Theorem icopnfhmeo 22049

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 22048	. . . 4 |- ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) /\ `' F = ( y e. ( 0 [,) +oo ) |-> ( y / ( 1 + y ) ) ) )
3	2	simpli 465	. . 3 |- F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )
4		0re 9661	. . . . . . . . . . 11 |- 0 e. RR
5		1re 9660	. . . . . . . . . . . 12 |- 1 e. RR
6	5	rexri 9711	. . . . . . . . . . 11 |- 1 e. RR*
7		elico2 11723	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) ) )
8	4, 6, 7	mp2an 686	. . . . . . . . . 10 |- ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) )
9	8	simp1bi 1045	. . . . . . . . 9 |- ( x e. ( 0 [,) 1 ) -> x e. RR )
10	9	ssriv 3422	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR
11	10	sseli 3414	. . . . . . 7 |- ( z e. ( 0 [,) 1 ) -> z e. RR )
12	11	adantr 472	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. RR )
13		elico2 11723	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) ) )
14	4, 6, 13	mp2an 686	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) )
15	14	simp3bi 1047	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> w < 1 )
16	10	sseli 3414	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) -> w e. RR )
17		difrp 11360	. . . . . . . . . 10 |- ( ( w e. RR /\ 1 e. RR ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
18	16, 5, 17	sylancl 675	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
19	15, 18	mpbid 215	. . . . . . . 8 |- ( w e. ( 0 [,) 1 ) -> ( 1 - w ) e. RR+ )
20	19	rpregt0d 11370	. . . . . . 7 |- ( w e. ( 0 [,) 1 ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
21	20	adantl 473	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
22	16	adantl 473	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. RR )
23		elico2 11723	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) ) )
24	4, 6, 23	mp2an 686	. . . . . . . . . 10 |- ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) )
25	24	simp3bi 1047	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> z < 1 )
26		difrp 11360	. . . . . . . . . 10 |- ( ( z e. RR /\ 1 e. RR ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
27	11, 5, 26	sylancl 675	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
28	25, 27	mpbid 215	. . . . . . . 8 |- ( z e. ( 0 [,) 1 ) -> ( 1 - z ) e. RR+ )
29	28	adantr 472	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( 1 - z ) e. RR+ )
30	29	rpregt0d 11370	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - z ) e. RR /\ 0 < ( 1 - z ) ) )
31		lt2mul2div 10505	. . . . . 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 1293	. . . . 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 9689	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. w ) e. RR )
34	12, 22, 33	ltsub1d 10243	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( z - ( z x. w ) ) < ( w - ( z x. w ) ) ) )
35	12	recnd 9687	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. CC )
36		1cnd 9677	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> 1 e. CC )
37	22	recnd 9687	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. CC )
38	35, 36, 37	subdid 10095	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( ( z x. 1 ) - ( z x. w ) ) )
39	35	mulid1d 9678	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. 1 ) = z )
40	39	oveq1d 6323	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( z x. 1 ) - ( z x. w ) ) = ( z - ( z x. w ) ) )
41	38, 40	eqtrd 2505	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( z - ( z x. w ) ) )
42	37, 36, 35	subdid 10095	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( ( w x. 1 ) - ( w x. z ) ) )
43	37	mulid1d 9678	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. 1 ) = w )
44	37, 35	mulcomd 9682	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. z ) = ( z x. w ) )
45	43, 44	oveq12d 6326	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( w x. 1 ) - ( w x. z ) ) = ( w - ( z x. w ) ) )
46	42, 45	eqtrd 2505	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( w - ( z x. w ) ) )
47	41, 46	breq12d 4408	. . . . . 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 264	. . . . 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 6316	. . . . . . . 8 |- ( x = z -> ( 1 - x ) = ( 1 - z ) )
51	49, 50	oveq12d 6326	. . . . . . 7 |- ( x = z -> ( x / ( 1 - x ) ) = ( z / ( 1 - z ) ) )
52		ovex 6336	. . . . . . 7 |- ( z / ( 1 - z ) ) e. _V
53	51, 1, 52	fvmpt 5963	. . . . . 6 |- ( z e. ( 0 [,) 1 ) -> ( F ` z ) = ( z / ( 1 - z ) ) )
54		id 22	. . . . . . . 8 |- ( x = w -> x = w )
55		oveq2 6316	. . . . . . . 8 |- ( x = w -> ( 1 - x ) = ( 1 - w ) )
56	54, 55	oveq12d 6326	. . . . . . 7 |- ( x = w -> ( x / ( 1 - x ) ) = ( w / ( 1 - w ) ) )
57		ovex 6336	. . . . . . 7 |- ( w / ( 1 - w ) ) e. _V
58	56, 1, 57	fvmpt 5963	. . . . . 6 |- ( w e. ( 0 [,) 1 ) -> ( F ` w ) = ( w / ( 1 - w ) ) )
59	53, 58	breqan12d 4411	. . . . 5 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( F ` z ) < ( F ` w ) <-> ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) ) )
60	32, 48, 59	3bitr4d 293	. . . 4 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( F ` z ) < ( F ` w ) ) )
61	60	rgen2a 2820	. . 3 |- A. z e. ( 0 [,) 1 ) A. w e. ( 0 [,) 1 ) ( z < w <-> ( F ` z ) < ( F ` w ) )
62		df-isom 5598	. . 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 934	. 2 |- F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
64		letsr 16551	. . . . . 6 |- <_ e. TosetRel
65	64	elexi 3041	. . . . 5 |- <_ e. _V
66	65	inex1 4537	. . . 4 |- ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) e. _V
67	65	inex1 4537	. . . 4 |- ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) e. _V
68		icossxr 11744	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR*
69		icossxr 11744	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR*
70		leiso 12663	. . . . . . . 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 686	. . . . . . 7 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
72	63, 71	mpbi 213	. . . . . 6 |- F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
73		isores1 6243	. . . . . 6 |- ( F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
74	72, 73	mpbi 213	. . . . 5 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
75		isores2 6242	. . . . 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 213	. . . 4 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
77		tsrps 16545	. . . . . . . 8 |- ( <_ e. TosetRel -> <_ e. PosetRel )
78	64, 77	ax-mp 5	. . . . . . 7 |- <_ e. PosetRel
79		ledm 16548	. . . . . . . 8 |- RR* = dom <_
80	79	psssdm 16540	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) 1 ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 ) )
81	78, 68, 80	mp2an 686	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 )
82	81	eqcomi 2480	. . . . 5 |- ( 0 [,) 1 ) = dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) )
83	79	psssdm 16540	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) +oo ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo ) )
84	78, 69, 83	mp2an 686	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
85	84	eqcomi 2480	. . . . 5 |- ( 0 [,) +oo ) = dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) )
86	82, 85	ordthmeo 20894	. . . 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 1390	. . 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 2471	. . . . . . 7 |- (ordTop ` <_ ) = (ordTop ` <_ )
90	88, 89	xrrest2 21904	. . . . . 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 11733	. . . . . 6 |- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) 1 ) ) -> ( x [,] y ) C_ ( 0 [,) 1 ) )
93	68, 92	ordtrestixx 20315	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
94	91, 93	eqtri 2493	. . . 4 |- ( J ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
95		rge0ssre 11766	. . . . . 6 |- ( 0 [,) +oo ) C_ RR
96	88, 89	xrrest2 21904	. . . . . 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 11733	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x [,] y ) C_ ( 0 [,) +oo ) )
99	69, 98	ordtrestixx 20315	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
100	97, 99	eqtri 2493	. . . 4 |- ( J ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
101	94, 100	oveq12i 6320	. . 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 2548	. 2 |- F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) )
103	63, 102	pm3.2i 462	1 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) /\ F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) ) )

Syntax hints:   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   i^i cin 3389   C_ wss 3390   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   `'ccnv 4838   dom cdm 4839   -1-1-onto->wf1o 5588   ` cfv 5589   Isom wiso 5590  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   +oocpnf 9690   RR*cxr 9692   < clt 9693   <_ cle 9694   - cmin 9880   / cdiv 10291   RR+crp 11325   [,)cico 11662   ↾_t crest 15397   TopOpenctopn 15398  ordTopcordt 15475   PosetRelcps 16522   TosetRel ctsr 16523  ℂ_fldccnfld 19047   Homeochmeo 20845

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-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

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  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-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-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  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-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-ordt 15477  df-ps 16524  df-tsr 16525  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cn 20320  df-hmeo 20847  df-xms 21413  df-ms 21414

This theorem is referenced by:  iccpnfhmeo  22051