Theorem icopnfhmeo 21971

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 21970	. . . 4 |- ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) /\ `' F = ( y e. ( 0 [,) +oo ) |-> ( y / ( 1 + y ) ) ) )
3	2	simpli 460	. . 3 |- F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )
4		0re 9643	. . . . . . . . . . 11 |- 0 e. RR
5		1re 9642	. . . . . . . . . . . 12 |- 1 e. RR
6	5	rexri 9693	. . . . . . . . . . 11 |- 1 e. RR*
7		elico2 11698	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) ) )
8	4, 6, 7	mp2an 678	. . . . . . . . . 10 |- ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) )
9	8	simp1bi 1023	. . . . . . . . 9 |- ( x e. ( 0 [,) 1 ) -> x e. RR )
10	9	ssriv 3436	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR
11	10	sseli 3428	. . . . . . 7 |- ( z e. ( 0 [,) 1 ) -> z e. RR )
12	11	adantr 467	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. RR )
13		elico2 11698	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) ) )
14	4, 6, 13	mp2an 678	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) )
15	14	simp3bi 1025	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> w < 1 )
16	10	sseli 3428	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) -> w e. RR )
17		difrp 11337	. . . . . . . . . 10 |- ( ( w e. RR /\ 1 e. RR ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
18	16, 5, 17	sylancl 668	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
19	15, 18	mpbid 214	. . . . . . . 8 |- ( w e. ( 0 [,) 1 ) -> ( 1 - w ) e. RR+ )
20	19	rpregt0d 11347	. . . . . . 7 |- ( w e. ( 0 [,) 1 ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
21	20	adantl 468	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
22	16	adantl 468	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. RR )
23		elico2 11698	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) ) )
24	4, 6, 23	mp2an 678	. . . . . . . . . 10 |- ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) )
25	24	simp3bi 1025	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> z < 1 )
26		difrp 11337	. . . . . . . . . 10 |- ( ( z e. RR /\ 1 e. RR ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
27	11, 5, 26	sylancl 668	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
28	25, 27	mpbid 214	. . . . . . . 8 |- ( z e. ( 0 [,) 1 ) -> ( 1 - z ) e. RR+ )
29	28	adantr 467	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( 1 - z ) e. RR+ )
30	29	rpregt0d 11347	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - z ) e. RR /\ 0 < ( 1 - z ) ) )
31		lt2mul2div 10483	. . . . . 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 1269	. . . . 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 9671	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. w ) e. RR )
34	12, 22, 33	ltsub1d 10222	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( z - ( z x. w ) ) < ( w - ( z x. w ) ) ) )
35	12	recnd 9669	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. CC )
36		1cnd 9659	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> 1 e. CC )
37	22	recnd 9669	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. CC )
38	35, 36, 37	subdid 10074	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( ( z x. 1 ) - ( z x. w ) ) )
39	35	mulid1d 9660	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. 1 ) = z )
40	39	oveq1d 6305	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( z x. 1 ) - ( z x. w ) ) = ( z - ( z x. w ) ) )
41	38, 40	eqtrd 2485	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( z - ( z x. w ) ) )
42	37, 36, 35	subdid 10074	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( ( w x. 1 ) - ( w x. z ) ) )
43	37	mulid1d 9660	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. 1 ) = w )
44	37, 35	mulcomd 9664	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. z ) = ( z x. w ) )
45	43, 44	oveq12d 6308	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( w x. 1 ) - ( w x. z ) ) = ( w - ( z x. w ) ) )
46	42, 45	eqtrd 2485	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( w - ( z x. w ) ) )
47	41, 46	breq12d 4415	. . . . . 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 260	. . . . 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 6298	. . . . . . . 8 |- ( x = z -> ( 1 - x ) = ( 1 - z ) )
51	49, 50	oveq12d 6308	. . . . . . 7 |- ( x = z -> ( x / ( 1 - x ) ) = ( z / ( 1 - z ) ) )
52		ovex 6318	. . . . . . 7 |- ( z / ( 1 - z ) ) e. _V
53	51, 1, 52	fvmpt 5948	. . . . . 6 |- ( z e. ( 0 [,) 1 ) -> ( F ` z ) = ( z / ( 1 - z ) ) )
54		id 22	. . . . . . . 8 |- ( x = w -> x = w )
55		oveq2 6298	. . . . . . . 8 |- ( x = w -> ( 1 - x ) = ( 1 - w ) )
56	54, 55	oveq12d 6308	. . . . . . 7 |- ( x = w -> ( x / ( 1 - x ) ) = ( w / ( 1 - w ) ) )
57		ovex 6318	. . . . . . 7 |- ( w / ( 1 - w ) ) e. _V
58	56, 1, 57	fvmpt 5948	. . . . . 6 |- ( w e. ( 0 [,) 1 ) -> ( F ` w ) = ( w / ( 1 - w ) ) )
59	53, 58	breqan12d 4418	. . . . 5 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( F ` z ) < ( F ` w ) <-> ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) ) )
60	32, 48, 59	3bitr4d 289	. . . 4 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( F ` z ) < ( F ` w ) ) )
61	60	rgen2a 2815	. . 3 |- A. z e. ( 0 [,) 1 ) A. w e. ( 0 [,) 1 ) ( z < w <-> ( F ` z ) < ( F ` w ) )
62		df-isom 5591	. . 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 931	. 2 |- F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
64		letsr 16473	. . . . . 6 |- <_ e. TosetRel
65	64	elexi 3055	. . . . 5 |- <_ e. _V
66	65	inex1 4544	. . . 4 |- ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) e. _V
67	65	inex1 4544	. . . 4 |- ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) e. _V
68		icossxr 11719	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR*
69		icossxr 11719	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR*
70		leiso 12622	. . . . . . . 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 678	. . . . . . 7 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
72	63, 71	mpbi 212	. . . . . 6 |- F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
73		isores1 6225	. . . . . 6 |- ( F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
74	72, 73	mpbi 212	. . . . 5 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
75		isores2 6224	. . . . 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 212	. . . 4 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
77		tsrps 16467	. . . . . . . 8 |- ( <_ e. TosetRel -> <_ e. PosetRel )
78	64, 77	ax-mp 5	. . . . . . 7 |- <_ e. PosetRel
79		ledm 16470	. . . . . . . 8 |- RR* = dom <_
80	79	psssdm 16462	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) 1 ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 ) )
81	78, 68, 80	mp2an 678	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 )
82	81	eqcomi 2460	. . . . 5 |- ( 0 [,) 1 ) = dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) )
83	79	psssdm 16462	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) +oo ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo ) )
84	78, 69, 83	mp2an 678	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
85	84	eqcomi 2460	. . . . 5 |- ( 0 [,) +oo ) = dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) )
86	82, 85	ordthmeo 20817	. . . 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 1364	. . 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 2451	. . . . . . 7 |- (ordTop ` <_ ) = (ordTop ` <_ )
90	88, 89	xrrest2 21826	. . . . . 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 11708	. . . . . 6 |- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) 1 ) ) -> ( x [,] y ) C_ ( 0 [,) 1 ) )
93	68, 92	ordtrestixx 20238	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
94	91, 93	eqtri 2473	. . . 4 |- ( J ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
95		rge0ssre 11740	. . . . . 6 |- ( 0 [,) +oo ) C_ RR
96	88, 89	xrrest2 21826	. . . . . 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 11708	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x [,] y ) C_ ( 0 [,) +oo ) )
99	69, 98	ordtrestixx 20238	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
100	97, 99	eqtri 2473	. . . 4 |- ( J ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
101	94, 100	oveq12i 6302	. . 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 2528	. 2 |- F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) )
103	63, 102	pm3.2i 457	1 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) /\ F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) ) )

Syntax hints:   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   _Vcvv 3045   i^i cin 3403   C_ wss 3404   class class class wbr 4402   |-> cmpt 4461   X. cxp 4832   `'ccnv 4833   dom cdm 4834   -1-1-onto->wf1o 5581   ` cfv 5582   Isom wiso 5583  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   +oocpnf 9672   RR*cxr 9674   < clt 9675   <_ cle 9676   - cmin 9860   / cdiv 10269   RR+crp 11302   [,)cico 11637   ↾_t crest 15319   TopOpenctopn 15320  ordTopcordt 15397   PosetRelcps 16444   TosetRel ctsr 16445  ℂ_fldccnfld 18970   Homeochmeo 20768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-topgen 15342  df-ordt 15399  df-ps 16446  df-tsr 16447  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cn 20243  df-hmeo 20770  df-xms 21335  df-ms 21336

This theorem is referenced by:  iccpnfhmeo  21973