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Theorem iccpnfhmeo 22028
Description: The defined bijection from  [ 0 ,  1 ] to  [ 0 , +oo ] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
Hypotheses
Ref Expression
iccpnfhmeo.f  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  1 , +oo , 
( x  /  (
1  -  x ) ) ) )
iccpnfhmeo.k  |-  K  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
Assertion
Ref Expression
iccpnfhmeo  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  /\  F  e.  ( II Homeo K ) )

Proof of Theorem iccpnfhmeo
Dummy variables  y  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iccssxr 11751 . . . 4  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 11474 . . . 4  |-  <  Or  RR*
3 soss 4795 . . . 4  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . . 3  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 11751 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
6 soss 4795 . . . . 5  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,] +oo ) ) )
75, 2, 6mp2 9 . . . 4  |-  <  Or  ( 0 [,] +oo )
8 sopo 4794 . . . 4  |-  (  < 
Or  ( 0 [,] +oo )  ->  <  Po  ( 0 [,] +oo ) )
97, 8ax-mp 5 . . 3  |-  <  Po  ( 0 [,] +oo )
10 iccpnfhmeo.f . . . . . 6  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  1 , +oo , 
( x  /  (
1  -  x ) ) ) )
1110iccpnfcnv 22027 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( y  e.  ( 0 [,] +oo )  |->  if ( y  = +oo ,  1 ,  ( y  /  (
1  +  y ) ) ) ) )
1211simpli 464 . . . 4  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
13 f1ofo 5848 . . . 4  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  ->  F : ( 0 [,] 1 )
-onto-> ( 0 [,] +oo ) )
1412, 13ax-mp 5 . . 3  |-  F :
( 0 [,] 1
) -onto-> ( 0 [,] +oo )
15 0re 9674 . . . . . . . . . . . . 13  |-  0  e.  RR
16 1re 9673 . . . . . . . . . . . . 13  |-  1  e.  RR
1715, 16elicc2i 11734 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
1817simp1bi 1029 . . . . . . . . . . 11  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
19183ad2ant1 1035 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  RR )
2015, 16elicc2i 11734 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 [,] 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <_  1
) )
2120simp1bi 1029 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  e.  RR )
22213ad2ant2 1036 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  RR )
23 1red 9689 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  e.  RR )
24 simp3 1016 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  w )
2520simp3bi 1031 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  <_  1 )
26253ad2ant2 1036 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  <_  1 )
2719, 22, 23, 24, 26ltletrd 9826 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  1 )
2819, 27gtned 9801 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  =/=  z )
2928necomd 2691 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  =/=  1 )
30 ifnefalse 3905 . . . . . . . 8  |-  ( z  =/=  1  ->  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  =  ( z  /  ( 1  -  z ) ) )
3129, 30syl 17 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  =  ( z  /  ( 1  -  z ) ) )
32 breq2 4422 . . . . . . . 8  |-  ( +oo  =  if ( w  =  1 , +oo , 
( w  /  (
1  -  w ) ) )  ->  (
( z  /  (
1  -  z ) )  < +oo  <->  ( z  /  ( 1  -  z ) )  < 
if ( w  =  1 , +oo , 
( w  /  (
1  -  w ) ) ) ) )
33 breq2 4422 . . . . . . . 8  |-  ( ( w  /  ( 1  -  w ) )  =  if ( w  =  1 , +oo ,  ( w  / 
( 1  -  w
) ) )  -> 
( ( z  / 
( 1  -  z
) )  <  (
w  /  ( 1  -  w ) )  <-> 
( z  /  (
1  -  z ) )  <  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) ) ) )
34 resubcl 9969 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  z  e.  RR )  ->  ( 1  -  z
)  e.  RR )
3516, 19, 34sylancr 674 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  e.  RR )
36 ax-1cn 9628 . . . . . . . . . . . . 13  |-  1  e.  CC
3719recnd 9700 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  CC )
38 subeq0 9931 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =  0  <->  1  =  z ) )
3938necon3bid 2680 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4036, 37, 39sylancr 674 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4128, 40mpbird 240 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  =/=  0 )
4219, 35, 41redivcld 10468 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  e.  RR )
43 ltpnf 11456 . . . . . . . . . 10  |-  ( ( z  /  ( 1  -  z ) )  e.  RR  ->  (
z  /  ( 1  -  z ) )  < +oo )
4442, 43syl 17 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  < +oo )
4544adantr 471 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  w  = 
1 )  ->  (
z  /  ( 1  -  z ) )  < +oo )
46 simpl3 1019 . . . . . . . . . 10  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  <  w )
47 eqid 2462 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) )
48 eqid 2462 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4947, 48icopnfhmeo 22026 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) )  Isom  <  ,  <  ( ( 0 [,) 1
) ,  ( 0 [,) +oo ) )  /\  ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) )  e.  ( ( ( TopOpen ` fld )t  (
0 [,) 1 ) ) Homeo ( ( TopOpen ` fld )t  (
0 [,) +oo )
) ) )
5049simpli 464 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) ) 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,) +oo ) )
5150a1i 11 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) )  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) ) )
52 simp1 1014 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( 0 [,] 1 ) )
53 0xr 9718 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR*
5416rexri 9724 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR*
55 0le1 10170 . . . . . . . . . . . . . . . . . . 19  |-  0  <_  1
56 snunico 11794 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  (
( 0 [,) 1
)  u.  { 1 } )  =  ( 0 [,] 1 ) )
5753, 54, 55, 56mp3an 1373 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,) 1 )  u.  { 1 } )  =  ( 0 [,] 1 )
5852, 57syl6eleqr 2551 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( ( 0 [,) 1 )  u.  { 1 } ) )
59 elun 3586 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6058, 59sylib 201 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6160ord 383 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  e.  { 1 } ) )
62 elsni 4005 . . . . . . . . . . . . . . 15  |-  ( z  e.  { 1 }  ->  z  =  1 )
6361, 62syl6 34 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  = 
1 ) )
6463necon1ad 2653 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  =/=  1  ->  z  e.  ( 0 [,) 1 ) ) )
6529, 64mpd 15 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( 0 [,) 1 ) )
6665adantr 471 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  e.  ( 0 [,) 1 ) )
67 simp2 1015 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( 0 [,] 1 ) )
6867, 57syl6eleqr 2551 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( (
0 [,) 1 )  u.  { 1 } ) )
69 elun 3586 . . . . . . . . . . . . . . . 16  |-  ( w  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7068, 69sylib 201 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7170ord 383 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  e.  { 1 } ) )
72 elsni 4005 . . . . . . . . . . . . . 14  |-  ( w  e.  { 1 }  ->  w  =  1 )
7371, 72syl6 34 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  = 
1 ) )
7473con1d 129 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  =  1  ->  w  e.  ( 0 [,) 1
) ) )
7574imp 435 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  ->  w  e.  ( 0 [,) 1 ) )
76 isorel 6247 . . . . . . . . . . 11  |-  ( ( ( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) )  Isom  <  ,  <  ( ( 0 [,) 1 ) ,  ( 0 [,) +oo ) )  /\  (
z  e.  ( 0 [,) 1 )  /\  w  e.  ( 0 [,) 1 ) ) )  ->  ( z  <  w  <->  ( ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) ) `
 z )  < 
( ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) ) `  w ) ) )
7751, 66, 75, 76syl12anc 1274 . . . . . . . . . 10  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( z  <  w  <->  ( ( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  z
)  <  ( (
x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) ) `  w ) ) )
7846, 77mpbid 215 . . . . . . . . 9  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) ) `  z )  <  (
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  w
) )
79 id 22 . . . . . . . . . . . 12  |-  ( x  =  z  ->  x  =  z )
80 oveq2 6328 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
8179, 80oveq12d 6338 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
82 ovex 6348 . . . . . . . . . . 11  |-  ( z  /  ( 1  -  z ) )  e. 
_V
8381, 47, 82fvmpt 5976 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  ->  (
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  z
)  =  ( z  /  ( 1  -  z ) ) )
8466, 83syl 17 . . . . . . . . 9  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) ) `  z )  =  ( z  /  ( 1  -  z ) ) )
85 id 22 . . . . . . . . . . . 12  |-  ( x  =  w  ->  x  =  w )
86 oveq2 6328 . . . . . . . . . . . 12  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
8785, 86oveq12d 6338 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
88 ovex 6348 . . . . . . . . . . 11  |-  ( w  /  ( 1  -  w ) )  e. 
_V
8987, 47, 88fvmpt 5976 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  w
)  =  ( w  /  ( 1  -  w ) ) )
9075, 89syl 17 . . . . . . . . 9  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) ) `  w )  =  ( w  /  ( 1  -  w ) ) )
9178, 84, 903brtr3d 4448 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( z  /  (
1  -  z ) )  <  ( w  /  ( 1  -  w ) ) )
9232, 33, 45, 91ifbothda 3928 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  <  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) ) )
9331, 92eqbrtrd 4439 . . . . . 6  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  <  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) ) )
94933expia 1217 . . . . 5  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  ->  ( z  < 
w  ->  if (
z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  <  if ( w  =  1 , +oo ,  ( w  / 
( 1  -  w
) ) ) ) )
95 eqeq1 2466 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  1  <->  z  =  1 ) )
9695, 81ifbieq2d 3918 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  1 , +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( z  =  1 , +oo ,  ( z  / 
( 1  -  z
) ) ) )
97 pnfex 11447 . . . . . . . 8  |- +oo  e.  _V
9897, 82ifex 3961 . . . . . . 7  |-  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  e.  _V
9996, 10, 98fvmpt 5976 . . . . . 6  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  z )  =  if ( z  =  1 , +oo , 
( z  /  (
1  -  z ) ) ) )
100 eqeq1 2466 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  1  <->  w  =  1 ) )
101100, 87ifbieq2d 3918 . . . . . . 7  |-  ( x  =  w  ->  if ( x  =  1 , +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( w  =  1 , +oo ,  ( w  / 
( 1  -  w
) ) ) )
10297, 88ifex 3961 . . . . . . 7  |-  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) )  e.  _V
103101, 10, 102fvmpt 5976 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  ->  ( F `  w )  =  if ( w  =  1 , +oo , 
( w  /  (
1  -  w ) ) ) )
10499, 103breqan12d 4434 . . . . 5  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  ->  ( ( F `
 z )  < 
( F `  w
)  <->  if ( z  =  1 , +oo , 
( z  /  (
1  -  z ) ) )  <  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) ) ) )
10594, 104sylibrd 242 . . . 4  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  ->  ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
) )
106105rgen2a 2827 . . 3  |-  A. z  e.  ( 0 [,] 1
) A. w  e.  ( 0 [,] 1
) ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
)
107 soisoi 6249 . . 3  |-  ( ( (  <  Or  (
0 [,] 1 )  /\  <  Po  (
0 [,] +oo )
)  /\  ( F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo )  /\  A. z  e.  ( 0 [,] 1 ) A. w  e.  ( 0 [,] 1 ) ( z  <  w  -> 
( F `  z
)  <  ( F `  w ) ) ) )  ->  F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) ) )
1084, 9, 14, 106, 107mp4an 684 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
109 letsr 16528 . . . . . 6  |-  <_  e.  TosetRel
110109elexi 3067 . . . . 5  |-  <_  e.  _V
111110inex1 4560 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  _V
112110inex1 4560 . . . 4  |-  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )  e.  _V
113 leiso 12661 . . . . . . . 8  |-  ( ( ( 0 [,] 1
)  C_  RR*  /\  (
0 [,] +oo )  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) ) )
1141, 5, 113mp2an 683 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) )
115108, 114mpbi 213 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
116 isores1 6255 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) )
117115, 116mpbi 213 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  <_  ( (
0 [,] 1 ) ,  ( 0 [,] +oo ) )
118 isores2 6254 . . . . 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 ) ) )
119117, 118mpbi 213 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
120 tsrps 16522 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
121109, 120ax-mp 5 . . . . . . 7  |-  <_  e.  PosetRel
122 ledm 16525 . . . . . . . 8  |-  RR*  =  dom  <_
123122psssdm 16517 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  =  ( 0 [,] 1 ) )
124121, 1, 123mp2an 683 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  =  ( 0 [,] 1
)
125124eqcomi 2471 . . . . 5  |-  ( 0 [,] 1 )  =  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
126122psssdm 16517 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] +oo )  C_ 
RR* )  ->  dom  (  <_  i^i  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  =  ( 0 [,] +oo ) )
127121, 5, 126mp2an 683 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  =  ( 0 [,] +oo )
128127eqcomi 2471 . . . . 5  |-  ( 0 [,] +oo )  =  dom  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )
129125, 128ordthmeo 20872 . . . 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 ) ) ) ) ) )
130111, 112, 119, 129mp3an 1373 . . 3  |-  F  e.  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) ) )
Homeo (ordTop `  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) ) )
131 dfii5 21972 . . . 4  |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
132 iccpnfhmeo.k . . . . 5  |-  K  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
133 ordtresticc 20294 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  (ordTop `  (  <_  i^i  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) )
134132, 133eqtri 2484 . . . 4  |-  K  =  (ordTop `  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) )
135131, 134oveq12i 6332 . . 3  |-  ( II
Homeo K )  =  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) Homeo (ordTop `  (  <_  i^i  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) ) )
136130, 135eleqtrri 2539 . 2  |-  F  e.  ( II Homeo K )
137108, 136pm3.2i 461 1  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  /\  F  e.  ( II Homeo K ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   _Vcvv 3057    u. cun 3414    i^i cin 3415    C_ wss 3416   ifcif 3893   {csn 3980   class class class wbr 4418    |-> cmpt 4477    Po wpo 4775    Or wor 4776    X. cxp 4854   `'ccnv 4855   dom cdm 4856   -onto->wfo 5603   -1-1-onto->wf1o 5604   ` cfv 5605    Isom wiso 5606  (class class class)co 6320   CCcc 9568   RRcr 9569   0cc0 9570   1c1 9571    + caddc 9573   +oocpnf 9703   RR*cxr 9705    < clt 9706    <_ cle 9707    - cmin 9891    / cdiv 10302   [,)cico 11671   [,]cicc 11672   ↾t crest 15374   TopOpenctopn 15375  ordTopcordt 15452   PosetRelcps 16499    TosetRel ctsr 16500  ℂfldccnfld 19025   Homeochmeo 20823   IIcii 21962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-fi 7956  df-sup 7987  df-inf 7988  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-7 10706  df-8 10707  df-9 10708  df-10 10709  df-n0 10904  df-z 10972  df-dec 11086  df-uz 11194  df-q 11299  df-rp 11337  df-xneg 11443  df-xadd 11444  df-xmul 11445  df-ioo 11673  df-ioc 11674  df-ico 11675  df-icc 11676  df-fz 11820  df-seq 12252  df-exp 12311  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-struct 15178  df-ndx 15179  df-slot 15180  df-base 15181  df-plusg 15258  df-mulr 15259  df-starv 15260  df-tset 15264  df-ple 15265  df-ds 15267  df-unif 15268  df-rest 15376  df-topn 15377  df-topgen 15397  df-ordt 15454  df-ps 16501  df-tsr 16502  df-psmet 19017  df-xmet 19018  df-met 19019  df-bl 19020  df-mopn 19021  df-cnfld 19026  df-top 19976  df-bases 19977  df-topon 19978  df-topsp 19979  df-cn 20298  df-hmeo 20825  df-xms 21390  df-ms 21391  df-ii 21964
This theorem is referenced by:  xrhmeo  22029  xrge0hmph  28789
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