( J ↾t ( 0 [,) 1 ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) ) )
91	10, 90	ax-mp 5	. . . . 5 |- ( J ↾t ( 0 [,) 1 ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) )
92		iccssico2 12247	. . . . . 6 |- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) 1 ) ) -> ( x [,] y ) C_ ( 0 [,) 1 ) )
93	68, 92	ordtrestixx 21026	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
94	91, 93	eqtri 2644	. . . 4 |- ( J ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
95		rge0ssre 12280	. . . . . 6 |- ( 0 [,) +oo ) C_ RR
96	88, 89	xrrest2 22611	. . . . . 6 |- ( ( 0 [,) +oo ) C_ RR -> ( J ↾t ( 0 [,) +oo ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) ) )
97	95, 96	ax-mp 5	. . . . 5 |- ( J ↾t ( 0 [,) +oo ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) )
98		iccssico2 12247	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x [,] y ) C_ ( 0 [,) +oo ) )
99	69, 98	ordtrestixx 21026	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
100	97, 99	eqtri 2644	. . . 4 |- ( J ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
101	94, 100	oveq12i 6662	. . 3 |- ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) ) = ( (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) ) Homeo (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ) )
102	87, 101	eleqtrri 2700	. 2 |- F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) )
103	63, 102	pm3.2i 471	1 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) /\ F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   RR+crp 11832   [,)cico 12177   ↾_t crest 16081   TopOpenctopn 16082  ordTopcordt 16159   PosetRelcps 17198   TosetRel ctsr 17199  ℂ_fldccnfld 19746   Homeochmeo 21556

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  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-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-ordt 16161  df-ps 17200  df-tsr 17201  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-hmeo 21558  df-xms 22125  df-ms 22126

This theorem is referenced by:  iccpnfhmeo  22744