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Theorem iccpnfhmeo 18443
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 10732 . . . 4  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 10475 . . . 4  |-  <  Or  RR*
3 soss 4332 . . . 4  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 17 . . 3  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 10732 . . . . 5  |-  ( 0 [,]  +oo )  C_  RR*
6 soss 4332 . . . . 5  |-  ( ( 0 [,]  +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,]  +oo ) ) )
75, 2, 6mp2 17 . . . 4  |-  <  Or  ( 0 [,]  +oo )
8 sopo 4331 . . . 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 18442 . . . . 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 5479 . . . 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 8838 . . . . . . . . . . . . 13  |-  0  e.  RR
16 1re 8837 . . . . . . . . . . . . 13  |-  1  e.  RR
1715, 16elicc2i 10716 . . . . . . . . . . . 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 10716 . . . . . . . . . . . . 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 8976 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  1 )
2819, 27gtned 8954 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  =/=  z )
2928necomd 2529 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  =/=  1 )
30 ifnefalse 3573 . . . . . . . 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 4027 . . . . . . . 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 4027 . . . . . . . 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 9111 . . . . . . . . . . . 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 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
3719recnd 8861 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  CC )
38 subeq0 9073 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =  0  <->  1  =  z ) )
3938necon3bid 2481 . . . . . . . . . . . . 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 9588 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  e.  RR )
43 ltpnf 10463 . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) )
48 eqid 2283 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4947, 48icopnfhmeo 18441 . . . . . . . . . . . . 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 8878 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR*
54 rexr 8877 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1  e.  RR  ->  1  e.  RR* )
5516, 54ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR*
56 0le1 9297 . . . . . . . . . . . . . . . . . . 19  |-  0  <_  1
57 snunico 10763 . . . . . . . . . . . . . . . . . . 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 2374 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( ( 0 [,) 1 )  u.  { 1 } ) )
60 elun 3316 . . . . . . . . . . . . . . . . 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 3664 . . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . 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 2374 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( (
0 [,) 1 )  u.  { 1 } ) )
70 elun 3316 . . . . . . . . . . . . . . . 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 3664 . . . . . . . . . . . . . 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 5823 . . . . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
8280, 81oveq12d 5876 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
83 ovex 5883 . . . . . . . . . . 11  |-  ( z  /  ( 1  -  z ) )  e. 
_V
8482, 47, 83fvmpt 5602 . . . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
8886, 87oveq12d 5876 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
89 ovex 5883 . . . . . . . . . . 11  |-  ( w  /  ( 1  -  w ) )  e. 
_V
9088, 47, 89fvmpt 5602 . . . . . . . . . 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 4052 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( z  /  (
1  -  z ) )  <  ( w  /  ( 1  -  w ) ) )
9332, 33, 45, 92ifbothda 3595 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  <  if ( w  =  1 , 
+oo ,  ( w  /  ( 1  -  w ) ) ) )
9431, 93eqbrtrd 4043 . . . . . 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 2289 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  1  <->  z  =  1 ) )
9796, 82ifbieq2d 3585 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( z  =  1 ,  +oo ,  ( z  /  (
1  -  z ) ) ) )
98 pnfxr 10455 . . . . . . . . 9  |-  +oo  e.  RR*
9998elexi 2797 . . . . . . . 8  |-  +oo  e.  _V
10099, 83ifex 3623 . . . . . . 7  |-  if ( z  =  1 , 
+oo ,  ( z  /  ( 1  -  z ) ) )  e.  _V
10197, 10, 100fvmpt 5602 . . . . . 6  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  z )  =  if ( z  =  1 ,  +oo , 
( z  /  (
1  -  z ) ) ) )
102 eqeq1 2289 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  1  <->  w  =  1 ) )
103102, 88ifbieq2d 3585 . . . . . . 7  |-  ( x  =  w  ->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( w  =  1 ,  +oo ,  ( w  /  (
1  -  w ) ) ) )
10499, 89ifex 3623 . . . . . . 7  |-  if ( w  =  1 , 
+oo ,  ( w  /  ( 1  -  w ) ) )  e.  _V
105103, 10, 104fvmpt 5602 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  ->  ( F `  w )  =  if ( w  =  1 ,  +oo , 
( w  /  (
1  -  w ) ) ) )
106101, 105breqan12d 4038 . . . . 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 2609 . . 3  |-  A. z  e.  ( 0 [,] 1
) A. w  e.  ( 0 [,] 1
) ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
)
109 soisoi 5825 . . 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 14349 . . . . . 6  |-  <_  e.  TosetRel
112111elexi 2797 . . . . 5  |-  <_  e.  _V
113112inex1 4155 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  _V
114112inex1 4155 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) )  e.  _V
115 leiso 11397 . . . . . . . 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 5831 . . . . . 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 5830 . . . . 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 14330 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
123111, 122ax-mp 8 . . . . . . 7  |-  <_  e.  PosetRel
124 ledm 14346 . . . . . . . 8  |-  RR*  =  dom  <_
125124psssdm 14325 . . . . . . 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 2287 . . . . 5  |-  ( 0 [,] 1 )  =  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
128124psssdm 14325 . . . . . . 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 2287 . . . . 5  |-  ( 0 [,]  +oo )  =  dom  (  <_  i^i  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) )
131127, 130ordthmeo 17493 . . . 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 18389 . . . 4  |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
134 iccpnfhmeo.k . . . . 5  |-  K  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
)
135 ordtresticc 16953 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) )
136134, 135eqtri 2303 . . . 4  |-  K  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) )
137133, 136oveq12i 5870 . . 3  |-  ( II 
Homeo  K )  =  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )  Homeo  (ordTop `  (  <_  i^i  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) ) ) ) )
138132, 137eleqtrri 2356 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077    Po wpo 4312    Or wor 4313    X. cxp 4687   `'ccnv 4688   dom cdm 4689   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   [,)cico 10658   [,]cicc 10659   ↾t crest 13325   TopOpenctopn 13326  ordTopcordt 13398   PosetRelcps 14301    TosetRel ctsr 14302  ℂfldccnfld 16377    Homeo chmeo 17444   IIcii 18379
This theorem is referenced by:  xrhmeo  18444  xrge0hmph  23314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-ordt 13402  df-ps 14306  df-tsr 14307  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-hmeo 17446  df-xms 17885  df-ms 17886  df-ii 18381
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