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Theorem iccpnfhmeo 21530
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 11528 . . . 4  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 11268 . . . 4  |-  <  Or  RR*
3 soss 4732 . . . 4  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . . 3  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 11528 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
6 soss 4732 . . . . 5  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,] +oo ) ) )
75, 2, 6mp2 9 . . . 4  |-  <  Or  ( 0 [,] +oo )
8 sopo 4731 . . . 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 21529 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( y  e.  ( 0 [,] +oo )  |->  if ( y  = +oo ,  1 ,  ( y  /  (
1  +  y ) ) ) ) )
1211simpli 456 . . . 4  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
13 f1ofo 5731 . . . 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 9507 . . . . . . . . . . . . 13  |-  0  e.  RR
16 1re 9506 . . . . . . . . . . . . 13  |-  1  e.  RR
1715, 16elicc2i 11511 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
1817simp1bi 1009 . . . . . . . . . . 11  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
19183ad2ant1 1015 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  RR )
2015, 16elicc2i 11511 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 [,] 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <_  1
) )
2120simp1bi 1009 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  e.  RR )
22213ad2ant2 1016 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  RR )
23 1red 9522 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  e.  RR )
24 simp3 996 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  w )
2520simp3bi 1011 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  <_  1 )
26253ad2ant2 1016 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  <_  1 )
2719, 22, 23, 24, 26ltletrd 9653 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  1 )
2819, 27gtned 9631 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  =/=  z )
2928necomd 2653 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  =/=  1 )
30 ifnefalse 3869 . . . . . . . 8  |-  ( z  =/=  1  ->  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  =  ( z  /  ( 1  -  z ) ) )
3129, 30syl 16 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  =  ( z  /  ( 1  -  z ) ) )
32 breq2 4371 . . . . . . . 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 4371 . . . . . . . 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 9796 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  z  e.  RR )  ->  ( 1  -  z
)  e.  RR )
3516, 19, 34sylancr 661 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  e.  RR )
36 ax-1cn 9461 . . . . . . . . . . . . 13  |-  1  e.  CC
3719recnd 9533 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  CC )
38 subeq0 9758 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =  0  <->  1  =  z ) )
3938necon3bid 2640 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4036, 37, 39sylancr 661 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4128, 40mpbird 232 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  =/=  0 )
4219, 35, 41redivcld 10289 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  e.  RR )
43 ltpnf 11252 . . . . . . . . . 10  |-  ( ( z  /  ( 1  -  z ) )  e.  RR  ->  (
z  /  ( 1  -  z ) )  < +oo )
4442, 43syl 16 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  < +oo )
4544adantr 463 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  w  = 
1 )  ->  (
z  /  ( 1  -  z ) )  < +oo )
46 simpl3 999 . . . . . . . . . 10  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  <  w )
47 eqid 2382 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) )
48 eqid 2382 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4947, 48icopnfhmeo 21528 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 994 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( 0 [,] 1 ) )
53 0xr 9551 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR*
5416rexri 9557 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR*
55 0le1 9993 . . . . . . . . . . . . . . . . . . 19  |-  0  <_  1
56 snunico 11568 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  (
( 0 [,) 1
)  u.  { 1 } )  =  ( 0 [,] 1 ) )
5753, 54, 55, 56mp3an 1322 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,) 1 )  u.  { 1 } )  =  ( 0 [,] 1 )
5852, 57syl6eleqr 2481 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( ( 0 [,) 1 )  u.  { 1 } ) )
59 elun 3559 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6058, 59sylib 196 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6160ord 375 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  e.  { 1 } ) )
62 elsni 3969 . . . . . . . . . . . . . . 15  |-  ( z  e.  { 1 }  ->  z  =  1 )
6361, 62syl6 33 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  = 
1 ) )
6463necon1ad 2598 . . . . . . . . . . . . 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 463 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  e.  ( 0 [,) 1 ) )
67 simp2 995 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( 0 [,] 1 ) )
6867, 57syl6eleqr 2481 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( (
0 [,) 1 )  u.  { 1 } ) )
69 elun 3559 . . . . . . . . . . . . . . . 16  |-  ( w  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7068, 69sylib 196 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7170ord 375 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  e.  { 1 } ) )
72 elsni 3969 . . . . . . . . . . . . . 14  |-  ( w  e.  { 1 }  ->  w  =  1 )
7371, 72syl6 33 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  = 
1 ) )
7473con1d 124 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  =  1  ->  w  e.  ( 0 [,) 1
) ) )
7574imp 427 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  ->  w  e.  ( 0 [,) 1 ) )
76 isorel 6123 . . . . . . . . . . 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 1224 . . . . . . . . . 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 210 . . . . . . . . 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 6204 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
8179, 80oveq12d 6214 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
82 ovex 6224 . . . . . . . . . . 11  |-  ( z  /  ( 1  -  z ) )  e. 
_V
8381, 47, 82fvmpt 5857 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  ->  (
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  z
)  =  ( z  /  ( 1  -  z ) ) )
8466, 83syl 16 . . . . . . . . 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 6204 . . . . . . . . . . . 12  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
8785, 86oveq12d 6214 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
88 ovex 6224 . . . . . . . . . . 11  |-  ( w  /  ( 1  -  w ) )  e. 
_V
8987, 47, 88fvmpt 5857 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  w
)  =  ( w  /  ( 1  -  w ) ) )
9075, 89syl 16 . . . . . . . . 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 4396 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( z  /  (
1  -  z ) )  <  ( w  /  ( 1  -  w ) ) )
9232, 33, 45, 91ifbothda 3892 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  <  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) ) )
9331, 92eqbrtrd 4387 . . . . . 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 1196 . . . . 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 2386 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  1  <->  z  =  1 ) )
9695, 81ifbieq2d 3882 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  1 , +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( z  =  1 , +oo ,  ( z  / 
( 1  -  z
) ) ) )
97 pnfex 11243 . . . . . . . 8  |- +oo  e.  _V
9897, 82ifex 3925 . . . . . . 7  |-  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  e.  _V
9996, 10, 98fvmpt 5857 . . . . . 6  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  z )  =  if ( z  =  1 , +oo , 
( z  /  (
1  -  z ) ) ) )
100 eqeq1 2386 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  1  <->  w  =  1 ) )
101100, 87ifbieq2d 3882 . . . . . . 7  |-  ( x  =  w  ->  if ( x  =  1 , +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( w  =  1 , +oo ,  ( w  / 
( 1  -  w
) ) ) )
10297, 88ifex 3925 . . . . . . 7  |-  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) )  e.  _V
103101, 10, 102fvmpt 5857 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  ->  ( F `  w )  =  if ( w  =  1 , +oo , 
( w  /  (
1  -  w ) ) ) )
10499, 103breqan12d 4382 . . . . 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 234 . . . 4  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  ->  ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
) )
106105rgen2a 2809 . . 3  |-  A. z  e.  ( 0 [,] 1
) A. w  e.  ( 0 [,] 1
) ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
)
107 soisoi 6125 . . 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 671 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
109 letsr 15974 . . . . . 6  |-  <_  e.  TosetRel
110109elexi 3044 . . . . 5  |-  <_  e.  _V
111110inex1 4506 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  _V
112110inex1 4506 . . . 4  |-  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )  e.  _V
113 leiso 12412 . . . . . . . 8  |-  ( ( ( 0 [,] 1
)  C_  RR*  /\  (
0 [,] +oo )  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) ) )
1141, 5, 113mp2an 670 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) )
115108, 114mpbi 208 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
116 isores1 6131 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) )
117115, 116mpbi 208 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  <_  ( (
0 [,] 1 ) ,  ( 0 [,] +oo ) )
118 isores2 6130 . . . . 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 208 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
120 tsrps 15968 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
121109, 120ax-mp 5 . . . . . . 7  |-  <_  e.  PosetRel
122 ledm 15971 . . . . . . . 8  |-  RR*  =  dom  <_
123122psssdm 15963 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  =  ( 0 [,] 1 ) )
124121, 1, 123mp2an 670 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  =  ( 0 [,] 1
)
125124eqcomi 2395 . . . . 5  |-  ( 0 [,] 1 )  =  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
126122psssdm 15963 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] +oo )  C_ 
RR* )  ->  dom  (  <_  i^i  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  =  ( 0 [,] +oo ) )
127121, 5, 126mp2an 670 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  =  ( 0 [,] +oo )
128127eqcomi 2395 . . . . 5  |-  ( 0 [,] +oo )  =  dom  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )
129125, 128ordthmeo 20388 . . . 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 1322 . . 3  |-  F  e.  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) ) )
Homeo (ordTop `  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) ) )
131 dfii5 21474 . . . 4  |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
132 iccpnfhmeo.k . . . . 5  |-  K  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
133 ordtresticc 19810 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  (ordTop `  (  <_  i^i  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) )
134132, 133eqtri 2411 . . . 4  |-  K  =  (ordTop `  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) )
135131, 134oveq12i 6208 . . 3  |-  ( II
Homeo K )  =  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) Homeo (ordTop `  (  <_  i^i  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) ) )
136130, 135eleqtrri 2469 . 2  |-  F  e.  ( II Homeo K )
137108, 136pm3.2i 453 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 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   _Vcvv 3034    u. cun 3387    i^i cin 3388    C_ wss 3389   ifcif 3857   {csn 3944   class class class wbr 4367    |-> cmpt 4425    Po wpo 4712    Or wor 4713    X. cxp 4911   `'ccnv 4912   dom cdm 4913   -onto->wfo 5494   -1-1-onto->wf1o 5495   ` cfv 5496    Isom wiso 5497  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406   +oocpnf 9536   RR*cxr 9538    < clt 9539    <_ cle 9540    - cmin 9718    / cdiv 10123   [,)cico 11452   [,]cicc 11453   ↾t crest 14828   TopOpenctopn 14829  ordTopcordt 14906   PosetRelcps 15945    TosetRel ctsr 15946  ℂfldccnfld 18533   Homeochmeo 20339   IIcii 21464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fi 7786  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-plusg 14715  df-mulr 14716  df-starv 14717  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-rest 14830  df-topn 14831  df-topgen 14851  df-ordt 14908  df-ps 15947  df-tsr 15948  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cn 19814  df-hmeo 20341  df-xms 20908  df-ms 20909  df-ii 21466
This theorem is referenced by:  xrhmeo  21531  xrge0hmph  28068
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