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Theorem iccpnfhmeo 18970
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 10993 . . . 4  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 10734 . . . 4  |-  <  Or  RR*
3 soss 4521 . . . 4  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . . 3  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 10993 . . . . 5  |-  ( 0 [,]  +oo )  C_  RR*
6 soss 4521 . . . . 5  |-  ( ( 0 [,]  +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,]  +oo ) ) )
75, 2, 6mp2 9 . . . 4  |-  <  Or  ( 0 [,]  +oo )
8 sopo 4520 . . . 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 18969 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( y  e.  ( 0 [,]  +oo )  |->  if ( y  = 
+oo ,  1 , 
( y  /  (
1  +  y ) ) ) ) )
1211simpli 445 . . . 4  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,]  +oo )
13 f1ofo 5681 . . . 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 9091 . . . . . . . . . . . . 13  |-  0  e.  RR
16 1re 9090 . . . . . . . . . . . . 13  |-  1  e.  RR
1715, 16elicc2i 10976 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
1817simp1bi 972 . . . . . . . . . . 11  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
19183ad2ant1 978 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  RR )
2015, 16elicc2i 10976 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 [,] 1 )  <->  ( w  e.  RR  /\  0  <_  w  /\  w  <_  1
) )
2120simp1bi 972 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  e.  RR )
22213ad2ant2 979 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  RR )
2316a1i 11 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  e.  RR )
24 simp3 959 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  w )
2520simp3bi 974 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 [,] 1 )  ->  w  <_  1 )
26253ad2ant2 979 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  <_  1 )
2719, 22, 23, 24, 26ltletrd 9230 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  <  1 )
2819, 27gtned 9208 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
1  =/=  z )
2928necomd 2687 . . . . . . . 8  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  =/=  1 )
30 ifnefalse 3747 . . . . . . . 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 4216 . . . . . . . 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 4216 . . . . . . . 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 9365 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  z  e.  RR )  ->  ( 1  -  z
)  e.  RR )
3516, 19, 34sylancr 645 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  e.  RR )
36 ax-1cn 9048 . . . . . . . . . . . . 13  |-  1  e.  CC
3719recnd 9114 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  CC )
38 subeq0 9327 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =  0  <->  1  =  z ) )
3938necon3bid 2636 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4036, 37, 39sylancr 645 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( ( 1  -  z )  =/=  0  <->  1  =/=  z ) )
4128, 40mpbird 224 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( 1  -  z
)  =/=  0 )
4219, 35, 41redivcld 9842 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  e.  RR )
43 ltpnf 10721 . . . . . . . . . 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 452 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  w  = 
1 )  ->  (
z  /  ( 1  -  z ) )  <  +oo )
46 simpl3 962 . . . . . . . . . 10  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  <  w )
47 eqid 2436 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 0 [,) 1 )  |->  ( x  /  ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,) 1
)  |->  ( x  / 
( 1  -  x
) ) )
48 eqid 2436 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4947, 48icopnfhmeo 18968 . . . . . . . . . . . . 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 445 . . . . . . . . . . . 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 957 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( 0 [,] 1 ) )
53 0xr 9131 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR*
5416rexri 9137 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR*
55 0le1 9551 . . . . . . . . . . . . . . . . . . 19  |-  0  <_  1
56 snunico 11024 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  (
( 0 [,) 1
)  u.  { 1 } )  =  ( 0 [,] 1 ) )
5753, 54, 55, 56mp3an 1279 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,) 1 )  u.  { 1 } )  =  ( 0 [,] 1 )
5852, 57syl6eleqr 2527 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
z  e.  ( ( 0 [,) 1 )  u.  { 1 } ) )
59 elun 3488 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6058, 59sylib 189 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  e.  ( 0 [,) 1 )  \/  z  e.  {
1 } ) )
6160ord 367 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  e.  { 1 } ) )
62 elsni 3838 . . . . . . . . . . . . . . 15  |-  ( z  e.  { 1 }  ->  z  =  1 )
6361, 62syl6 31 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  z  e.  ( 0 [,) 1
)  ->  z  = 
1 ) )
6463necon1ad 2671 . . . . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
z  e.  ( 0 [,) 1 ) )
67 simp2 958 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( 0 [,] 1 ) )
6867, 57syl6eleqr 2527 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  ->  w  e.  ( (
0 [,) 1 )  u.  { 1 } ) )
69 elun 3488 . . . . . . . . . . . . . . . 16  |-  ( w  e.  ( ( 0 [,) 1 )  u. 
{ 1 } )  <-> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7068, 69sylib 189 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( w  e.  ( 0 [,) 1 )  \/  w  e.  {
1 } ) )
7170ord 367 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  e.  { 1 } ) )
72 elsni 3838 . . . . . . . . . . . . . 14  |-  ( w  e.  { 1 }  ->  w  =  1 )
7371, 72syl6 31 . . . . . . . . . . . . 13  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  e.  ( 0 [,) 1
)  ->  w  = 
1 ) )
7473con1d 118 . . . . . . . . . . . 12  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( -.  w  =  1  ->  w  e.  ( 0 [,) 1
) ) )
7574imp 419 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  ->  w  e.  ( 0 [,) 1 ) )
76 isorel 6046 . . . . . . . . . . 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 1182 . . . . . . . . . 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 202 . . . . . . . . 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 20 . . . . . . . . . . . 12  |-  ( x  =  z  ->  x  =  z )
80 oveq2 6089 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
1  -  x )  =  ( 1  -  z ) )
8179, 80oveq12d 6099 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  /  ( 1  -  x ) )  =  ( z  / 
( 1  -  z
) ) )
82 ovex 6106 . . . . . . . . . . 11  |-  ( z  /  ( 1  -  z ) )  e. 
_V
8381, 47, 82fvmpt 5806 . . . . . . . . . 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 20 . . . . . . . . . . . 12  |-  ( x  =  w  ->  x  =  w )
86 oveq2 6089 . . . . . . . . . . . 12  |-  ( x  =  w  ->  (
1  -  x )  =  ( 1  -  w ) )
8785, 86oveq12d 6099 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  /  ( 1  -  x ) )  =  ( w  / 
( 1  -  w
) ) )
88 ovex 6106 . . . . . . . . . . 11  |-  ( w  /  ( 1  -  w ) )  e. 
_V
8987, 47, 88fvmpt 5806 . . . . . . . . . 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 4241 . . . . . . . 8  |-  ( ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w
)  /\  -.  w  =  1 )  -> 
( z  /  (
1  -  z ) )  <  ( w  /  ( 1  -  w ) ) )
9232, 33, 45, 91ifbothda 3769 . . . . . . 7  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 )  /\  z  <  w )  -> 
( z  /  (
1  -  z ) )  <  if ( w  =  1 , 
+oo ,  ( w  /  ( 1  -  w ) ) ) )
9331, 92eqbrtrd 4232 . . . . . 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 1155 . . . . 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 2442 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  1  <->  z  =  1 ) )
9695, 81ifbieq2d 3759 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( z  =  1 ,  +oo ,  ( z  /  (
1  -  z ) ) ) )
97 pnfxr 10713 . . . . . . . . 9  |-  +oo  e.  RR*
9897elexi 2965 . . . . . . . 8  |-  +oo  e.  _V
9998, 82ifex 3797 . . . . . . 7  |-  if ( z  =  1 , 
+oo ,  ( z  /  ( 1  -  z ) ) )  e.  _V
10096, 10, 99fvmpt 5806 . . . . . 6  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  z )  =  if ( z  =  1 ,  +oo , 
( z  /  (
1  -  z ) ) ) )
101 eqeq1 2442 . . . . . . . 8  |-  ( x  =  w  ->  (
x  =  1  <->  w  =  1 ) )
102101, 87ifbieq2d 3759 . . . . . . 7  |-  ( x  =  w  ->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) )  =  if ( w  =  1 ,  +oo ,  ( w  /  (
1  -  w ) ) ) )
10398, 88ifex 3797 . . . . . . 7  |-  if ( w  =  1 , 
+oo ,  ( w  /  ( 1  -  w ) ) )  e.  _V
104102, 10, 103fvmpt 5806 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  ->  ( F `  w )  =  if ( w  =  1 ,  +oo , 
( w  /  (
1  -  w ) ) ) )
105100, 104breqan12d 4227 . . . . 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 ) ) ) ) )
10694, 105sylibrd 226 . . . 4  |-  ( ( z  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  ->  ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
) )
107106rgen2a 2772 . . 3  |-  A. z  e.  ( 0 [,] 1
) A. w  e.  ( 0 [,] 1
) ( z  < 
w  ->  ( F `  z )  <  ( F `  w )
)
108 soisoi 6048 . . 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 ) ) )
1094, 9, 14, 107, 108mp4an 655 . 2  |-  F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )
110 letsr 14672 . . . . . 6  |-  <_  e.  TosetRel
111110elexi 2965 . . . . 5  |-  <_  e.  _V
112111inex1 4344 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  _V
113111inex1 4344 . . . 4  |-  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) )  e.  _V
114 leiso 11708 . . . . . . . 8  |-  ( ( ( 0 [,] 1
)  C_  RR*  /\  (
0 [,]  +oo )  C_  RR* )  ->  ( F  Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,]  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) ) ) )
1151, 5, 114mp2an 654 . . . . . . 7  |-  ( F 
Isom  <  ,  <  (
( 0 [,] 1
) ,  ( 0 [,]  +oo ) )  <->  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) ) )
116109, 115mpbi 200 . . . . . 6  |-  F  Isom  <_  ,  <_  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )
117 isores1 6054 . . . . . 6  |-  ( F 
Isom  <_  ,  <_  (
( 0 [,] 1
) ,  ( 0 [,]  +oo ) )  <->  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  <_  ( (
0 [,] 1 ) ,  ( 0 [,] 
+oo ) ) )
118116, 117mpbi 200 . . . . 5  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  <_  ( (
0 [,] 1 ) ,  ( 0 [,] 
+oo ) )
119 isores2 6053 . . . . 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 ) ) )
120118, 119mpbi 200 . . . 4  |-  F  Isom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ,  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )
121 tsrps 14653 . . . . . . . 8  |-  (  <_  e. 
TosetRel  ->  <_  e.  PosetRel )
122110, 121ax-mp 8 . . . . . . 7  |-  <_  e.  PosetRel
123 ledm 14669 . . . . . . . 8  |-  RR*  =  dom  <_
124123psssdm 14648 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,] 1 ) 
C_  RR* )  ->  dom  (  <_  i^i  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  =  ( 0 [,] 1 ) )
125122, 1, 124mp2an 654 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  =  ( 0 [,] 1
)
126125eqcomi 2440 . . . . 5  |-  ( 0 [,] 1 )  =  dom  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
127123psssdm 14648 . . . . . . 7  |-  ( (  <_  e.  PosetRel  /\  (
0 [,]  +oo )  C_  RR* )  ->  dom  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) )  =  ( 0 [,]  +oo ) )
128122, 5, 127mp2an 654 . . . . . 6  |-  dom  (  <_  i^i  ( ( 0 [,]  +oo )  X.  (
0 [,]  +oo ) ) )  =  ( 0 [,]  +oo )
129128eqcomi 2440 . . . . 5  |-  ( 0 [,]  +oo )  =  dom  (  <_  i^i  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) )
130126, 129ordthmeo 17834 . . . 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 ) ) ) ) ) )
131112, 113, 120, 130mp3an 1279 . . 3  |-  F  e.  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) ) ) 
Homeo  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) ) )
132 dfii5 18915 . . . 4  |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
133 iccpnfhmeo.k . . . . 5  |-  K  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
)
134 ordtresticc 17287 . . . . 5  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) )
135133, 134eqtri 2456 . . . 4  |-  K  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) )
136132, 135oveq12i 6093 . . 3  |-  ( II 
Homeo  K )  =  ( (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )  Homeo  (ordTop `  (  <_  i^i  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) ) ) ) )
137131, 136eleqtrri 2509 . 2  |-  F  e.  ( II  Homeo  K )
138109, 137pm3.2i 442 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956    u. cun 3318    i^i cin 3319    C_ wss 3320   ifcif 3739   {csn 3814   class class class wbr 4212    e. cmpt 4266    Po wpo 4501    Or wor 4502    X. cxp 4876   `'ccnv 4877   dom cdm 4878   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   [,)cico 10918   [,]cicc 10919   ↾t crest 13648   TopOpenctopn 13649  ordTopcordt 13721   PosetRelcps 14624    TosetRel ctsr 14625  ℂfldccnfld 16703    Homeo chmeo 17785   IIcii 18905
This theorem is referenced by:  xrhmeo  18971  xrge0hmph  24318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-rest 13650  df-topn 13651  df-topgen 13667  df-ordt 13725  df-ps 14629  df-tsr 14630  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cn 17291  df-hmeo 17787  df-xms 18350  df-ms 18351  df-ii 18907
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