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Theorem icopnfhmeo 20640
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 20639 . . . 4  |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo )  /\  `' F  =  ( y  e.  ( 0 [,) +oo )  |->  ( y  /  (
1  +  y ) ) ) )
32simpli 458 . . 3  |-  F :
( 0 [,) 1
)
-1-1-onto-> ( 0 [,) +oo )
4 0re 9490 . . . . . . . . . . 11  |-  0  e.  RR
5 1re 9489 . . . . . . . . . . . 12  |-  1  e.  RR
65rexri 9540 . . . . . . . . . . 11  |-  1  e.  RR*
7 elico2 11463 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( x  e.  ( 0 [,) 1 )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <  1 ) ) )
84, 6, 7mp2an 672 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <  1
) )
98simp1bi 1003 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 1 )  ->  x  e.  RR )
109ssriv 3461 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR
1110sseli 3453 . . . . . . 7  |-  ( z  e.  ( 0 [,) 1 )  ->  z  e.  RR )
1211adantr 465 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  RR )
13 elico2 11463 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( w  e.  ( 0 [,) 1 )  <-> 
( w  e.  RR  /\  0  <_  w  /\  w  <  1 ) ) )
144, 6, 13mp2an 672 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <  1
) )
1514simp3bi 1005 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  w  <  1 )
1610sseli 3453 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  w  e.  RR )
17 difrp 11128 . . . . . . . . . 10  |-  ( ( w  e.  RR  /\  1  e.  RR )  ->  ( w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1816, 5, 17sylancl 662 . . . . . . . . 9  |-  ( w  e.  ( 0 [,) 1 )  ->  (
w  <  1  <->  ( 1  -  w )  e.  RR+ ) )
1915, 18mpbid 210 . . . . . . . 8  |-  ( w  e.  ( 0 [,) 1 )  ->  (
1  -  w )  e.  RR+ )
2019rpregt0d 11137 . . . . . . 7  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( 1  -  w
)  e.  RR  /\  0  <  ( 1  -  w ) ) )
2120adantl 466 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  w )  e.  RR  /\  0  < 
( 1  -  w
) ) )
2216adantl 466 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  RR )
23 elico2 11463 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR* )  -> 
( z  e.  ( 0 [,) 1 )  <-> 
( z  e.  RR  /\  0  <_  z  /\  z  <  1 ) ) )
244, 6, 23mp2an 672 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <  1 ) )
2524simp3bi 1005 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  z  <  1 )
26 difrp 11128 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2711, 5, 26sylancl 662 . . . . . . . . 9  |-  ( z  e.  ( 0 [,) 1 )  ->  (
z  <  1  <->  ( 1  -  z )  e.  RR+ ) )
2825, 27mpbid 210 . . . . . . . 8  |-  ( z  e.  ( 0 [,) 1 )  ->  (
1  -  z )  e.  RR+ )
2928adantr 465 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( 1  -  z )  e.  RR+ )
3029rpregt0d 11137 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( 1  -  z )  e.  RR  /\  0  < 
( 1  -  z
) ) )
31 lt2mul2div 10312 . . . . . 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 1220 . . . . 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 9518 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  w )  e.  RR )
3412, 22, 33ltsub1d 10052 . . . . . 6  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  -  ( z  x.  w
) )  <  (
w  -  ( z  x.  w ) ) ) )
3512recnd 9516 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  z  e.  CC )
36 ax-1cn 9444 . . . . . . . . . 10  |-  1  e.  CC
3736a1i 11 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  1  e.  CC )
3822recnd 9516 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  w  e.  CC )
3935, 37, 38subdid 9904 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( ( z  x.  1 )  -  ( z  x.  w ) ) )
4035mulid1d 9507 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  1 )  =  z )
4140oveq1d 6208 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( z  x.  1 )  -  ( z  x.  w
) )  =  ( z  -  ( z  x.  w ) ) )
4239, 41eqtrd 2492 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  x.  ( 1  -  w
) )  =  ( z  -  ( z  x.  w ) ) )
4338, 37, 35subdid 9904 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( ( w  x.  1 )  -  ( w  x.  z ) ) )
4438mulid1d 9507 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  1 )  =  w )
4538, 35mulcomd 9511 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  z )  =  ( z  x.  w ) )
4644, 45oveq12d 6211 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( w  x.  1 )  -  ( w  x.  z
) )  =  ( w  -  ( z  x.  w ) ) )
4743, 46eqtrd 2492 . . . . . . 7  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( w  x.  ( 1  -  z
) )  =  ( w  -  ( z  x.  w ) ) )
4842, 47breq12d 4406 . . . . . 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 256 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( z  x.  ( 1  -  w
) )  <  (
w  x.  ( 1  -  z ) ) ) )
50 id 22 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
51 oveq2 6201 . . . . . . . 8  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
5250, 51oveq12d 6211 . . . . . . 7  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
53 ovex 6218 . . . . . . 7  |-  ( z  /  ( 1  -  z ) )  e. 
_V
5452, 1, 53fvmpt 5876 . . . . . 6  |-  ( z  e.  ( 0 [,) 1 )  ->  ( F `  z )  =  ( z  / 
( 1  -  z
) ) )
55 id 22 . . . . . . . 8  |-  ( x  =  w  ->  x  =  w )
56 oveq2 6201 . . . . . . . 8  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
5755, 56oveq12d 6211 . . . . . . 7  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
58 ovex 6218 . . . . . . 7  |-  ( w  /  ( 1  -  w ) )  e. 
_V
5957, 1, 58fvmpt 5876 . . . . . 6  |-  ( w  e.  ( 0 [,) 1 )  ->  ( F `  w )  =  ( w  / 
( 1  -  w
) ) )
6054, 59breqan12d 4408 . . . . 5  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( ( F `
 z )  < 
( F `  w
)  <->  ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) ) ) )
6132, 49, 603bitr4d 285 . . . 4  |-  ( ( z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) )  ->  ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
) )
6261rgen2a 2893 . . 3  |-  A. z  e.  ( 0 [,) 1
) A. w  e.  ( 0 [,) 1
) ( z  < 
w  <->  ( F `  z )  <  ( F `  w )
)
63 df-isom 5528 . . 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 911 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
65 letsr 15508 . . . . . 6  |-  <_  e.  TosetRel
6665elexi 3081 . . . . 5  |-  <_  e.  _V
6766inex1 4534 . . . 4  |-  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )  e.  _V
6866inex1 4534 . . . 4  |-  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) )  e.  _V
69 icossxr 11484 . . . . . . . 8  |-  ( 0 [,) 1 )  C_  RR*
70 icossxr 11484 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR*
71 leiso 12323 . . . . . . . 8  |-  ( ( ( 0 [,) 1
)  C_  RR*  /\  (
0 [,) +oo )  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) ) )
7269, 70, 71mp2an 672 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) )
7364, 72mpbi 208 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
74 isores1 6127 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )  <-> 
F  Isom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) ,  <_  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) ) )
7573, 74mpbi 208 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  <_  ( (
0 [,) 1 ) ,  ( 0 [,) +oo ) )
76 isores2 6126 . . . . 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 208 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) ) ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )
78 tsrps 15502 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
7965, 78ax-mp 5 . . . . . . 7  |-  <_  e.  PosetRel
80 ledm 15505 . . . . . . . 8  |-  RR*  =  dom  <_
8180psssdm 15497 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) )  =  ( 0 [,) 1 ) )
8279, 69, 81mp2an 672 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  ( 0 [,) 1
) ) )  =  ( 0 [,) 1
)
8382eqcomi 2464 . . . . 5  |-  ( 0 [,) 1 )  =  dom  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) )
8480psssdm 15497 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,) +oo )  C_ 
RR* )  ->  dom  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo ) )
8579, 70, 84mp2an 672 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) )  =  ( 0 [,) +oo )
8685eqcomi 2464 . . . . 5  |-  ( 0 [,) +oo )  =  dom  (  <_  i^i  ( ( 0 [,) +oo )  X.  (
0 [,) +oo )
) )
8783, 86ordthmeo 19500 . . . 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 1315 . . 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 2451 . . . . . . 7  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
9189, 90xrrest2 20510 . . . . . 6  |-  ( ( 0 [,) 1 ) 
C_  RR  ->  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) ) )
9210, 91ax-mp 5 . . . . 5  |-  ( Jt  ( 0 [,) 1 ) )  =  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )
93 iccssico2 11473 . . . . . 6  |-  ( ( x  e.  ( 0 [,) 1 )  /\  y  e.  ( 0 [,) 1 ) )  ->  ( x [,] y )  C_  (
0 [,) 1 ) )
9469, 93ordtrestixx 18951 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,) 1 )  X.  (
0 [,) 1 ) ) ) )
9592, 94eqtri 2480 . . . 4  |-  ( Jt  ( 0 [,) 1 ) )  =  (ordTop `  (  <_  i^i  ( (
0 [,) 1 )  X.  ( 0 [,) 1 ) ) ) )
96 elrege0 11502 . . . . . . . 8  |-  ( y  e.  ( 0 [,) +oo )  <->  ( y  e.  RR  /\  0  <_ 
y ) )
9796simplbi 460 . . . . . . 7  |-  ( y  e.  ( 0 [,) +oo )  ->  y  e.  RR )
9897ssriv 3461 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
9989, 90xrrest2 20510 . . . . . 6  |-  ( ( 0 [,) +oo )  C_  RR  ->  ( Jt  (
0 [,) +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) +oo ) ) )
10098, 99ax-mp 5 . . . . 5  |-  ( Jt  ( 0 [,) +oo )
)  =  ( (ordTop `  <_  )t  ( 0 [,) +oo ) )
101 iccssico2 11473 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x [,] y )  C_  (
0 [,) +oo )
)
10270, 101ordtrestixx 18951 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,) +oo ) )  =  (ordTop `  (  <_  i^i  (
( 0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
103100, 102eqtri 2480 . . . 4  |-  ( Jt  ( 0 [,) +oo )
)  =  (ordTop `  (  <_  i^i  ( (
0 [,) +oo )  X.  ( 0 [,) +oo ) ) ) )
10495, 103oveq12i 6205 . . 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 2538 . 2  |-  F  e.  ( ( Jt  ( 0 [,) 1 ) )
Homeo ( Jt  ( 0 [,) +oo ) ) )
10664, 105pm3.2i 455 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071    i^i cin 3428    C_ wss 3429   class class class wbr 4393    |-> cmpt 4451    X. cxp 4939   `'ccnv 4940   dom cdm 4941   -1-1-onto->wf1o 5518   ` cfv 5519    Isom wiso 5520  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391   +oocpnf 9519   RR*cxr 9521    < clt 9522    <_ cle 9523    - cmin 9699    / cdiv 10097   RR+crp 11095   [,)cico 11406   ↾t crest 14470   TopOpenctopn 14471  ordTopcordt 14548   PosetRelcps 15479    TosetRel ctsr 15480  ℂfldccnfld 17936   Homeochmeo 19451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fi 7765  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-plusg 14362  df-mulr 14363  df-starv 14364  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-rest 14472  df-topn 14473  df-topgen 14493  df-ordt 14550  df-ps 15481  df-tsr 15482  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cn 18956  df-hmeo 19453  df-xms 20020  df-ms 20021
This theorem is referenced by:  iccpnfhmeo  20642
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