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Theorem iccpnfhmeo 18496
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 10779 . . . 4  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 10522 . . . 4  |-  <  Or  RR*
3 soss 4369 . . . 4  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 17 . . 3  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 10779 . . . . 5  |-  ( 0 [,]  +oo )  C_  RR*
6 soss 4369 . . . . 5  |-  ( ( 0 [,]  +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,]  +oo ) ) )
75, 2, 6mp2 17 . . . 4  |-  <  Or  ( 0 [,]  +oo )
8 sopo 4368 . . . 4  |-  (  < 
Or  ( 0 [,] 
+oo )  ->  <  Po  ( 0 [,]  +oo ) )
97, 8ax-mp 8 . . 3  |-  <  Po  ( 0 [,]  +oo )
10 iccpnfhmeo.f . . . . . 6  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  1 ,  +oo , 
( x  /  (
1  -  x ) ) ) )
1110iccpnfcnv 18495 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( y  e.  ( 0 [,]  +oo )  |->  if ( y  = 
+oo ,  1 , 
( y  /  (
1  +  y ) ) ) ) )
1211simpli 444 . . . 4  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,]  +oo )
13 f1ofo 5517 . . . 4  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  ->  F : ( 0 [,] 1 )
-onto-> ( 0 [,]  +oo ) )
1412, 13ax-mp 8 . . 3  |-  F :
( 0 [,] 1
) -onto-> ( 0 [,] 
+oo )
15 0re 8883 . . . . . . . . . . . . 13  |-  0  e.  RR
16 1re 8882 . . . . . . . . . . . . 13  |-  1  e.  RR
1715, 16elicc2i 10763 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
1817simp1bi 970 . . . . . . . . . . 11  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
19183ad2ant1 976 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  RR )
2015, 16elicc2i 10763 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 [,] 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <_  1
) )
2120simp1bi 970 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  e.  RR )
22213ad2ant2 977 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  RR )
2316a1i 10 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  e.  RR )
24 simp3 957 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  w )
2520simp3bi 972 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  <_  1 )
26253ad2ant2 977 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  <_  1 )
2719, 22, 23, 24, 26ltletrd 9021 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  1 )
2819, 27gtned 8999 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  =/=  z )
2928necomd 2562 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  =/=  1 )
30 ifnefalse 3607 . . . . . . . 8  |-  ( z  =/=  1  ->  if ( z  =  1 ,  +oo ,  ( z  /  ( 1  -  z ) ) )  =  ( z  /  ( 1  -  z ) ) )
3129, 30syl 15 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  if ( z  =  1 ,  +oo ,  ( z  /  ( 1  -  z ) ) )  =  ( z  /  ( 1  -  z ) ) )
32 breq2 4064 . . . . . . . 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 4064 . . . . . . . 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 9156 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  z  e.  RR )  ->  ( 1  -  z
)  e.  RR )
3516, 19, 34sylancr 644 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  e.  RR )
36 ax-1cn 8840 . . . . . . . . . . . . 13  |-  1  e.  CC
3719recnd 8906 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  CC )
38 subeq0 9118 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =  0  <->  1  =  z ) )
3938necon3bid 2514 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4036, 37, 39sylancr 644 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4128, 40mpbird 223 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  =/=  0 )
4219, 35, 41redivcld 9633 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  e.  RR )
43 ltpnf 10510 . . . . . . . . . 10  |-  ( ( z  /  ( 1  -  z ) )  e.  RR  ->  (
z  /  ( 1  -  z ) )  <  +oo )
4442, 43syl 15 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  <  +oo )
4544adantr 451 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  w  = 
1 )  ->  (
z  /  ( 1  -  z ) )  <  +oo )
46 simpl3 960 . . . . . . . . . 10  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  <  w )
47 eqid 2316 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) )
48 eqid 2316 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4947, 48icopnfhmeo 18494 . . . . . . . . . . . . 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 444 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) ) 
Isom  <  ,  <  (
( 0 [,) 1
) ,  ( 0 [,)  +oo ) )
5150a1i 10 . . . . . . . . . . 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 955 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( 0 [,] 1 ) )
53 0xr 8923 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR*
54 rexr 8922 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1  e.  RR  ->  1  e.  RR* )
5516, 54ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR*
56 0le1 9342 . . . . . . . . . . . . . . . . . . 19  |-  0  <_  1
57 snunico 10810 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  (
( 0 [,) 1
)  u.  { 1 } )  =  ( 0 [,] 1 ) )
5853, 55, 56, 57mp3an 1277 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,) 1 )  u.  { 1 } )  =  ( 0 [,] 1 )
5952, 58syl6eleqr 2407 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( ( 0 [,) 1 )  u.  { 1 } ) )
60 elun 3350 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6159, 60sylib 188 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6261ord 366 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  e.  { 1 } ) )
63 elsni 3698 . . . . . . . . . . . . . . 15  |-  ( z  e.  { 1 }  ->  z  =  1 )
6462, 63syl6 29 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  = 
1 ) )
6564necon1ad 2546 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  =/=  1  ->  z  e.  ( 0 [,) 1 ) ) )
6629, 65mpd 14 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( 0 [,) 1 ) )
6766adantr 451 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  e.  ( 0 [,) 1 ) )
68 simp2 956 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( 0 [,] 1 ) )
6968, 58syl6eleqr 2407 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( (
0 [,) 1 )  u.  { 1 } ) )
70 elun 3350 . . . . . . . . . . . . . . . 16  |-  ( w  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7169, 70sylib 188 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7271ord 366 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  e.  { 1 } ) )
73 elsni 3698 . . . . . . . . . . . . . 14  |-  ( w  e.  { 1 }  ->  w  =  1 )
7472, 73syl6 29 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  = 
1 ) )
7574con1d 116 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  =  1  ->  w  e.  ( 0 [,) 1
) ) )
7675imp 418 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  ->  w  e.  ( 0 [,) 1 ) )
77 isorel 5865 . . . . . . . . . . 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 ) ) )
7851, 67, 76, 77syl12anc 1180 . . . . . . . . . 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 ) ) )
7946, 78mpbid 201 . . . . . . . . 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
) )
80 id 19 . . . . . . . . . . . 12  |-  ( x  =  z  ->  x  =  z )
81 oveq2 5908 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
8280, 81oveq12d 5918 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
83 ovex 5925 . . . . . . . . . . 11  |-  ( z  /  ( 1  -  z ) )  e. 
_V
8482, 47, 83fvmpt 5640 . . . . . . . . . 10  |-  ( z  e.  ( 0 [,) 1 )  ->  (
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  z
)  =  ( z  /  ( 1  -  z ) ) )
8567, 84syl 15 . . . . . . . . 9  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) ) `  z )  =  ( z  /  ( 1  -  z ) ) )
86 id 19 . . . . . . . . . . . 12  |-  ( x  =  w  ->  x  =  w )
87 oveq2 5908 . . . . . . . . . . . 12  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
8886, 87oveq12d 5918 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
89 ovex 5925 . . . . . . . . . . 11  |-  ( w  /  ( 1  -  w ) )  e. 
_V
9088, 47, 89fvmpt 5640 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,) 1 )  ->  (
( x  e.  ( 0 [,) 1 ) 
|->  ( x  /  (
1  -  x ) ) ) `  w
)  =  ( w  /  ( 1  -  w ) ) )
9176, 90syl 15 . . . . . . . . 9  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) ) `  w )  =  ( w  /  ( 1  -  w ) ) )
9279, 85, 913brtr3d 4089 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( z  /  (
1  -  z ) )  <  ( w  /  ( 1  -  w ) ) )
9332, 33, 45, 92ifbothda 3629 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  <  if ( w  =  1 , 
+oo ,  ( w  /  ( 1  -  w ) ) ) )
9431, 93eqbrtrd 4080 . . . . . 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 ) ) ) )
95943expia 1153 . . . . 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 ) ) ) ) )
96 eqeq1 2322 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  1  <->  z  =  1 ) )
9796, 82ifbieq2d 3619 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( z  =  1 ,  +oo ,  ( z  /  (
1  -  z ) ) ) )
98 pnfxr 10502 . . . . . . . . 9  |-  +oo  e.  RR*
9998elexi 2831 . . . . . . . 8  |-  +oo  e.  _V
10099, 83ifex 3657 . . . . . . 7  |-  if ( z  =  1 , 
+oo ,  ( z  /  ( 1  -  z ) ) )  e.  _V
10197, 10, 100fvmpt 5640 . . . . . 6  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  z )  =  if ( z  =  1 ,  +oo , 
( z  /  (
1  -  z ) ) ) )
102 eqeq1 2322 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  1  <->  w  =  1 ) )
103102, 88ifbieq2d 3619 . . . . . . 7  |-  ( x  =  w  ->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( w  =  1 ,  +oo ,  ( w  /  (
1  -  w ) ) ) )
10499, 89ifex 3657 . . . . . . 7  |-  if ( w  =  1 , 
+oo ,  ( w  /  ( 1  -  w ) ) )  e.  _V
105103, 10, 104fvmpt 5640 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  ->  ( F `  w )  =  if ( w  =  1 ,  +oo , 
( w  /  (
1  -  w ) ) ) )
106101, 105breqan12d 4075 . . . . 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 ) ) ) ) )
10795, 106sylibrd 225 . . . 4  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  ->  ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
) )
108107rgen2a 2643 . . 3  |-  A. z  e.  ( 0 [,] 1
) A. w  e.  ( 0 [,] 1
) ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
)
109 soisoi 5867 . . 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 ) ) )
1104, 9, 14, 108, 109mp4an 654 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )
111 letsr 14398 . . . . . 6  |-  <_  e.  TosetRel
112111elexi 2831 . . . . 5  |-  <_  e.  _V
113112inex1 4192 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  _V
114112inex1 4192 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) )  e.  _V
115 leiso 11444 . . . . . . . 8  |-  ( ( ( 0 [,] 1
)  C_  RR*  /\  (
0 [,]  +oo )  C_  RR* )  ->  ( F  Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,]  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) ) ) )
1161, 5, 115mp2an 653 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,]  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) ) )
117110, 116mpbi 199 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )
118 isores1 5873 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,] 1
) ,  ( 0 [,]  +oo ) )  <->  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  <_  ( (
0 [,] 1 ) ,  ( 0 [,] 
+oo ) ) )
119117, 118mpbi 199 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  <_  ( (
0 [,] 1 ) ,  ( 0 [,] 
+oo ) )
120 isores2 5872 . . . . 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 ) ) )
121119, 120mpbi 199 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )
122 tsrps 14379 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
123111, 122ax-mp 8 . . . . . . 7  |-  <_  e.  PosetRel
124 ledm 14395 . . . . . . . 8  |-  RR*  =  dom  <_
125124psssdm 14374 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  =  ( 0 [,] 1 ) )
126123, 1, 125mp2an 653 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  =  ( 0 [,] 1
)
127126eqcomi 2320 . . . . 5  |-  ( 0 [,] 1 )  =  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
128124psssdm 14374 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,]  +oo )  C_  RR* )  ->  dom  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) )  =  ( 0 [,]  +oo ) )
129123, 5, 128mp2an 653 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,]  +oo )  X.  (
0 [,]  +oo ) ) )  =  ( 0 [,]  +oo )
130129eqcomi 2320 . . . . 5  |-  ( 0 [,]  +oo )  =  dom  (  <_  i^i  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) )
131127, 130ordthmeo 17549 . . . 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 ) ) ) ) ) )
132113, 114, 121, 131mp3an 1277 . . 3  |-  F  e.  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) ) ) 
Homeo  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) ) )
133 dfii5 18441 . . . 4  |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
134 iccpnfhmeo.k . . . . 5  |-  K  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
)
135 ordtresticc 17009 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) )
136134, 135eqtri 2336 . . . 4  |-  K  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) )
137133, 136oveq12i 5912 . . 3  |-  ( II 
Homeo  K )  =  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )  Homeo  (ordTop `  (  <_  i^i  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) ) ) ) )
138132, 137eleqtrri 2389 . 2  |-  F  e.  ( II  Homeo  K )
139110, 138pm3.2i 441 1  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,]  +oo ) )  /\  F  e.  ( II  Homeo  K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   _Vcvv 2822    u. cun 3184    i^i cin 3185    C_ wss 3186   ifcif 3599   {csn 3674   class class class wbr 4060    e. cmpt 4114    Po wpo 4349    Or wor 4350    X. cxp 4724   `'ccnv 4725   dom cdm 4726   -onto->wfo 5290   -1-1-onto->wf1o 5291   ` cfv 5292    Isom wiso 5293  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    +oocpnf 8909   RR*cxr 8911    < clt 8912    <_ cle 8913    - cmin 9082    / cdiv 9468   [,)cico 10705   [,]cicc 10706   ↾t crest 13374   TopOpenctopn 13375  ordTopcordt 13447   PosetRelcps 14350    TosetRel ctsr 14351  ℂfldccnfld 16432    Homeo chmeo 17500   IIcii 18431
This theorem is referenced by:  xrhmeo  18497  xrge0hmph  23387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ioc 10708  df-ico 10709  df-icc 10710  df-fz 10830  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-plusg 13268  df-mulr 13269  df-starv 13270  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-rest 13376  df-topn 13377  df-topgen 13393  df-ordt 13451  df-ps 14355  df-tsr 14356  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cn 17013  df-hmeo 17502  df-xms 17937  df-ms 17938  df-ii 18433
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