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Theorem icopnfhmeo 18960
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 18959 . . . 4  |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,)  +oo )  /\  `' F  =  ( y  e.  ( 0 [,)  +oo )  |->  ( y  /  (
1  +  y ) ) ) )
32simpli 445 . . 3  |-  F :
( 0 [,) 1
)
-1-1-onto-> ( 0 [,)  +oo )
4 0re 9083 . . . . . . . . . . 11  |-  0  e.  RR
5 1re 9082 . . . . . . . . . . . 12  |-  1  e.  RR
65rexri 9129 . . . . . . . . . . 11  |-  1  e.  RR*
7 elico2 10966 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( x  e.  ( 0 [,) 1 )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <  1 ) ) )
84, 6, 7mp2an 654 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <  1
) )
98simp1bi 972 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 1 )  ->  x  e.  RR )
109ssriv 3344 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR
1110sseli 3336 . . . . . . 7  |-  ( z  e.  ( 0 [,) 1 )  ->  z  e.  RR )
1211adantr 452 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  RR )
13 elico2 10966 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( w  e.  ( 0 [,) 1 )  <-> 
( w  e.  RR  /\  0  <_  w  /\  w  <  1 ) ) )
144, 6, 13mp2an 654 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  1
) )
1514simp3bi 974 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  w  <  1 )
1610sseli 3336 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  w  e.  RR )
17 difrp 10637 . . . . . . . . . 10  |-  ( ( w  e.  RR  /\  1  e.  RR )  ->  ( w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1816, 5, 17sylancl 644 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  (
w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1915, 18mpbid 202 . . . . . . . 8  |-  ( w  e.  ( 0 [,) 1 )  ->  (
1  -  w )  e.  RR+ )
2019rpregt0d 10646 . . . . . . 7  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( 1  -  w
)  e.  RR  /\  0  <  ( 1  -  w ) ) )
2120adantl 453 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  w )  e.  RR  /\  0  < 
( 1  -  w
) ) )
2216adantl 453 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  RR )
23 elico2 10966 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( z  e.  ( 0 [,) 1 )  <-> 
( z  e.  RR  /\  0  <_  z  /\  z  <  1 ) ) )
244, 6, 23mp2an 654 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <  1 ) )
2524simp3bi 974 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  z  <  1 )
26 difrp 10637 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2711, 5, 26sylancl 644 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  (
z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2825, 27mpbid 202 . . . . . . . 8  |-  ( z  e.  ( 0 [,) 1 )  ->  (
1  -  z )  e.  RR+ )
2928adantr 452 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( 1  -  z )  e.  RR+ )
3029rpregt0d 10646 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  z )  e.  RR  /\  0  < 
( 1  -  z
) ) )
31 lt2mul2div 9878 . . . . . 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 1185 . . . . 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 9108 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  w )  e.  RR )
3412, 22, 33ltsub1d 9627 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  -  ( z  x.  w
) )  <  (
w  -  ( z  x.  w ) ) ) )
3512recnd 9106 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  CC )
36 ax-1cn 9040 . . . . . . . . . 10  |-  1  e.  CC
3736a1i 11 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  1  e.  CC )
3822recnd 9106 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  CC )
3935, 37, 38subdid 9481 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( ( z  x.  1 )  -  ( z  x.  w ) ) )
4035mulid1d 9097 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  1 )  =  z )
4140oveq1d 6088 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  1 )  -  ( z  x.  w
) )  =  ( z  -  ( z  x.  w ) ) )
4239, 41eqtrd 2467 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( z  -  ( z  x.  w ) ) )
4338, 37, 35subdid 9481 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( ( w  x.  1 )  -  ( w  x.  z ) ) )
4438mulid1d 9097 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  1 )  =  w )
4538, 35mulcomd 9101 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  z )  =  ( z  x.  w ) )
4644, 45oveq12d 6091 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( w  x.  1 )  -  ( w  x.  z
) )  =  ( w  -  ( z  x.  w ) ) )
4743, 46eqtrd 2467 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( w  -  ( z  x.  w ) ) )
4842, 47breq12d 4217 . . . . . 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 248 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  x.  ( 1  -  w
) )  <  (
w  x.  ( 1  -  z ) ) ) )
50 id 20 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
51 oveq2 6081 . . . . . . . 8  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
5250, 51oveq12d 6091 . . . . . . 7  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
53 ovex 6098 . . . . . . 7  |-  ( z  /  ( 1  -  z ) )  e. 
_V
5452, 1, 53fvmpt 5798 . . . . . 6  |-  ( z  e.  ( 0 [,) 1 )  ->  ( F `  z )  =  ( z  / 
( 1  -  z
) ) )
55 id 20 . . . . . . . 8  |-  ( x  =  w  ->  x  =  w )
56 oveq2 6081 . . . . . . . 8  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
5755, 56oveq12d 6091 . . . . . . 7  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
58 ovex 6098 . . . . . . 7  |-  ( w  /  ( 1  -  w ) )  e. 
_V
5957, 1, 58fvmpt 5798 . . . . . 6  |-  ( w  e.  ( 0 [,) 1 )  ->  ( F `  w )  =  ( w  / 
( 1  -  w
) ) )
6054, 59breqan12d 4219 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( F `
 z )  < 
( F `  w
)  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
6132, 49, 603bitr4d 277 . . . 4  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
) )
6261rgen2a 2764 . . 3  |-  A. z  e.  ( 0 [,) 1
) A. w  e.  ( 0 [,) 1
) ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
)
63 df-isom 5455 . . 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 887 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) )
65 letsr 14664 . . . . . 6  |-  <_  e.  TosetRel
6665elexi 2957 . . . . 5  |-  <_  e.  _V
6766inex1 4336 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )  e.  _V
6866inex1 4336 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) )  e.  _V
69 icossxr 10987 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR*
70 icossxr 10987 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  RR*
71 leiso 11700 . . . . . . . 8  |-  ( ( ( 0 [,) 1
)  C_  RR*  /\  (
0 [,)  +oo )  C_  RR* )  ->  ( F  Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) ) ) )
7269, 70, 71mp2an 654 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) ) )
7364, 72mpbi 200 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) )
74 isores1 6046 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )  <->  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  <_  ( (
0 [,) 1 ) ,  ( 0 [,) 
+oo ) ) )
7573, 74mpbi 200 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  <_  ( (
0 [,) 1 ) ,  ( 0 [,) 
+oo ) )
76 isores2 6045 . . . . 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 200 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) ) ( ( 0 [,) 1 ) ,  ( 0 [,)  +oo ) )
78 tsrps 14645 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
7965, 78ax-mp 8 . . . . . . 7  |-  <_  e.  PosetRel
80 ledm 14661 . . . . . . . 8  |-  RR*  =  dom  <_
8180psssdm 14640 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) )  =  ( 0 [,) 1 ) )
8279, 69, 81mp2an 654 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) )  =  ( 0 [,) 1
)
8382eqcomi 2439 . . . . 5  |-  ( 0 [,) 1 )  =  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )
8480psssdm 14640 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,)  +oo )  C_  RR* )  ->  dom  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) )  =  ( 0 [,)  +oo ) )
8579, 70, 84mp2an 654 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,)  +oo )  X.  (
0 [,)  +oo ) ) )  =  ( 0 [,)  +oo )
8685eqcomi 2439 . . . . 5  |-  ( 0 [,)  +oo )  =  dom  (  <_  i^i  ( (
0 [,)  +oo )  X.  ( 0 [,)  +oo ) ) )
8783, 86ordthmeo 17826 . . . 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 1279 . . 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 2435 . . . . . . 7  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
9189, 90xrrest2 18831 . . . . . 6  |-  ( ( 0 [,) 1 ) 
C_  RR  ->  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) ) )
9210, 91ax-mp 8 . . . . 5  |-  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )
93 iccssico2 10976 . . . . . 6  |-  ( ( x  e.  ( 0 [,) 1 )  /\  y  e.  ( 0 [,) 1 ) )  ->  ( x [,] y )  C_  (
0 [,) 1 ) )
9469, 93ordtrestixx 17278 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) )
9592, 94eqtri 2455 . . . 4  |-  ( Jt  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) )
96 elrege0 10999 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  <->  ( y  e.  RR  /\  0  <_ 
y ) )
9796simplbi 447 . . . . . . 7  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  RR )
9897ssriv 3344 . . . . . 6  |-  ( 0 [,)  +oo )  C_  RR
9989, 90xrrest2 18831 . . . . . 6  |-  ( ( 0 [,)  +oo )  C_  RR  ->  ( Jt  (
0 [,)  +oo ) )  =  ( (ordTop `  <_  )t  ( 0 [,)  +oo ) ) )
10098, 99ax-mp 8 . . . . 5  |-  ( Jt  ( 0 [,)  +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) 
+oo ) )
101 iccssico2 10976 . . . . . 6  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x [,] y ) 
C_  ( 0 [,) 
+oo ) )
10270, 101ordtrestixx 17278 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 
+oo ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 
+oo )  X.  (
0 [,)  +oo ) ) ) )
103100, 102eqtri 2455 . . . 4  |-  ( Jt  ( 0 [,)  +oo )
)  =  (ordTop `  (  <_  i^i  ( (
0 [,)  +oo )  X.  ( 0 [,)  +oo ) ) ) )
10495, 103oveq12i 6085 . . 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 2508 . 2  |-  F  e.  ( ( Jt  ( 0 [,) 1 ) ) 
Homeo  ( Jt  ( 0 [,) 
+oo ) ) )
10664, 105pm3.2i 442 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   dom cdm 4870   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   RR+crp 10604   [,)cico 10910   ↾t crest 13640   TopOpenctopn 13641  ordTopcordt 13713   PosetRelcps 14616    TosetRel ctsr 14617  ℂfldccnfld 16695    Homeo chmeo 17777
This theorem is referenced by:  iccpnfhmeo  18962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-rest 13642  df-topn 13643  df-topgen 13659  df-ordt 13717  df-ps 14621  df-tsr 14622  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cn 17283  df-hmeo 17779  df-xms 18342  df-ms 18343
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