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Theorem iccpnfhmeo 21971
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 11724 . . . 4  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 11447 . . . 4  |-  <  Or  RR*
3 soss 4792 . . . 4  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . . 3  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 11724 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
6 soss 4792 . . . . 5  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,] +oo ) ) )
75, 2, 6mp2 9 . . . 4  |-  <  Or  ( 0 [,] +oo )
8 sopo 4791 . . . 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 21970 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( y  e.  ( 0 [,] +oo )  |->  if ( y  = +oo ,  1 ,  ( y  /  (
1  +  y ) ) ) ) )
1211simpli 459 . . . 4  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
13 f1ofo 5838 . . . 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 9650 . . . . . . . . . . . . 13  |-  0  e.  RR
16 1re 9649 . . . . . . . . . . . . 13  |-  1  e.  RR
1715, 16elicc2i 11707 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
1817simp1bi 1020 . . . . . . . . . . 11  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
19183ad2ant1 1026 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  RR )
2015, 16elicc2i 11707 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 [,] 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <_  1
) )
2120simp1bi 1020 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  e.  RR )
22213ad2ant2 1027 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  RR )
23 1red 9665 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  e.  RR )
24 simp3 1007 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  w )
2520simp3bi 1022 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  <_  1 )
26253ad2ant2 1027 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  <_  1 )
2719, 22, 23, 24, 26ltletrd 9802 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  1 )
2819, 27gtned 9777 . . . . . . . . 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 3923 . . . . . . . 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 4427 . . . . . . . 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 4427 . . . . . . . 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 9945 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  z  e.  RR )  ->  ( 1  -  z
)  e.  RR )
3516, 19, 34sylancr 667 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  e.  RR )
36 ax-1cn 9604 . . . . . . . . . . . . 13  |-  1  e.  CC
3719recnd 9676 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  CC )
38 subeq0 9907 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =  0  <->  1  =  z ) )
3938necon3bid 2678 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4036, 37, 39sylancr 667 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4128, 40mpbird 235 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  =/=  0 )
4219, 35, 41redivcld 10442 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  e.  RR )
43 ltpnf 11429 . . . . . . . . . 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 466 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  w  = 
1 )  ->  (
z  /  ( 1  -  z ) )  < +oo )
46 simpl3 1010 . . . . . . . . . 10  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  <  w )
47 eqid 2422 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) )
48 eqid 2422 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4947, 48icopnfhmeo 21969 . . . . . . . . . . . . 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 459 . . . . . . . . . . . 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 1005 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( 0 [,] 1 ) )
53 0xr 9694 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR*
5416rexri 9700 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR*
55 0le1 10144 . . . . . . . . . . . . . . . . . . 19  |-  0  <_  1
56 snunico 11766 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  (
( 0 [,) 1
)  u.  { 1 } )  =  ( 0 [,] 1 ) )
5753, 54, 55, 56mp3an 1360 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,) 1 )  u.  { 1 } )  =  ( 0 [,] 1 )
5852, 57syl6eleqr 2518 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( ( 0 [,) 1 )  u.  { 1 } ) )
59 elun 3606 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6058, 59sylib 199 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6160ord 378 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  e.  { 1 } ) )
62 elsni 4023 . . . . . . . . . . . . . . 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 2636 . . . . . . . . . . . . 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 466 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  e.  ( 0 [,) 1 ) )
67 simp2 1006 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( 0 [,] 1 ) )
6867, 57syl6eleqr 2518 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( (
0 [,) 1 )  u.  { 1 } ) )
69 elun 3606 . . . . . . . . . . . . . . . 16  |-  ( w  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7068, 69sylib 199 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7170ord 378 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  e.  { 1 } ) )
72 elsni 4023 . . . . . . . . . . . . . 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 127 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  =  1  ->  w  e.  ( 0 [,) 1
) ) )
7574imp 430 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  ->  w  e.  ( 0 [,) 1 ) )
76 isorel 6232 . . . . . . . . . . 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 1262 . . . . . . . . . 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 213 . . . . . . . . 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 6313 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
8179, 80oveq12d 6323 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
82 ovex 6333 . . . . . . . . . . 11  |-  ( z  /  ( 1  -  z ) )  e. 
_V
8381, 47, 82fvmpt 5964 . . . . . . . . . 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 6313 . . . . . . . . . . . 12  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
8785, 86oveq12d 6323 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
88 ovex 6333 . . . . . . . . . . 11  |-  ( w  /  ( 1  -  w ) )  e. 
_V
8987, 47, 88fvmpt 5964 . . . . . . . . . 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 4453 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( z  /  (
1  -  z ) )  <  ( w  /  ( 1  -  w ) ) )
9232, 33, 45, 91ifbothda 3946 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  <  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) ) )
9331, 92eqbrtrd 4444 . . . . . 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 1207 . . . . 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 2426 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  1  <->  z  =  1 ) )
9695, 81ifbieq2d 3936 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  1 , +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( z  =  1 , +oo ,  ( z  / 
( 1  -  z
) ) ) )
97 pnfex 11420 . . . . . . . 8  |- +oo  e.  _V
9897, 82ifex 3979 . . . . . . 7  |-  if ( z  =  1 , +oo ,  ( z  /  ( 1  -  z ) ) )  e.  _V
9996, 10, 98fvmpt 5964 . . . . . 6  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  z )  =  if ( z  =  1 , +oo , 
( z  /  (
1  -  z ) ) ) )
100 eqeq1 2426 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  1  <->  w  =  1 ) )
101100, 87ifbieq2d 3936 . . . . . . 7  |-  ( x  =  w  ->  if ( x  =  1 , +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( w  =  1 , +oo ,  ( w  / 
( 1  -  w
) ) ) )
10297, 88ifex 3979 . . . . . . 7  |-  if ( w  =  1 , +oo ,  ( w  /  ( 1  -  w ) ) )  e.  _V
103101, 10, 102fvmpt 5964 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  ->  ( F `  w )  =  if ( w  =  1 , +oo , 
( w  /  (
1  -  w ) ) ) )
10499, 103breqan12d 4439 . . . . 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 237 . . . 4  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  ->  ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
) )
106105rgen2a 2849 . . 3  |-  A. z  e.  ( 0 [,] 1
) A. w  e.  ( 0 [,] 1
) ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
)
107 soisoi 6234 . . 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 677 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
109 letsr 16472 . . . . . 6  |-  <_  e.  TosetRel
110109elexi 3090 . . . . 5  |-  <_  e.  _V
111110inex1 4565 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  _V
112110inex1 4565 . . . 4  |-  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )  e.  _V
113 leiso 12626 . . . . . . . 8  |-  ( ( ( 0 [,] 1
)  C_  RR*  /\  (
0 [,] +oo )  C_ 
RR* )  ->  ( F  Isom  <  ,  <  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) ) )
1141, 5, 113mp2an 676 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  <_  ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) )
115108, 114mpbi 211 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
116 isores1 6240 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,] 1
) ,  ( 0 [,] +oo ) )  <-> 
F  Isom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ,  <_  ( ( 0 [,] 1
) ,  ( 0 [,] +oo ) ) )
117115, 116mpbi 211 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  <_  ( (
0 [,] 1 ) ,  ( 0 [,] +oo ) )
118 isores2 6239 . . . . 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 211 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
120 tsrps 16466 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
121109, 120ax-mp 5 . . . . . . 7  |-  <_  e.  PosetRel
122 ledm 16469 . . . . . . . 8  |-  RR*  =  dom  <_
123122psssdm 16461 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  =  ( 0 [,] 1 ) )
124121, 1, 123mp2an 676 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  =  ( 0 [,] 1
)
125124eqcomi 2435 . . . . 5  |-  ( 0 [,] 1 )  =  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
126122psssdm 16461 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] +oo )  C_ 
RR* )  ->  dom  (  <_  i^i  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  =  ( 0 [,] +oo ) )
127121, 5, 126mp2an 676 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  =  ( 0 [,] +oo )
128127eqcomi 2435 . . . . 5  |-  ( 0 [,] +oo )  =  dom  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )
129125, 128ordthmeo 20815 . . . 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 1360 . . 3  |-  F  e.  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) ) )
Homeo (ordTop `  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) ) )
131 dfii5 21915 . . . 4  |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
132 iccpnfhmeo.k . . . . 5  |-  K  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
133 ordtresticc 20237 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  (ordTop `  (  <_  i^i  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) )
134132, 133eqtri 2451 . . . 4  |-  K  =  (ordTop `  (  <_  i^i  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) )
135131, 134oveq12i 6317 . . 3  |-  ( II
Homeo K )  =  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) Homeo (ordTop `  (  <_  i^i  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) ) )
136130, 135eleqtrri 2506 . 2  |-  F  e.  ( II Homeo K )
137108, 136pm3.2i 456 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   _Vcvv 3080    u. cun 3434    i^i cin 3435    C_ wss 3436   ifcif 3911   {csn 3998   class class class wbr 4423    |-> cmpt 4482    Po wpo 4772    Or wor 4773    X. cxp 4851   `'ccnv 4852   dom cdm 4853   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601    Isom wiso 5602  (class class class)co 6305   CCcc 9544   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549   +oocpnf 9679   RR*cxr 9681    < clt 9682    <_ cle 9683    - cmin 9867    / cdiv 10276   [,)cico 11644   [,]cicc 11645   ↾t crest 15318   TopOpenctopn 15319  ordTopcordt 15396   PosetRelcps 16443    TosetRel ctsr 16444  ℂfldccnfld 18969   Homeochmeo 20766   IIcii 21905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fi 7934  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-ico 11648  df-icc 11649  df-fz 11792  df-seq 12220  df-exp 12279  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-plusg 15202  df-mulr 15203  df-starv 15204  df-tset 15208  df-ple 15209  df-ds 15211  df-unif 15212  df-rest 15320  df-topn 15321  df-topgen 15341  df-ordt 15398  df-ps 16445  df-tsr 16446  df-psmet 18961  df-xmet 18962  df-met 18963  df-bl 18964  df-mopn 18965  df-cnfld 18970  df-top 19919  df-bases 19920  df-topon 19921  df-topsp 19922  df-cn 20241  df-hmeo 20768  df-xms 21333  df-ms 21334  df-ii 21907
This theorem is referenced by:  xrhmeo  21972  xrge0hmph  28746
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