Theorem iccpnfhmeo 22744

Description: The defined bijection from [ 0 , 1 ] to [ 0 , +oo ] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)

Hypotheses

Ref	Expression
iccpnfhmeo.f	|- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) )
iccpnfhmeo.k	|- K = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )

Assertion

Ref	Expression
iccpnfhmeo	|- ( F Isom < , < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) /\ F e. ( II Homeo K ) )

Proof of Theorem iccpnfhmeo

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

Step	Hyp	Ref	Expression
1		iccssxr 12256	. . . 4 |- ( 0 [,] 1 ) C_ RR*
2		xrltso 11974	. . . 4 |- < Or RR*
3		soss 5053	. . . 4 |- ( ( 0 [,] 1 ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] 1 ) ) )
4	1, 2, 3	mp2 9	. . 3 |- < Or ( 0 [,] 1 )
5		iccssxr 12256	. . . . 5 |- ( 0 [,] +oo ) C_ RR*
6		soss 5053	. . . . 5 |- ( ( 0 [,] +oo ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] +oo ) ) )
7	5, 2, 6	mp2 9	. . . 4 |- < Or ( 0 [,] +oo )
8		sopo 5052	. . . 4 |- ( < Or ( 0 [,] +oo ) -> < Po ( 0 [,] +oo ) )
9	7, 8	ax-mp 5	. . 3 |- < Po ( 0 [,] +oo )
10		iccpnfhmeo.f	. . . . . 6 |- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) )
11	10	iccpnfcnv 22743	. . . . 5 |- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) ) )
12	11	simpli 474	. . . 4 |- F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )
13		f1ofo 6144	. . . 4 |- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) -> F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo ) )
14	12, 13	ax-mp 5	. . 3 |- F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo )
15		0re 10040	. . . . . . . . . . . . 13 |- 0 e. RR
16		1re 10039	. . . . . . . . . . . . 13 |- 1 e. RR
17	15, 16	elicc2i 12239	. . . . . . . . . . . 12 |- ( z e. ( 0 [,] 1 ) <-> ( z e. RR /\ 0 <_ z /\ z <_ 1 ) )
18	17	simp1bi 1076	. . . . . . . . . . 11 |- ( z e. ( 0 [,] 1 ) -> z e. RR )
19	18	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z e. RR )
20	15, 16	elicc2i 12239	. . . . . . . . . . . . 13 |- ( w e. ( 0 [,] 1 ) <-> ( w e. RR /\ 0 <_ w /\ w <_ 1 ) )
21	20	simp1bi 1076	. . . . . . . . . . . 12 |- ( w e. ( 0 [,] 1 ) -> w e. RR )
22	21	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> w e. RR )
23		1red 10055	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> 1 e. RR )
24		simp3 1063	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z < w )
25	20	simp3bi 1078	. . . . . . . . . . . 12 |- ( w e. ( 0 [,] 1 ) -> w <_ 1 )
26	25	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> w <_ 1 )
27	19, 22, 23, 24, 26	ltletrd 10197	. . . . . . . . . 10 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z < 1 )
28	19, 27	gtned 10172	. . . . . . . . 9 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> 1 =/= z )
29	28	necomd 2849	. . . . . . . 8 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z =/= 1 )
30		ifnefalse 4098	. . . . . . . 8 |- ( z =/= 1 -> if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) = ( z / ( 1 - z ) ) )
31	29, 30	syl 17	. . . . . . 7 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) = ( z / ( 1 - z ) ) )
32		breq2 4657	. . . . . . . 8 |- ( +oo = if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) -> ( ( z / ( 1 - z ) ) < +oo <-> ( z / ( 1 - z ) ) < if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) ) )
33		breq2 4657	. . . . . . . 8 |- ( ( w / ( 1 - w ) ) = if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) -> ( ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) <-> ( z / ( 1 - z ) ) < if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) ) )
34		resubcl 10345	. . . . . . . . . . . 12 |- ( ( 1 e. RR /\ z e. RR ) -> ( 1 - z ) e. RR )
35	16, 19, 34	sylancr 695	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( 1 - z ) e. RR )
36		ax-1cn 9994	. . . . . . . . . . . . 13 |- 1 e. CC
37	19	recnd 10068	. . . . . . . . . . . . 13 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z e. CC )
38		subeq0 10307	. . . . . . . . . . . . . 14 |- ( ( 1 e. CC /\ z e. CC ) -> ( ( 1 - z ) = 0 <-> 1 = z ) )
39	38	necon3bid 2838	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ z e. CC ) -> ( ( 1 - z ) =/= 0 <-> 1 =/= z ) )
40	36, 37, 39	sylancr 695	. . . . . . . . . . . 12 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( ( 1 - z ) =/= 0 <-> 1 =/= z ) )
41	28, 40	mpbird 247	. . . . . . . . . . 11 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( 1 - z ) =/= 0 )
42	19, 35, 41	redivcld 10853	. . . . . . . . . 10 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( z / ( 1 - z ) ) e. RR )
43		ltpnf 11954	. . . . . . . . . 10 |- ( ( z / ( 1 - z ) ) e. RR -> ( z / ( 1 - z ) ) < +oo )
44	42, 43	syl 17	. . . . . . . . 9 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( z / ( 1 - z ) ) < +oo )
45	44	adantr 481	. . . . . . . 8 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ w = 1 ) -> ( z / ( 1 - z ) ) < +oo )
46		simpl3 1066	. . . . . . . . . 10 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> z < w )
47		eqid 2622	. . . . . . . . . . . . . 14 |- ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) = ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) )
48		eqid 2622	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
49	47, 48	icopnfhmeo 22742	. . . . . . . . . . . . 13 |- ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) /\ ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) 1 ) ) Homeo ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) ) )
50	49	simpli 474	. . . . . . . . . . . 12 |- ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) )
51	50	a1i 11	. . . . . . . . . . 11 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) )
52		simp1 1061	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z e. ( 0 [,] 1 ) )
53		0xr 10086	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR*
54	16	rexri 10097	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR*
55		0le1 10551	. . . . . . . . . . . . . . . . . . 19 |- 0 <_ 1
56		snunico 12299	. . . . . . . . . . . . . . . . . . 19 |- ( ( 0 e. RR* /\ 1 e. RR* /\ 0 <_ 1 ) -> ( ( 0 [,) 1 ) u. { 1 } ) = ( 0 [,] 1 ) )
57	53, 54, 55, 56	mp3an 1424	. . . . . . . . . . . . . . . . . 18 |- ( ( 0 [,) 1 ) u. { 1 } ) = ( 0 [,] 1 )
58	52, 57	syl6eleqr 2712	. . . . . . . . . . . . . . . . 17 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z e. ( ( 0 [,) 1 ) u. { 1 } ) )
59		elun 3753	. . . . . . . . . . . . . . . . 17 |- ( z e. ( ( 0 [,) 1 ) u. { 1 } ) <-> ( z e. ( 0 [,) 1 ) \/ z e. { 1 } ) )
60	58, 59	sylib 208	. . . . . . . . . . . . . . . 16 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( z e. ( 0 [,) 1 ) \/ z e. { 1 } ) )
61	60	ord 392	. . . . . . . . . . . . . . 15 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( -. z e. ( 0 [,) 1 ) -> z e. { 1 } ) )
62		elsni 4194	. . . . . . . . . . . . . . 15 |- ( z e. { 1 } -> z = 1 )
63	61, 62	syl6 35	. . . . . . . . . . . . . 14 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( -. z e. ( 0 [,) 1 ) -> z = 1 ) )
64	63	necon1ad 2811	. . . . . . . . . . . . 13 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( z =/= 1 -> z e. ( 0 [,) 1 ) ) )
65	29, 64	mpd 15	. . . . . . . . . . . 12 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> z e. ( 0 [,) 1 ) )
66	65	adantr 481	. . . . . . . . . . 11 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> z e. ( 0 [,) 1 ) )
67		simp2 1062	. . . . . . . . . . . . . . . . 17 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> w e. ( 0 [,] 1 ) )
68	67, 57	syl6eleqr 2712	. . . . . . . . . . . . . . . 16 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> w e. ( ( 0 [,) 1 ) u. { 1 } ) )
69		elun 3753	. . . . . . . . . . . . . . . 16 |- ( w e. ( ( 0 [,) 1 ) u. { 1 } ) <-> ( w e. ( 0 [,) 1 ) \/ w e. { 1 } ) )
70	68, 69	sylib 208	. . . . . . . . . . . . . . 15 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( w e. ( 0 [,) 1 ) \/ w e. { 1 } ) )
71	70	ord 392	. . . . . . . . . . . . . 14 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( -. w e. ( 0 [,) 1 ) -> w e. { 1 } ) )
72		elsni 4194	. . . . . . . . . . . . . 14 |- ( w e. { 1 } -> w = 1 )
73	71, 72	syl6 35	. . . . . . . . . . . . 13 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( -. w e. ( 0 [,) 1 ) -> w = 1 ) )
74	73	con1d 139	. . . . . . . . . . . 12 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( -. w = 1 -> w e. ( 0 [,) 1 ) ) )
75	74	imp 445	. . . . . . . . . . 11 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> w e. ( 0 [,) 1 ) )
76		isorel 6576	. . . . . . . . . . 11 |- ( ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +oo ) ) /\ ( z e. ( 0 [,) 1 ) /\ w e. ( 0 [,) 1 ) ) ) -> ( z < w <-> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` z ) < ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` w ) ) )
77	51, 66, 75, 76	syl12anc 1324	. . . . . . . . . 10 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> ( z < w <-> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` z ) < ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` w ) ) )
78	46, 77	mpbid 222	. . . . . . . . 9 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` z ) < ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` w ) )
79		id 22	. . . . . . . . . . . 12 |- ( x = z -> x = z )
80		oveq2 6658	. . . . . . . . . . . 12 |- ( x = z -> ( 1 - x ) = ( 1 - z ) )
81	79, 80	oveq12d 6668	. . . . . . . . . . 11 |- ( x = z -> ( x / ( 1 - x ) ) = ( z / ( 1 - z ) ) )
82		ovex 6678	. . . . . . . . . . 11 |- ( z / ( 1 - z ) ) e. _V
83	81, 47, 82	fvmpt 6282	. . . . . . . . . 10 |- ( z e. ( 0 [,) 1 ) -> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` z ) = ( z / ( 1 - z ) ) )
84	66, 83	syl 17	. . . . . . . . 9 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` z ) = ( z / ( 1 - z ) ) )
85		id 22	. . . . . . . . . . . 12 |- ( x = w -> x = w )
86		oveq2 6658	. . . . . . . . . . . 12 |- ( x = w -> ( 1 - x ) = ( 1 - w ) )
87	85, 86	oveq12d 6668	. . . . . . . . . . 11 |- ( x = w -> ( x / ( 1 - x ) ) = ( w / ( 1 - w ) ) )
88		ovex 6678	. . . . . . . . . . 11 |- ( w / ( 1 - w ) ) e. _V
89	87, 47, 88	fvmpt 6282	. . . . . . . . . 10 |- ( w e. ( 0 [,) 1 ) -> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` w ) = ( w / ( 1 - w ) ) )
90	75, 89	syl 17	. . . . . . . . 9 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` w ) = ( w / ( 1 - w ) ) )
91	78, 84, 90	3brtr3d 4684	. . . . . . . 8 |- ( ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) /\ -. w = 1 ) -> ( z / ( 1 - z ) ) < ( w / ( 1 - w ) ) )
92	32, 33, 45, 91	ifbothda 4123	. . . . . . 7 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> ( z / ( 1 - z ) ) < if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) )
93	31, 92	eqbrtrd 4675	. . . . . 6 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) /\ z < w ) -> if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) < if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) )
94	93	3expia 1267	. . . . 5 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) -> ( z < w -> if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) < if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) ) )
95		eqeq1 2626	. . . . . . . 8 |- ( x = z -> ( x = 1 <-> z = 1 ) )
96	95, 81	ifbieq2d 4111	. . . . . . 7 |- ( x = z -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) = if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) )
97		pnfex 10093	. . . . . . . 8 |- +oo e. _V
98	97, 82	ifex 4156	. . . . . . 7 |- if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) e. _V
99	96, 10, 98	fvmpt 6282	. . . . . 6 |- ( z e. ( 0 [,] 1 ) -> ( F ` z ) = if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) )
100		eqeq1 2626	. . . . . . . 8 |- ( x = w -> ( x = 1 <-> w = 1 ) )
101	100, 87	ifbieq2d 4111	. . . . . . 7 |- ( x = w -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) = if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) )
102	97, 88	ifex 4156	. . . . . . 7 |- if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) e. _V
103	101, 10, 102	fvmpt 6282	. . . . . 6 |- ( w e. ( 0 [,] 1 ) -> ( F ` w ) = if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) )
104	99, 103	breqan12d 4669	. . . . 5 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) -> ( ( F ` z ) < ( F ` w ) <-> if ( z = 1 , +oo , ( z / ( 1 - z ) ) ) < if ( w = 1 , +oo , ( w / ( 1 - w ) ) ) ) )
105	94, 104	sylibrd 249	. . . 4 |- ( ( z e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) -> ( z < w -> ( F ` z ) < ( F ` w ) ) )
106	105	rgen2a 2977	. . 3 |- A. z e. ( 0 [,] 1 ) A. w e. ( 0 [,] 1 ) ( z < w -> ( F ` z ) < ( F ` w ) )
107		soisoi 6578	. . 3 |- ( ( ( < Or ( 0 [,] 1 ) /\ < Po ( 0 [,] +oo ) ) /\ ( F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo ) /\ A. z e. ( 0 [,] 1 ) A. w e. ( 0 [,] 1 ) ( z < w -> ( F ` z ) < ( F ` w ) ) ) ) -> F Isom < , < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) )
108	4, 9, 14, 106, 107	mp4an 709	. 2 |- F Isom < , < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) )
109		letsr 17227	. . . . . 6 |- <_ e. TosetRel
110	109	elexi 3213	. . . . 5 |- <_ e. _V
111	110	inex1 4799	. . . 4 |- ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) e. _V
112	110	inex1 4799	. . . 4 |- ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) e. _V
113		leiso 13243	. . . . . . . 8 |- ( ( ( 0 [,] 1 ) C_ RR* /\ ( 0 [,] +oo ) C_ RR* ) -> ( F Isom < , < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) ) )
114	1, 5, 113	mp2an 708	. . . . . . 7 |- ( F Isom < , < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) <-> F Isom <_ , <_ ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) )
115	108, 114	mpbi 220	. . . . . 6 |- F Isom <_ , <_ ( ( 0 [,] 1 ) , ( 0 [,] +oo ) )
116		isores1 6584	. . . . . 6 |- ( F Isom <_ , <_ ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) <-> F Isom ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) , <_ ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) )
117	115, 116	mpbi 220	. . . . 5 |- F Isom ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) , <_ ( ( 0 [,] 1 ) , ( 0 [,] +oo ) )
118		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 ) ) )
119	117, 118	mpbi 220	. . . 4 |- F Isom ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) , ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) ( ( 0 [,] 1 ) , ( 0 [,] +oo ) )
120		tsrps 17221	. . . . . . . 8 |- ( <_ e. TosetRel -> <_ e. PosetRel )
121	109, 120	ax-mp 5	. . . . . . 7 |- <_ e. PosetRel
122		ledm 17224	. . . . . . . 8 |- RR* = dom <_
123	122	psssdm 17216	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,] 1 ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) = ( 0 [,] 1 ) )
124	121, 1, 123	mp2an 708	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) = ( 0 [,] 1 )
125	124	eqcomi 2631	. . . . 5 |- ( 0 [,] 1 ) = dom ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
126	122	psssdm 17216	. . . . . . 7 |- ( ( <_ e. PosetRel /\ ( 0 [,] +oo ) C_ RR* ) -> dom ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) = ( 0 [,] +oo ) )
127	121, 5, 126	mp2an 708	. . . . . 6 |- dom ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) = ( 0 [,] +oo )
128	127	eqcomi 2631	. . . . 5 |- ( 0 [,] +oo ) = dom ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) )
129	125, 128	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 ) ) ) ) ) )
130	111, 112, 119, 129	mp3an 1424	. . 3 |- F e. ( (ordTop ` ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) ) Homeo (ordTop ` ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) ) )
131		dfii5 22688	. . . 4 |- II = (ordTop ` ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) )
132		iccpnfhmeo.k	. . . . 5 |- K = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
133		ordtresticc 21027	. . . . 5 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = (ordTop ` ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) )
134	132, 133	eqtri 2644	. . . 4 |- K = (ordTop ` ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) )
135	131, 134	oveq12i 6662	. . 3 |- ( II Homeo K ) = ( (ordTop ` ( <_ i^i ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) ) Homeo (ordTop ` ( <_ i^i ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) ) )
136	130, 135	eleqtrri 2700	. 2 |- F e. ( II Homeo K )
137	108, 136	pm3.2i 471	1 |- ( F Isom < , < ( ( 0 [,] 1 ) , ( 0 [,] +oo ) ) /\ F e. ( II Homeo K ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   Po wpo 5033   Or wor 5034   X. cxp 5112   `'ccnv 5113   dom cdm 5114   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   [,)cico 12177   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082  ordTopcordt 16159   PosetRelcps 17198   TosetRel ctsr 17199  ℂ_fldccnfld 19746   Homeochmeo 21556   IIcii 22678

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  df-ii 22680

This theorem is referenced by:  xrhmeo  22745  xrge0hmph  29978