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Theorem icopnfhmeo 22049
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 22048 . . . 4  |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )  /\  `' F  =  ( y  e.  ( 0 [,) +oo )  |->  ( y  /  (
1  +  y ) ) ) )
32simpli 465 . . 3  |-  F :
( 0 [,) 1
)
-1-1-onto-> ( 0 [,) +oo )
4 0re 9661 . . . . . . . . . . 11  |-  0  e.  RR
5 1re 9660 . . . . . . . . . . . 12  |-  1  e.  RR
65rexri 9711 . . . . . . . . . . 11  |-  1  e.  RR*
7 elico2 11723 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( x  e.  ( 0 [,) 1 )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <  1 ) ) )
84, 6, 7mp2an 686 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <  1
) )
98simp1bi 1045 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 1 )  ->  x  e.  RR )
109ssriv 3422 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR
1110sseli 3414 . . . . . . 7  |-  ( z  e.  ( 0 [,) 1 )  ->  z  e.  RR )
1211adantr 472 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  RR )
13 elico2 11723 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( w  e.  ( 0 [,) 1 )  <-> 
( w  e.  RR  /\  0  <_  w  /\  w  <  1 ) ) )
144, 6, 13mp2an 686 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  1
) )
1514simp3bi 1047 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  w  <  1 )
1610sseli 3414 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  w  e.  RR )
17 difrp 11360 . . . . . . . . . 10  |-  ( ( w  e.  RR  /\  1  e.  RR )  ->  ( w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1816, 5, 17sylancl 675 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  (
w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1915, 18mpbid 215 . . . . . . . 8  |-  ( w  e.  ( 0 [,) 1 )  ->  (
1  -  w )  e.  RR+ )
2019rpregt0d 11370 . . . . . . 7  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( 1  -  w
)  e.  RR  /\  0  <  ( 1  -  w ) ) )
2120adantl 473 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  w )  e.  RR  /\  0  < 
( 1  -  w
) ) )
2216adantl 473 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  RR )
23 elico2 11723 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( z  e.  ( 0 [,) 1 )  <-> 
( z  e.  RR  /\  0  <_  z  /\  z  <  1 ) ) )
244, 6, 23mp2an 686 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <  1 ) )
2524simp3bi 1047 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  z  <  1 )
26 difrp 11360 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2711, 5, 26sylancl 675 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  (
z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2825, 27mpbid 215 . . . . . . . 8  |-  ( z  e.  ( 0 [,) 1 )  ->  (
1  -  z )  e.  RR+ )
2928adantr 472 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( 1  -  z )  e.  RR+ )
3029rpregt0d 11370 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  z )  e.  RR  /\  0  < 
( 1  -  z
) ) )
31 lt2mul2div 10505 . . . . . 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 1293 . . . . 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 9689 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  w )  e.  RR )
3412, 22, 33ltsub1d 10243 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  -  ( z  x.  w
) )  <  (
w  -  ( z  x.  w ) ) ) )
3512recnd 9687 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  CC )
36 1cnd 9677 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  1  e.  CC )
3722recnd 9687 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  CC )
3835, 36, 37subdid 10095 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( ( z  x.  1 )  -  ( z  x.  w ) ) )
3935mulid1d 9678 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  1 )  =  z )
4039oveq1d 6323 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  1 )  -  ( z  x.  w
) )  =  ( z  -  ( z  x.  w ) ) )
4138, 40eqtrd 2505 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( z  -  ( z  x.  w ) ) )
4237, 36, 35subdid 10095 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( ( w  x.  1 )  -  ( w  x.  z ) ) )
4337mulid1d 9678 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  1 )  =  w )
4437, 35mulcomd 9682 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  z )  =  ( z  x.  w ) )
4543, 44oveq12d 6326 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( w  x.  1 )  -  ( w  x.  z
) )  =  ( w  -  ( z  x.  w ) ) )
4642, 45eqtrd 2505 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( w  -  ( z  x.  w ) ) )
4741, 46breq12d 4408 . . . . . 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 264 . . . . 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 6316 . . . . . . . 8  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
5149, 50oveq12d 6326 . . . . . . 7  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
52 ovex 6336 . . . . . . 7  |-  ( z  /  ( 1  -  z ) )  e. 
_V
5351, 1, 52fvmpt 5963 . . . . . 6  |-  ( z  e.  ( 0 [,) 1 )  ->  ( F `  z )  =  ( z  / 
( 1  -  z
) ) )
54 id 22 . . . . . . . 8  |-  ( x  =  w  ->  x  =  w )
55 oveq2 6316 . . . . . . . 8  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
5654, 55oveq12d 6326 . . . . . . 7  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
57 ovex 6336 . . . . . . 7  |-  ( w  /  ( 1  -  w ) )  e. 
_V
5856, 1, 57fvmpt 5963 . . . . . 6  |-  ( w  e.  ( 0 [,) 1 )  ->  ( F `  w )  =  ( w  / 
( 1  -  w
) ) )
5953, 58breqan12d 4411 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( F `
 z )  < 
( F `  w
)  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
6032, 48, 593bitr4d 293 . . . 4  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
) )
6160rgen2a 2820 . . 3  |-  A. z  e.  ( 0 [,) 1
) A. w  e.  ( 0 [,) 1
) ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
)
62 df-isom 5598 . . 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 934 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
64 letsr 16551 . . . . . 6  |-  <_  e.  TosetRel
6564elexi 3041 . . . . 5  |-  <_  e.  _V
6665inex1 4537 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )  e.  _V
6765inex1 4537 . . . 4  |-  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) )  e.  _V
68 icossxr 11744 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR*
69 icossxr 11744 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR*
70 leiso 12663 . . . . . . . 8  |-  ( ( ( 0 [,) 1
)  C_  RR*  /\  (
0 [,) +oo )  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) ) )
7168, 69, 70mp2an 686 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) )
7263, 71mpbi 213 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
73 isores1 6243 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) )
7472, 73mpbi 213 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  <_  ( (
0 [,) 1 ) ,  ( 0 [,) +oo ) )
75 isores2 6242 . . . . 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 213 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) ) ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
77 tsrps 16545 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
7864, 77ax-mp 5 . . . . . . 7  |-  <_  e.  PosetRel
79 ledm 16548 . . . . . . . 8  |-  RR*  =  dom  <_
8079psssdm 16540 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) )  =  ( 0 [,) 1 ) )
8178, 68, 80mp2an 686 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) )  =  ( 0 [,) 1
)
8281eqcomi 2480 . . . . 5  |-  ( 0 [,) 1 )  =  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )
8379psssdm 16540 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) +oo )  C_ 
RR* )  ->  dom  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo ) )
8478, 69, 83mp2an 686 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo )
8584eqcomi 2480 . . . . 5  |-  ( 0 [,) +oo )  =  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) )
8682, 85ordthmeo 20894 . . . 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 1390 . . 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 2471 . . . . . . 7  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
9088, 89xrrest2 21904 . . . . . 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 11733 . . . . . 6  |-  ( ( x  e.  ( 0 [,) 1 )  /\  y  e.  ( 0 [,) 1 ) )  ->  ( x [,] y )  C_  (
0 [,) 1 ) )
9368, 92ordtrestixx 20315 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) )
9491, 93eqtri 2493 . . . 4  |-  ( Jt  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) )
95 rge0ssre 11766 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
9688, 89xrrest2 21904 . . . . . 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 11733 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x [,] y )  C_  (
0 [,) +oo )
)
9969, 98ordtrestixx 20315 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) +oo ) )  =  (ordTop `  (  <_  i^i  (
( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
10097, 99eqtri 2493 . . . 4  |-  ( Jt  ( 0 [,) +oo )
)  =  (ordTop `  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
10194, 100oveq12i 6320 . . 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 2548 . 2  |-  F  e.  ( ( Jt  ( 0 [,) 1 ) )
Homeo ( Jt  ( 0 [,) +oo ) ) )
10363, 102pm3.2i 462 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    i^i cin 3389    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   dom cdm 4839   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   RR+crp 11325   [,)cico 11662   ↾t crest 15397   TopOpenctopn 15398  ordTopcordt 15475   PosetRelcps 16522    TosetRel ctsr 16523  ℂfldccnfld 19047   Homeochmeo 20845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-ordt 15477  df-ps 16524  df-tsr 16525  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cn 20320  df-hmeo 20847  df-xms 21413  df-ms 21414
This theorem is referenced by:  iccpnfhmeo  22051
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