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Theorem icopnfhmeo 18457
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 18456 . . . 4  |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,)  +oo )  /\  `' F  =  ( y  e.  ( 0 [,)  +oo )  |->  ( y  /  (
1  +  y ) ) ) )
32simpli 444 . . 3  |-  F :
( 0 [,) 1
)
-1-1-onto-> ( 0 [,)  +oo )
4 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
5 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
6 rexr 8893 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  1  e.  RR* )
75, 6ax-mp 8 . . . . . . . . . . 11  |-  1  e.  RR*
8 elico2 10730 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( x  e.  ( 0 [,) 1 )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <  1 ) ) )
94, 7, 8mp2an 653 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <  1
) )
109simp1bi 970 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 1 )  ->  x  e.  RR )
1110ssriv 3197 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR
1211sseli 3189 . . . . . . 7  |-  ( z  e.  ( 0 [,) 1 )  ->  z  e.  RR )
1312adantr 451 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  RR )
14 elico2 10730 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( w  e.  ( 0 [,) 1 )  <-> 
( w  e.  RR  /\  0  <_  w  /\  w  <  1 ) ) )
154, 7, 14mp2an 653 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  1
) )
1615simp3bi 972 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  w  <  1 )
1711sseli 3189 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  w  e.  RR )
18 difrp 10403 . . . . . . . . . 10  |-  ( ( w  e.  RR  /\  1  e.  RR )  ->  ( w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1917, 5, 18sylancl 643 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  (
w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
2016, 19mpbid 201 . . . . . . . 8  |-  ( w  e.  ( 0 [,) 1 )  ->  (
1  -  w )  e.  RR+ )
2120rpregt0d 10412 . . . . . . 7  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( 1  -  w
)  e.  RR  /\  0  <  ( 1  -  w ) ) )
2221adantl 452 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  w )  e.  RR  /\  0  < 
( 1  -  w
) ) )
2317adantl 452 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  RR )
24 elico2 10730 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( z  e.  ( 0 [,) 1 )  <-> 
( z  e.  RR  /\  0  <_  z  /\  z  <  1 ) ) )
254, 7, 24mp2an 653 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <  1 ) )
2625simp3bi 972 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  z  <  1 )
27 difrp 10403 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2812, 5, 27sylancl 643 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  (
z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2926, 28mpbid 201 . . . . . . . 8  |-  ( z  e.  ( 0 [,) 1 )  ->  (
1  -  z )  e.  RR+ )
3029adantr 451 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( 1  -  z )  e.  RR+ )
3130rpregt0d 10412 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  z )  e.  RR  /\  0  < 
( 1  -  z
) ) )
32 lt2mul2div 9648 . . . . . 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 ) ) ) )
3313, 22, 23, 31, 32syl22anc 1183 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  ( 1  -  w ) )  < 
( w  x.  (
1  -  z ) )  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
3413, 23remulcld 8879 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  w )  e.  RR )
3513, 23, 34ltsub1d 9397 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  -  ( z  x.  w
) )  <  (
w  -  ( z  x.  w ) ) ) )
3613recnd 8877 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  CC )
37 ax-1cn 8811 . . . . . . . . . 10  |-  1  e.  CC
3837a1i 10 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  1  e.  CC )
3923recnd 8877 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  CC )
4036, 38, 39subdid 9251 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( ( z  x.  1 )  -  ( z  x.  w ) ) )
4136mulid1d 8868 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  1 )  =  z )
4241oveq1d 5889 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  1 )  -  ( z  x.  w
) )  =  ( z  -  ( z  x.  w ) ) )
4340, 42eqtrd 2328 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( z  -  ( z  x.  w ) ) )
4439, 38, 36subdid 9251 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( ( w  x.  1 )  -  ( w  x.  z ) ) )
4539mulid1d 8868 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  1 )  =  w )
4639, 36mulcomd 8872 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  z )  =  ( z  x.  w ) )
4745, 46oveq12d 5892 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( w  x.  1 )  -  ( w  x.  z
) )  =  ( w  -  ( z  x.  w ) ) )
4844, 47eqtrd 2328 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( w  -  ( z  x.  w ) ) )
4943, 48breq12d 4052 . . . . . 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 ) ) ) )
5035, 49bitr4d 247 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  x.  ( 1  -  w
) )  <  (
w  x.  ( 1  -  z ) ) ) )
51 id 19 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
52 oveq2 5882 . . . . . . . 8  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
5351, 52oveq12d 5892 . . . . . . 7  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
54 ovex 5899 . . . . . . 7  |-  ( z  /  ( 1  -  z ) )  e. 
_V
5553, 1, 54fvmpt 5618 . . . . . 6  |-  ( z  e.  ( 0 [,) 1 )  ->  ( F `  z )  =  ( z  / 
( 1  -  z
) ) )
56 id 19 . . . . . . . 8  |-  ( x  =  w  ->  x  =  w )
57 oveq2 5882 . . . . . . . 8  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
5856, 57oveq12d 5892 . . . . . . 7  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
59 ovex 5899 . . . . . . 7  |-  ( w  /  ( 1  -  w ) )  e. 
_V
6058, 1, 59fvmpt 5618 . . . . . 6  |-  ( w  e.  ( 0 [,) 1 )  ->  ( F `  w )  =  ( w  / 
( 1  -  w
) ) )
6155, 60breqan12d 4054 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( F `
 z )  < 
( F `  w
)  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
6233, 50, 613bitr4d 276 . . . 4  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
) )
6362rgen2a 2622 . . 3  |-  A. z  e.  ( 0 [,) 1
) A. w  e.  ( 0 [,) 1
) ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
)
64 df-isom 5280 . . 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 )
) ) )
653, 63, 64mpbir2an 886 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) )
66 letsr 14365 . . . . . 6  |-  <_  e.  TosetRel
6766elexi 2810 . . . . 5  |-  <_  e.  _V
6867inex1 4171 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )  e.  _V
6967inex1 4171 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) )  e.  _V
70 icossxr 10750 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR*
71 icossxr 10750 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  RR*
72 leiso 11413 . . . . . . . 8  |-  ( ( ( 0 [,) 1
)  C_  RR*  /\  (
0 [,)  +oo )  C_  RR* )  ->  ( F  Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) ) ) )
7370, 71, 72mp2an 653 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) ) )
7465, 73mpbi 199 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) )
75 isores1 5847 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )  <->  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  <_  ( (
0 [,) 1 ) ,  ( 0 [,) 
+oo ) ) )
7674, 75mpbi 199 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  <_  ( (
0 [,) 1 ) ,  ( 0 [,) 
+oo ) )
77 isores2 5846 . . . . 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 ) ) )
7876, 77mpbi 199 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) ) ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) )
79 tsrps 14346 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
8066, 79ax-mp 8 . . . . . . 7  |-  <_  e.  PosetRel
81 ledm 14362 . . . . . . . 8  |-  RR*  =  dom  <_
8281psssdm 14341 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) )  =  ( 0 [,) 1 ) )
8380, 70, 82mp2an 653 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) )  =  ( 0 [,) 1
)
8483eqcomi 2300 . . . . 5  |-  ( 0 [,) 1 )  =  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )
8581psssdm 14341 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,)  +oo )  C_  RR* )  ->  dom  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) )  =  ( 0 [,)  +oo ) )
8680, 71, 85mp2an 653 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,)  +oo )  X.  (
0 [,)  +oo ) ) )  =  ( 0 [,)  +oo )
8786eqcomi 2300 . . . . 5  |-  ( 0 [,)  +oo )  =  dom  (  <_  i^i  ( (
0 [,)  +oo )  X.  ( 0 [,)  +oo ) ) )
8884, 87ordthmeo 17509 . . . 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 ) ) ) ) ) )
8968, 69, 78, 88mp3an 1277 . . 3  |-  F  e.  ( (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) ) ) 
Homeo  (ordTop `  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) ) ) )
90 icopnfhmeo.j . . . . . . 7  |-  J  =  ( TopOpen ` fld )
91 eqid 2296 . . . . . . 7  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
9290, 91xrrest2 18330 . . . . . 6  |-  ( ( 0 [,) 1 ) 
C_  RR  ->  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) ) )
9311, 92ax-mp 8 . . . . 5  |-  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )
94 iccssico2 10739 . . . . . 6  |-  ( ( x  e.  ( 0 [,) 1 )  /\  y  e.  ( 0 [,) 1 ) )  ->  ( x [,] y )  C_  (
0 [,) 1 ) )
9570, 94ordtrestixx 16968 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) )
9693, 95eqtri 2316 . . . 4  |-  ( Jt  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) )
97 elrege0 10762 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  <->  ( y  e.  RR  /\  0  <_ 
y ) )
9897simplbi 446 . . . . . . 7  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  RR )
9998ssriv 3197 . . . . . 6  |-  ( 0 [,)  +oo )  C_  RR
10090, 91xrrest2 18330 . . . . . 6  |-  ( ( 0 [,)  +oo )  C_  RR  ->  ( Jt  (
0 [,)  +oo ) )  =  ( (ordTop `  <_  )t  ( 0 [,)  +oo ) ) )
10199, 100ax-mp 8 . . . . 5  |-  ( Jt  ( 0 [,)  +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) 
+oo ) )
102 iccssico2 10739 . . . . . 6  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x [,] y ) 
C_  ( 0 [,) 
+oo ) )
10371, 102ordtrestixx 16968 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 
+oo ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) ) )
104101, 103eqtri 2316 . . . 4  |-  ( Jt  ( 0 [,)  +oo )
)  =  (ordTop `  (  <_  i^i  ( (
0 [,)  +oo )  X.  ( 0 [,)  +oo ) ) ) )
10596, 104oveq12i 5886 . . 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 ) ) ) ) )
10689, 105eleqtrri 2369 . 2  |-  F  e.  ( ( Jt  ( 0 [,) 1 ) ) 
Homeo  ( Jt  ( 0 [,) 
+oo ) ) )
10765, 106pm3.2i 441 1  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )  /\  F  e.  ( ( Jt  ( 0 [,) 1
) )  Homeo  ( Jt  ( 0 [,)  +oo )
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   RR+crp 10370   [,)cico 10674   ↾t crest 13341   TopOpenctopn 13342  ordTopcordt 13414   PosetRelcps 14317    TosetRel ctsr 14318  ℂfldccnfld 16393    Homeo chmeo 17460
This theorem is referenced by:  iccpnfhmeo  18459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-ordt 13418  df-ps 14322  df-tsr 14323  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-hmeo 17462  df-xms 17901  df-ms 17902
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