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Theorem icopnfhmeo 21971
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 21970 . . . 4  |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )  /\  `' F  =  ( y  e.  ( 0 [,) +oo )  |->  ( y  /  (
1  +  y ) ) ) )
32simpli 460 . . 3  |-  F :
( 0 [,) 1
)
-1-1-onto-> ( 0 [,) +oo )
4 0re 9643 . . . . . . . . . . 11  |-  0  e.  RR
5 1re 9642 . . . . . . . . . . . 12  |-  1  e.  RR
65rexri 9693 . . . . . . . . . . 11  |-  1  e.  RR*
7 elico2 11698 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( x  e.  ( 0 [,) 1 )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <  1 ) ) )
84, 6, 7mp2an 678 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <  1
) )
98simp1bi 1023 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 1 )  ->  x  e.  RR )
109ssriv 3436 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR
1110sseli 3428 . . . . . . 7  |-  ( z  e.  ( 0 [,) 1 )  ->  z  e.  RR )
1211adantr 467 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  RR )
13 elico2 11698 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( w  e.  ( 0 [,) 1 )  <-> 
( w  e.  RR  /\  0  <_  w  /\  w  <  1 ) ) )
144, 6, 13mp2an 678 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  1
) )
1514simp3bi 1025 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  w  <  1 )
1610sseli 3428 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  w  e.  RR )
17 difrp 11337 . . . . . . . . . 10  |-  ( ( w  e.  RR  /\  1  e.  RR )  ->  ( w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1816, 5, 17sylancl 668 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  (
w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1915, 18mpbid 214 . . . . . . . 8  |-  ( w  e.  ( 0 [,) 1 )  ->  (
1  -  w )  e.  RR+ )
2019rpregt0d 11347 . . . . . . 7  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( 1  -  w
)  e.  RR  /\  0  <  ( 1  -  w ) ) )
2120adantl 468 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  w )  e.  RR  /\  0  < 
( 1  -  w
) ) )
2216adantl 468 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  RR )
23 elico2 11698 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( z  e.  ( 0 [,) 1 )  <-> 
( z  e.  RR  /\  0  <_  z  /\  z  <  1 ) ) )
244, 6, 23mp2an 678 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <  1 ) )
2524simp3bi 1025 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  z  <  1 )
26 difrp 11337 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2711, 5, 26sylancl 668 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  (
z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2825, 27mpbid 214 . . . . . . . 8  |-  ( z  e.  ( 0 [,) 1 )  ->  (
1  -  z )  e.  RR+ )
2928adantr 467 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( 1  -  z )  e.  RR+ )
3029rpregt0d 11347 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  z )  e.  RR  /\  0  < 
( 1  -  z
) ) )
31 lt2mul2div 10483 . . . . . 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 1269 . . . . 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 9671 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  w )  e.  RR )
3412, 22, 33ltsub1d 10222 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  -  ( z  x.  w
) )  <  (
w  -  ( z  x.  w ) ) ) )
3512recnd 9669 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  CC )
36 1cnd 9659 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  1  e.  CC )
3722recnd 9669 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  CC )
3835, 36, 37subdid 10074 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( ( z  x.  1 )  -  ( z  x.  w ) ) )
3935mulid1d 9660 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  1 )  =  z )
4039oveq1d 6305 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  1 )  -  ( z  x.  w
) )  =  ( z  -  ( z  x.  w ) ) )
4138, 40eqtrd 2485 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( z  -  ( z  x.  w ) ) )
4237, 36, 35subdid 10074 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( ( w  x.  1 )  -  ( w  x.  z ) ) )
4337mulid1d 9660 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  1 )  =  w )
4437, 35mulcomd 9664 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  z )  =  ( z  x.  w ) )
4543, 44oveq12d 6308 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( w  x.  1 )  -  ( w  x.  z
) )  =  ( w  -  ( z  x.  w ) ) )
4642, 45eqtrd 2485 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( w  -  ( z  x.  w ) ) )
4741, 46breq12d 4415 . . . . . 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 ) ) ) )
4834, 47bitr4d 260 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  x.  ( 1  -  w
) )  <  (
w  x.  ( 1  -  z ) ) ) )
49 id 22 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
50 oveq2 6298 . . . . . . . 8  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
5149, 50oveq12d 6308 . . . . . . 7  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
52 ovex 6318 . . . . . . 7  |-  ( z  /  ( 1  -  z ) )  e. 
_V
5351, 1, 52fvmpt 5948 . . . . . 6  |-  ( z  e.  ( 0 [,) 1 )  ->  ( F `  z )  =  ( z  / 
( 1  -  z
) ) )
54 id 22 . . . . . . . 8  |-  ( x  =  w  ->  x  =  w )
55 oveq2 6298 . . . . . . . 8  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
5654, 55oveq12d 6308 . . . . . . 7  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
57 ovex 6318 . . . . . . 7  |-  ( w  /  ( 1  -  w ) )  e. 
_V
5856, 1, 57fvmpt 5948 . . . . . 6  |-  ( w  e.  ( 0 [,) 1 )  ->  ( F `  w )  =  ( w  / 
( 1  -  w
) ) )
5953, 58breqan12d 4418 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( F `
 z )  < 
( F `  w
)  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
6032, 48, 593bitr4d 289 . . . 4  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
) )
6160rgen2a 2815 . . 3  |-  A. z  e.  ( 0 [,) 1
) A. w  e.  ( 0 [,) 1
) ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
)
62 df-isom 5591 . . 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 ) ) ) )
633, 61, 62mpbir2an 931 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
64 letsr 16473 . . . . . 6  |-  <_  e.  TosetRel
6564elexi 3055 . . . . 5  |-  <_  e.  _V
6665inex1 4544 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )  e.  _V
6765inex1 4544 . . . 4  |-  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) )  e.  _V
68 icossxr 11719 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR*
69 icossxr 11719 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR*
70 leiso 12622 . . . . . . . 8  |-  ( ( ( 0 [,) 1
)  C_  RR*  /\  (
0 [,) +oo )  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) ) )
7168, 69, 70mp2an 678 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) )
7263, 71mpbi 212 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
73 isores1 6225 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) )
7472, 73mpbi 212 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  <_  ( (
0 [,) 1 ) ,  ( 0 [,) +oo ) )
75 isores2 6224 . . . . 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 ) ) )
7674, 75mpbi 212 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) ) ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
77 tsrps 16467 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
7864, 77ax-mp 5 . . . . . . 7  |-  <_  e.  PosetRel
79 ledm 16470 . . . . . . . 8  |-  RR*  =  dom  <_
8079psssdm 16462 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) )  =  ( 0 [,) 1 ) )
8178, 68, 80mp2an 678 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) )  =  ( 0 [,) 1
)
8281eqcomi 2460 . . . . 5  |-  ( 0 [,) 1 )  =  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )
8379psssdm 16462 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) +oo )  C_ 
RR* )  ->  dom  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo ) )
8478, 69, 83mp2an 678 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo )
8584eqcomi 2460 . . . . 5  |-  ( 0 [,) +oo )  =  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) )
8682, 85ordthmeo 20817 . . . 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 ) ) ) ) ) )
8766, 67, 76, 86mp3an 1364 . . 3  |-  F  e.  ( (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) ) )
Homeo (ordTop `  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) ) ) )
88 icopnfhmeo.j . . . . . . 7  |-  J  =  ( TopOpen ` fld )
89 eqid 2451 . . . . . . 7  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
9088, 89xrrest2 21826 . . . . . 6  |-  ( ( 0 [,) 1 ) 
C_  RR  ->  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) ) )
9110, 90ax-mp 5 . . . . 5  |-  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )
92 iccssico2 11708 . . . . . 6  |-  ( ( x  e.  ( 0 [,) 1 )  /\  y  e.  ( 0 [,) 1 ) )  ->  ( x [,] y )  C_  (
0 [,) 1 ) )
9368, 92ordtrestixx 20238 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) )
9491, 93eqtri 2473 . . . 4  |-  ( Jt  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) )
95 rge0ssre 11740 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
9688, 89xrrest2 21826 . . . . . 6  |-  ( ( 0 [,) +oo )  C_  RR  ->  ( Jt  (
0 [,) +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) +oo ) ) )
9795, 96ax-mp 5 . . . . 5  |-  ( Jt  ( 0 [,) +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) +oo ) )
98 iccssico2 11708 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x [,] y )  C_  (
0 [,) +oo )
)
9969, 98ordtrestixx 20238 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) +oo ) )  =  (ordTop `  (  <_  i^i  (
( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
10097, 99eqtri 2473 . . . 4  |-  ( Jt  ( 0 [,) +oo )
)  =  (ordTop `  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
10194, 100oveq12i 6302 . . 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 ) ) ) ) )
10287, 101eleqtrri 2528 . 2  |-  F  e.  ( ( Jt  ( 0 [,) 1 ) )
Homeo ( Jt  ( 0 [,) +oo ) ) )
10363, 102pm3.2i 457 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    i^i cin 3403    C_ wss 3404   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   `'ccnv 4833   dom cdm 4834   -1-1-onto->wf1o 5581   ` cfv 5582    Isom wiso 5583  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   RR+crp 11302   [,)cico 11637   ↾t crest 15319   TopOpenctopn 15320  ordTopcordt 15397   PosetRelcps 16444    TosetRel ctsr 16445  ℂfldccnfld 18970   Homeochmeo 20768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-topgen 15342  df-ordt 15399  df-ps 16446  df-tsr 16447  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cn 20243  df-hmeo 20770  df-xms 21335  df-ms 21336
This theorem is referenced by:  iccpnfhmeo  21973
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