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Theorem icopnfhmeo 18441
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 18440 . . . 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 8838 . . . . . . . . . . 11  |-  0  e.  RR
5 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
6 rexr 8877 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  1  e.  RR* )
75, 6ax-mp 8 . . . . . . . . . . 11  |-  1  e.  RR*
8 elico2 10714 . . . . . . . . . . 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 3184 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR
1211sseli 3176 . . . . . . 7  |-  ( z  e.  ( 0 [,) 1 )  ->  z  e.  RR )
1312adantr 451 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  RR )
14 elico2 10714 . . . . . . . . . . 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 3176 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  w  e.  RR )
18 difrp 10387 . . . . . . . . . 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 10396 . . . . . . 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 10714 . . . . . . . . . . 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 10387 . . . . . . . . . 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 10396 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  z )  e.  RR  /\  0  < 
( 1  -  z
) ) )
32 lt2mul2div 9632 . . . . . 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 8863 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  w )  e.  RR )
3513, 23, 34ltsub1d 9381 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  -  ( z  x.  w
) )  <  (
w  -  ( z  x.  w ) ) ) )
3613recnd 8861 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  CC )
37 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
3837a1i 10 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  1  e.  CC )
3923recnd 8861 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  CC )
4036, 38, 39subdid 9235 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( ( z  x.  1 )  -  ( z  x.  w ) ) )
4136mulid1d 8852 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  1 )  =  z )
4241oveq1d 5873 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  1 )  -  ( z  x.  w
) )  =  ( z  -  ( z  x.  w ) ) )
4340, 42eqtrd 2315 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( z  -  ( z  x.  w ) ) )
4439, 38, 36subdid 9235 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( ( w  x.  1 )  -  ( w  x.  z ) ) )
4539mulid1d 8852 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  1 )  =  w )
4639, 36mulcomd 8856 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  z )  =  ( z  x.  w ) )
4745, 46oveq12d 5876 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( w  x.  1 )  -  ( w  x.  z
) )  =  ( w  -  ( z  x.  w ) ) )
4844, 47eqtrd 2315 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( w  -  ( z  x.  w ) ) )
4943, 48breq12d 4036 . . . . . 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 5866 . . . . . . . 8  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
5351, 52oveq12d 5876 . . . . . . 7  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
54 ovex 5883 . . . . . . 7  |-  ( z  /  ( 1  -  z ) )  e. 
_V
5553, 1, 54fvmpt 5602 . . . . . 6  |-  ( z  e.  ( 0 [,) 1 )  ->  ( F `  z )  =  ( z  / 
( 1  -  z
) ) )
56 id 19 . . . . . . . 8  |-  ( x  =  w  ->  x  =  w )
57 oveq2 5866 . . . . . . . 8  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
5856, 57oveq12d 5876 . . . . . . 7  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
59 ovex 5883 . . . . . . 7  |-  ( w  /  ( 1  -  w ) )  e. 
_V
6058, 1, 59fvmpt 5602 . . . . . 6  |-  ( w  e.  ( 0 [,) 1 )  ->  ( F `  w )  =  ( w  / 
( 1  -  w
) ) )
6155, 60breqan12d 4038 . . . . 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 2609 . . 3  |-  A. z  e.  ( 0 [,) 1
) A. w  e.  ( 0 [,) 1
) ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
)
64 df-isom 5264 . . 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 14349 . . . . . 6  |-  <_  e.  TosetRel
6766elexi 2797 . . . . 5  |-  <_  e.  _V
6867inex1 4155 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )  e.  _V
6967inex1 4155 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) )  e.  _V
70 icossxr 10734 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR*
71 icossxr 10734 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  RR*
72 leiso 11397 . . . . . . . 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 5831 . . . . . 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 5830 . . . . 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 14330 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
8066, 79ax-mp 8 . . . . . . 7  |-  <_  e.  PosetRel
81 ledm 14346 . . . . . . . 8  |-  RR*  =  dom  <_
8281psssdm 14325 . . . . . . 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 2287 . . . . 5  |-  ( 0 [,) 1 )  =  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )
8581psssdm 14325 . . . . . . 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 2287 . . . . 5  |-  ( 0 [,)  +oo )  =  dom  (  <_  i^i  ( (
0 [,)  +oo )  X.  ( 0 [,)  +oo ) ) )
8884, 87ordthmeo 17493 . . . 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 2283 . . . . . . 7  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
9290, 91xrrest2 18314 . . . . . 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 10723 . . . . . 6  |-  ( ( x  e.  ( 0 [,) 1 )  /\  y  e.  ( 0 [,) 1 ) )  ->  ( x [,] y )  C_  (
0 [,) 1 ) )
9570, 94ordtrestixx 16952 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) )
9693, 95eqtri 2303 . . . 4  |-  ( Jt  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) )
97 elrege0 10746 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  <->  ( y  e.  RR  /\  0  <_ 
y ) )
9897simplbi 446 . . . . . . 7  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  RR )
9998ssriv 3184 . . . . . 6  |-  ( 0 [,)  +oo )  C_  RR
10090, 91xrrest2 18314 . . . . . 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 10723 . . . . . 6  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x [,] y ) 
C_  ( 0 [,) 
+oo ) )
10371, 102ordtrestixx 16952 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 
+oo ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) ) )
104101, 103eqtri 2303 . . . 4  |-  ( Jt  ( 0 [,)  +oo )
)  =  (ordTop `  (  <_  i^i  ( (
0 [,)  +oo )  X.  ( 0 [,)  +oo ) ) ) )
10596, 104oveq12i 5870 . . 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 2356 . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   RR+crp 10354   [,)cico 10658   ↾t crest 13325   TopOpenctopn 13326  ordTopcordt 13398   PosetRelcps 14301    TosetRel ctsr 14302  ℂfldccnfld 16377    Homeo chmeo 17444
This theorem is referenced by:  iccpnfhmeo  18443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-ordt 13402  df-ps 14306  df-tsr 14307  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-hmeo 17446  df-xms 17885  df-ms 17886
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