Theorem icopnfhmeo 21178

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 21177	. . . 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 9592	. . . . . . . . . . 11 |- 0 e. RR
5		1re 9591	. . . . . . . . . . . 12 |- 1 e. RR
6	5	rexri 9642	. . . . . . . . . . 11 |- 1 e. RR*
7		elico2 11584	. . . . . . . . . . 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 1011	. . . . . . . . 9 |- ( x e. ( 0 [,) 1 ) -> x e. RR )
10	9	ssriv 3508	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR
11	10	sseli 3500	. . . . . . 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 11584	. . . . . . . . . . 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 1013	. . . . . . . . 9 |- ( w e. ( 0 [,) 1 ) -> w < 1 )
16	10	sseli 3500	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) -> w e. RR )
17		difrp 11249	. . . . . . . . . 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 11258	. . . . . . 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 11584	. . . . . . . . . . 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 1013	. . . . . . . . 9 |- ( z e. ( 0 [,) 1 ) -> z < 1 )
26		difrp 11249	. . . . . . . . . 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 11258	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( 1 - z ) e. RR /\ 0 < ( 1 - z ) ) )
31		lt2mul2div 10417	. . . . . 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 1229	. . . . 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 9620	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. w ) e. RR )
34	12, 22, 33	ltsub1d 10157	. . . . . 6 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z < w <-> ( z - ( z x. w ) ) < ( w - ( z x. w ) ) ) )
35	12	recnd 9618	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> z e. CC )
36		ax-1cn 9546	. . . . . . . . . 10 |- 1 e. CC
37	36	a1i 11	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> 1 e. CC )
38	22	recnd 9618	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> w e. CC )
39	35, 37, 38	subdid 10008	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( ( z x. 1 ) - ( z x. w ) ) )
40	35	mulid1d 9609	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. 1 ) = z )
41	40	oveq1d 6297	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( z x. 1 ) - ( z x. w ) ) = ( z - ( z x. w ) ) )
42	39, 41	eqtrd 2508	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( z x. ( 1 - w ) ) = ( z - ( z x. w ) ) )
43	38, 37, 35	subdid 10008	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( ( w x. 1 ) - ( w x. z ) ) )
44	38	mulid1d 9609	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. 1 ) = w )
45	38, 35	mulcomd 9613	. . . . . . . . 9 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. z ) = ( z x. w ) )
46	44, 45	oveq12d 6300	. . . . . . . 8 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( ( w x. 1 ) - ( w x. z ) ) = ( w - ( z x. w ) ) )
47	43, 46	eqtrd 2508	. . . . . . 7 |- ( ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) -> ( w x. ( 1 - z ) ) = ( w - ( z x. w ) ) )
48	42, 47	breq12d 4460	. . . . . 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 6290	. . . . . . . 8 |- ( x = z -> ( 1 - x ) = ( 1 - z ) )
52	50, 51	oveq12d 6300	. . . . . . 7 |- ( x = z -> ( x / ( 1 - x ) ) = ( z / ( 1 - z ) ) )
53		ovex 6307	. . . . . . 7 |- ( z / ( 1 - z ) ) e. _V
54	52, 1, 53	fvmpt 5948	. . . . . 6 |- ( z e. ( 0 [,) 1 ) -> ( F ` z ) = ( z / ( 1 - z ) ) )
55		id 22	. . . . . . . 8 |- ( x = w -> x = w )
56		oveq2 6290	. . . . . . . 8 |- ( x = w -> ( 1 - x ) = ( 1 - w ) )
57	55, 56	oveq12d 6300	. . . . . . 7 |- ( x = w -> ( x / ( 1 - x ) ) = ( w / ( 1 - w ) ) )
58		ovex 6307	. . . . . . 7 |- ( w / ( 1 - w ) ) e. _V
59	57, 1, 58	fvmpt 5948	. . . . . 6 |- ( w e. ( 0 [,) 1 ) -> ( F ` w ) = ( w / ( 1 - w ) ) )
60	54, 59	breqan12d 4462	. . . . 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 2891	. . 3 |- A. z e. ( 0 [,) 1 ) A. w e. ( 0 [,) 1 ) ( z < w <-> ( F ` z ) < ( F ` w ) )
63		df-isom 5595	. . 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 918	. 2 |- F Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
65		letsr 15710	. . . . . 6 |- <_ e. TosetRel
66	65	elexi 3123	. . . . 5 |- <_ e. _V
67	66	inex1 4588	. . . 4 |- ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) e. _V
68	66	inex1 4588	. . . 4 |- ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) e. _V
69		icossxr 11605	. . . . . . . 8 |- ( 0 [,) 1 ) C_ RR*
70		icossxr 11605	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR*
71		leiso 12470	. . . . . . . 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 6216	. . . . . 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 6215	. . . . 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 15704	. . . . . . . 8 |- ( <_ e. TosetRel -> <_ e. PosetRel )
79	65, 78	ax-mp 5	. . . . . . 7 |- <_ e. PosetRel
80		ledm 15707	. . . . . . . 8 |- RR* = dom <_
81	80	psssdm 15699	. . . . . . 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 2480	. . . . 5 |- ( 0 [,) 1 ) = dom ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) )
84	80	psssdm 15699	. . . . . . 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 2480	. . . . 5 |- ( 0 [,) +oo ) = dom ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) )
87	83, 86	ordthmeo 20038	. . . 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 1324	. . 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 2467	. . . . . . 7 |- (ordTop ` <_ ) = (ordTop ` <_ )
91	89, 90	xrrest2 21048	. . . . . 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 11594	. . . . . 6 |- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) 1 ) ) -> ( x [,] y ) C_ ( 0 [,) 1 ) )
94	69, 93	ordtrestixx 19489	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
95	92, 94	eqtri 2496	. . . 4 |- ( J ↾t ( 0 [,) 1 ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) 1 ) X. ( 0 [,) 1 ) ) ) )
96		elrege0 11623	. . . . . . . 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 3508	. . . . . 6 |- ( 0 [,) +oo ) C_ RR
99	89, 90	xrrest2 21048	. . . . . 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 11594	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x [,] y ) C_ ( 0 [,) +oo ) )
102	70, 101	ordtrestixx 19489	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
103	100, 102	eqtri 2496	. . . 4 |- ( J ↾t ( 0 [,) +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,) +oo ) X. ( 0 [,) +oo ) ) ) )
104	95, 103	oveq12i 6294	. . 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 2554	. 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 973   = wceq 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   i^i cin 3475   C_ wss 3476   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   `'ccnv 4998   dom cdm 4999   -1-1-onto->wf1o 5585   ` cfv 5586   Isom wiso 5587  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   + caddc 9491   x. cmul 9493   +oocpnf 9621   RR*cxr 9623   < clt 9624   <_ cle 9625   - cmin 9801   / cdiv 10202   RR+crp 11216   [,)cico 11527   ↾_t crest 14672   TopOpenctopn 14673  ordTopcordt 14750   PosetRelcps 15681   TosetRel ctsr 15682  ℂ_fldccnfld 18191   Homeochmeo 19989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-rest 14674  df-topn 14675  df-topgen 14695  df-ordt 14752  df-ps 15683  df-tsr 15684  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cn 19494  df-hmeo 19991  df-xms 20558  df-ms 20559

This theorem is referenced by:  iccpnfhmeo  21180