Theorem icopnfhmeo 20490

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 20489	. . . 4 |- ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) /\ `' F = ( y e. ( 0 [,) +oo ) |-> ( y / ( 1 + y ) ) ) )
3	2	simpli 458	. . 3 |- F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )
4		0re 9378	. . . . . . . . . . 11 |- 0 e. RR
5		1re 9377	. . . . . . . . . . . 12 |- 1 e. RR
6	5	rexri 9428	. . . . . . . . . . 11 |- 1 e. RR*
7		elico2 11351	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) ) )
8	4, 6, 7	mp2an 672	. . . . . . . . . 10 |- ( x e. ( 0 [,) 1 ) <-> ( x e. RR /\ 0 <_ x /\ x < 1 ) )
9	8	simp1bi 1003	. . . . . . . . 9 |- ( x e. ( 0 [,) 1 ) -> x e. RR )
10	9	ssriv 3355	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR
11	10	sseli 3347	. . . . . . 7 |- ( z e. ( 0 [,) 1 ) -> z e. RR )
12	11	adantr 465	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. RR )
13		elico2 11351	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) ) )
14	4, 6, 13	mp2an 672	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) <-> ( w e. RR /\ 0 <_ w /\ w < 1 ) )
15	14	simp3bi 1005	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> w < 1 )
16	10	sseli 3347	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) -> w e. RR )
17		difrp 11016	. . . . . . . . . 10 |- ( ( w e. RR /\ 1 e. RR ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
18	16, 5, 17	sylancl 662	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> ( w < 1 <-> ( 1 - w ) e. RR+ ) )
19	15, 18	mpbid 210	. . . . . . . 8 |- ( w e. ( 0 [,) 1 ) -> ( 1 - w ) e. RR+ )
20	19	rpregt0d 11025	. . . . . . 7 |- ( w e. ( 0 [,) 1 ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
21	20	adantl 466	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - w ) e. RR /\ 0 < ( 1 - w ) ) )
22	16	adantl 466	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. RR )
23		elico2 11351	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR* ) -> ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) ) )
24	4, 6, 23	mp2an 672	. . . . . . . . . 10 |- ( z e. ( 0 [,) 1 ) <-> ( z e. RR /\ 0 <_ z /\ z < 1 ) )
25	24	simp3bi 1005	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> z < 1 )
26		difrp 11016	. . . . . . . . . 10 |- ( ( z e. RR /\ 1 e. RR ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
27	11, 5, 26	sylancl 662	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> ( z < 1 <-> ( 1 - z ) e. RR+ ) )
28	25, 27	mpbid 210	. . . . . . . 8 |- ( z e. ( 0 [,) 1 ) -> ( 1 - z ) e. RR+ )
29	28	adantr 465	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( 1 - z ) e. RR+ )
30	29	rpregt0d 11025	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - z ) e. RR /\ 0 < ( 1 - z ) ) )
31		lt2mul2div 10200	. . . . . 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 1219	. . . . 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 9406	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. w ) e. RR )
34	12, 22, 33	ltsub1d 9940	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( z - ( z x. w ) ) < ( w - ( z x. w ) ) ) )
35	12	recnd 9404	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. CC )
36		ax-1cn 9332	. . . . . . . . . 10 |- 1 e. CC
37	36	a1i 11	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> 1 e. CC )
38	22	recnd 9404	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. CC )
39	35, 37, 38	subdid 9792	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( ( z x. 1 ) - ( z x. w ) ) )
40	35	mulid1d 9395	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. 1 ) = z )
41	40	oveq1d 6101	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( z x. 1 ) - ( z x. w ) ) = ( z - ( z x. w ) ) )
42	39, 41	eqtrd 2470	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( z - ( z x. w ) ) )
43	38, 37, 35	subdid 9792	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( ( w x. 1 ) - ( w x. z ) ) )
44	38	mulid1d 9395	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. 1 ) = w )
45	38, 35	mulcomd 9399	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. z ) = ( z x. w ) )
46	44, 45	oveq12d 6104	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( w x. 1 ) - ( w x. z ) ) = ( w - ( z x. w ) ) )
47	43, 46	eqtrd 2470	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( w - ( z x. w ) ) )
48	42, 47	breq12d 4300	. . . . . 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 ) ) ) )
49	34, 48	bitr4d 256	. . . . 5 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( z x. ( 1 - w ) ) < ( w x. ( 1 - z ) ) ) )
50		id 22	. . . . . . . 8 |- ( x = z -> x = z )
51		oveq2 6094	. . . . . . . 8 |- ( x = z -> ( 1 - x ) = ( 1 - z ) )
52	50, 51	oveq12d 6104	. . . . . . 7 |- ( x = z -> ( x / ( 1 - x ) ) = ( z / ( 1 - z ) ) )
53		ovex 6111	. . . . . . 7 |- ( z / ( 1 - z ) ) e. _V
54	52, 1, 53	fvmpt 5769	. . . . . 6 |- ( z e. ( 0 [,) 1 ) -> ( F ` z ) = ( z / ( 1 - z ) ) )
55		id 22	. . . . . . . 8 |- ( x = w -> x = w )
56		oveq2 6094	. . . . . . . 8 |- ( x = w -> ( 1 - x ) = ( 1 - w ) )
57	55, 56	oveq12d 6104	. . . . . . 7 |- ( x = w -> ( x / ( 1 - x ) ) = ( w / ( 1 - w ) ) )
58		ovex 6111	. . . . . . 7 |- ( w / ( 1 - w ) ) e. _V
59	57, 1, 58	fvmpt 5769	. . . . . 6 |- ( w e. ( 0 [,) 1 ) -> ( F ` w ) = ( w / ( 1 - w ) ) )
60	54, 59	breqan12d 4302	. . . . 5 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( F ` z ) < ( F ` w ) <-> ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) ) )
61	32, 49, 60	3bitr4d 285	. . . 4 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( F ` z ) < ( F ` w ) ) )
62	61	rgen2a 2777	. . 3 |- A. z e. ( 0 [,) 1 ) A. w e. ( 0 [,) 1 ) ( z < w <-> ( F ` z ) < ( F ` w ) )
63		df-isom 5422	. . 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 ) ) ) )
64	3, 62, 63	mpbir2an 911	. 2 |- F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
65		letsr 15389	. . . . . 6 |- <_ e. TosetRel
66	65	elexi 2977	. . . . 5 |- <_ e. _V
67	66	inex1 4428	. . . 4 |- ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) e. _V
68	66	inex1 4428	. . . 4 |- ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) e. _V
69		icossxr 11372	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR*
70		icossxr 11372	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR*
71		leiso 12204	. . . . . . . 8 |- ( ( ( 0 [,) 1 ) C_ RR* /\ ( 0 [,) +oo ) C_ RR* ) -> ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) ) )
72	69, 70, 71	mp2an 672	. . . . . . 7 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
73	64, 72	mpbi 208	. . . . . 6 |- F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
74		isores1 6020	. . . . . 6 |- ( F Isom <_ , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) <-> F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
75	73, 74	mpbi 208	. . . . 5 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , <_ ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
76		isores2 6019	. . . . 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 ) ) )
77	75, 76	mpbi 208	. . . 4 |- F Isom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) , ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
78		tsrps 15383	. . . . . . . 8 |- ( <_ e. TosetRel -> <_ e. PosetRel )
79	65, 78	ax-mp 5	. . . . . . 7 |- <_ e. PosetRel
80		ledm 15386	. . . . . . . 8 |- RR* = dom <_
81	80	psssdm 15378	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) 1 ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 ) )
82	79, 69, 81	mp2an 672	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) = ( 0 [,) 1 )
83	82	eqcomi 2442	. . . . 5 |- ( 0 [,) 1 ) = dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) )
84	80	psssdm 15378	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,) +oo ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo ) )
85	79, 70, 84	mp2an 672	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
86	85	eqcomi 2442	. . . . 5 |- ( 0 [,) +oo ) = dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) )
87	83, 86	ordthmeo 19350	. . . 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 ) ) ) ) ) )
88	67, 68, 77, 87	mp3an 1314	. . 3 |- F e. ( (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) ) Homeo (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) ) )
89		icopnfhmeo.j	. . . . . . 7 |- J = ( TopOpen ` ℂfld )
90		eqid 2438	. . . . . . 7 |- (ordTop ` <_ ) = (ordTop ` <_ )
91	89, 90	xrrest2 20360	. . . . . 6 |- ( ( 0 [,) 1 ) C_ RR -> ( J ↾t ( 0 [,) 1 ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) ) )
92	10, 91	ax-mp 5	. . . . 5 |- ( J ↾t ( 0 [,) 1 ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) )
93		iccssico2 11361	. . . . . 6 |- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) 1 ) ) -> ( x [,] y ) C_ ( 0 [,) 1 ) )
94	69, 93	ordtrestixx 18801	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
95	92, 94	eqtri 2458	. . . 4 |- ( J ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
96		elrege0 11384	. . . . . . . 8 |- ( y e. ( 0 [,) +oo ) <-> ( y e. RR /\ 0 <_ y ) )
97	96	simplbi 460	. . . . . . 7 |- ( y e. ( 0 [,) +oo ) -> y e. RR )
98	97	ssriv 3355	. . . . . 6 |- ( 0 [,) +oo ) C_ RR
99	89, 90	xrrest2 20360	. . . . . 6 |- ( ( 0 [,) +oo ) C_ RR -> ( J ↾t ( 0 [,) +oo ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) ) )
100	98, 99	ax-mp 5	. . . . 5 |- ( J ↾t ( 0 [,) +oo ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) )
101		iccssico2 11361	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x [,] y ) C_ ( 0 [,) +oo ) )
102	70, 101	ordtrestixx 18801	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
103	100, 102	eqtri 2458	. . . 4 |- ( J ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
104	95, 103	oveq12i 6098	. . 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 ) ) ) ) )
105	88, 104	eleqtrri 2511	. 2 |- F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) )
106	64, 105	pm3.2i 455	1 |- ( F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) /\ F e. ( ( J ↾t ( 0 [,) 1 ) ) Homeo ( J ↾t ( 0 [,) +oo ) ) ) )

Syntax hints:   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2710   _Vcvv 2967   i^i cin 3322   C_ wss 3323   class class class wbr 4287   e. cmpt 4345   X. cxp 4833   `'ccnv 4834   dom cdm 4835   -1-1-onto->wf1o 5412   ` cfv 5413   Isom wiso 5414  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   + caddc 9277   x. cmul 9279   +oocpnf 9407   RR*cxr 9409   < clt 9410   <_ cle 9411   - cmin 9587   / cdiv 9985   RR+crp 10983   [,)cico 11294   ↾_t crest 14351   TopOpenctopn 14352  ordTopcordt 14429   PosetRelcps 15360   TosetRel ctsr 15361  ℂ_fldccnfld 17793   Homeochmeo 19301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-mulr 14244  df-starv 14245  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-rest 14353  df-topn 14354  df-topgen 14374  df-ordt 14431  df-ps 15362  df-tsr 15363  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cn 18806  df-hmeo 19303  df-xms 19870  df-ms 19871

This theorem is referenced by:  iccpnfhmeo  20492