MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  icopnfhmeo Structured version   Unicode version

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 ` fld )
Assertion
Ref Expression
icopnfhmeo  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  /\  F  e.  ( ( Jt  ( 0 [,) 1 ) ) Homeo ( Jt  ( 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.
StepHypRef Expression
1 icopnfhmeo.f . . . . 5  |-  F  =  ( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) )
21icopnfcnv 20489 . . . 4  |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )  /\  `' F  =  ( y  e.  ( 0 [,) +oo )  |->  ( y  /  (
1  +  y ) ) ) )
32simpli 458 . . 3  |-  F :
( 0 [,) 1
)
-1-1-onto-> ( 0 [,) +oo )
4 0re 9378 . . . . . . . . . . 11  |-  0  e.  RR
5 1re 9377 . . . . . . . . . . . 12  |-  1  e.  RR
65rexri 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 ) ) )
84, 6, 7mp2an 672 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <  1
) )
98simp1bi 1003 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 1 )  ->  x  e.  RR )
109ssriv 3355 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR
1110sseli 3347 . . . . . . 7  |-  ( z  e.  ( 0 [,) 1 )  ->  z  e.  RR )
1211adantr 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 ) ) )
144, 6, 13mp2an 672 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  1
) )
1514simp3bi 1005 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  w  <  1 )
1610sseli 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+ ) )
1816, 5, 17sylancl 662 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  (
w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1915, 18mpbid 210 . . . . . . . 8  |-  ( w  e.  ( 0 [,) 1 )  ->  (
1  -  w )  e.  RR+ )
2019rpregt0d 11025 . . . . . . 7  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( 1  -  w
)  e.  RR  /\  0  <  ( 1  -  w ) ) )
2120adantl 466 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  w )  e.  RR  /\  0  < 
( 1  -  w
) ) )
2216adantl 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 ) ) )
244, 6, 23mp2an 672 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <  1 ) )
2524simp3bi 1005 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  z  <  1 )
26 difrp 11016 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2711, 5, 26sylancl 662 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  (
z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2825, 27mpbid 210 . . . . . . . 8  |-  ( z  e.  ( 0 [,) 1 )  ->  (
1  -  z )  e.  RR+ )
2928adantr 465 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( 1  -  z )  e.  RR+ )
3029rpregt0d 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 ) ) ) )
3212, 21, 22, 30, 31syl22anc 1219 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  ( 1  -  w ) )  < 
( w  x.  (
1  -  z ) )  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
3312, 22remulcld 9406 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  w )  e.  RR )
3412, 22, 33ltsub1d 9940 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  -  ( z  x.  w
) )  <  (
w  -  ( z  x.  w ) ) ) )
3512recnd 9404 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  CC )
36 ax-1cn 9332 . . . . . . . . . 10  |-  1  e.  CC
3736a1i 11 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  1  e.  CC )
3822recnd 9404 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  CC )
3935, 37, 38subdid 9792 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( ( z  x.  1 )  -  ( z  x.  w ) ) )
4035mulid1d 9395 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  1 )  =  z )
4140oveq1d 6101 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  1 )  -  ( z  x.  w
) )  =  ( z  -  ( z  x.  w ) ) )
4239, 41eqtrd 2470 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( z  -  ( z  x.  w ) ) )
4338, 37, 35subdid 9792 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( ( w  x.  1 )  -  ( w  x.  z ) ) )
4438mulid1d 9395 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  1 )  =  w )
4538, 35mulcomd 9399 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  z )  =  ( z  x.  w ) )
4644, 45oveq12d 6104 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( w  x.  1 )  -  ( w  x.  z
) )  =  ( w  -  ( z  x.  w ) ) )
4743, 46eqtrd 2470 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( w  -  ( z  x.  w ) ) )
4842, 47breq12d 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 ) ) ) )
4934, 48bitr4d 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 ) )
5250, 51oveq12d 6104 . . . . . . 7  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
53 ovex 6111 . . . . . . 7  |-  ( z  /  ( 1  -  z ) )  e. 
_V
5452, 1, 53fvmpt 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 ) )
5755, 56oveq12d 6104 . . . . . . 7  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
58 ovex 6111 . . . . . . 7  |-  ( w  /  ( 1  -  w ) )  e. 
_V
5957, 1, 58fvmpt 5769 . . . . . 6  |-  ( w  e.  ( 0 [,) 1 )  ->  ( F `  w )  =  ( w  / 
( 1  -  w
) ) )
6054, 59breqan12d 4302 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( F `
 z )  < 
( F `  w
)  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
6132, 49, 603bitr4d 285 . . . 4  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
) )
6261rgen2a 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 ) ) ) )
643, 62, 63mpbir2an 911 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
65 letsr 15389 . . . . . 6  |-  <_  e.  TosetRel
6665elexi 2977 . . . . 5  |-  <_  e.  _V
6766inex1 4428 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )  e.  _V
6866inex1 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 ) ) ) )
7269, 70, 71mp2an 672 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) )
7364, 72mpbi 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 ) ) )
7573, 74mpbi 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 ) ) )
7775, 76mpbi 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 )
7965, 78ax-mp 5 . . . . . . 7  |-  <_  e.  PosetRel
80 ledm 15386 . . . . . . . 8  |-  RR*  =  dom  <_
8180psssdm 15378 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) )  =  ( 0 [,) 1 ) )
8279, 69, 81mp2an 672 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) )  =  ( 0 [,) 1
)
8382eqcomi 2442 . . . . 5  |-  ( 0 [,) 1 )  =  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )
8480psssdm 15378 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) +oo )  C_ 
RR* )  ->  dom  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo ) )
8579, 70, 84mp2an 672 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo )
8685eqcomi 2442 . . . . 5  |-  ( 0 [,) +oo )  =  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) )
8783, 86ordthmeo 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 ) ) ) ) ) )
8867, 68, 77, 87mp3an 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 ` fld )
90 eqid 2438 . . . . . . 7  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
9189, 90xrrest2 20360 . . . . . 6  |-  ( ( 0 [,) 1 ) 
C_  RR  ->  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) ) )
9210, 91ax-mp 5 . . . . 5  |-  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )
93 iccssico2 11361 . . . . . 6  |-  ( ( x  e.  ( 0 [,) 1 )  /\  y  e.  ( 0 [,) 1 ) )  ->  ( x [,] y )  C_  (
0 [,) 1 ) )
9469, 93ordtrestixx 18801 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) )
9592, 94eqtri 2458 . . . 4  |-  ( Jt  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) )
96 elrege0 11384 . . . . . . . 8  |-  ( y  e.  ( 0 [,) +oo )  <->  ( y  e.  RR  /\  0  <_ 
y ) )
9796simplbi 460 . . . . . . 7  |-  ( y  e.  ( 0 [,) +oo )  ->  y  e.  RR )
9897ssriv 3355 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
9989, 90xrrest2 20360 . . . . . 6  |-  ( ( 0 [,) +oo )  C_  RR  ->  ( Jt  (
0 [,) +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) +oo ) ) )
10098, 99ax-mp 5 . . . . 5  |-  ( Jt  ( 0 [,) +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) +oo ) )
101 iccssico2 11361 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x [,] y )  C_  (
0 [,) +oo )
)
10270, 101ordtrestixx 18801 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) +oo ) )  =  (ordTop `  (  <_  i^i  (
( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
103100, 102eqtri 2458 . . . 4  |-  ( Jt  ( 0 [,) +oo )
)  =  (ordTop `  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
10495, 103oveq12i 6098 . . 3  |-  ( ( Jt  ( 0 [,) 1
) ) Homeo ( Jt  ( 0 [,) +oo )
) )  =  ( (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) ) Homeo (ordTop `  (  <_  i^i  (
( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) ) )
10588, 104eleqtrri 2511 . 2  |-  F  e.  ( ( Jt  ( 0 [,) 1 ) )
Homeo ( Jt  ( 0 [,) +oo ) ) )
10664, 105pm3.2i 455 1  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  /\  F  e.  ( ( Jt  ( 0 [,) 1 ) ) Homeo ( Jt  ( 0 [,) +oo ) ) ) )
Colors of variables: wff setvar class
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  ℂfldccnfld 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
  Copyright terms: Public domain W3C validator