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Theorem xrge0iifcnv 23317
Description: Define a bijection from  [ 0 ,  1 ] to  [
0 ,  +oo ]. (Contributed by Thierry Arnoux, 29-Mar-2017.)
Hypothesis
Ref Expression
xrge0iifhmeo.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) )
Assertion
Ref Expression
xrge0iifcnv  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( y  e.  ( 0 [,]  +oo )  |->  if ( y  = 
+oo ,  0 , 
( exp `  -u y
) ) ) )
Distinct variable group:    x, y
Allowed substitution hints:    F( x, y)

Proof of Theorem xrge0iifcnv
StepHypRef Expression
1 xrge0iifhmeo.1 . . 3  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) )
2 0xr 8880 . . . . . . 7  |-  0  e.  RR*
3 pnfxr 10457 . . . . . . 7  |-  +oo  e.  RR*
4 pnfge 10471 . . . . . . . 8  |-  ( 0  e.  RR*  ->  0  <_  +oo )
52, 4ax-mp 8 . . . . . . 7  |-  0  <_  +oo
6 ubicc2 10755 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <_  +oo )  ->  +oo  e.  ( 0 [,]  +oo ) )
72, 3, 5, 6mp3an 1277 . . . . . 6  |-  +oo  e.  ( 0 [,]  +oo )
87a1i 10 . . . . 5  |-  ( ( x  e.  ( 0 [,] 1 )  /\  x  =  0 )  ->  +oo  e.  (
0 [,]  +oo ) )
9 icossicc 23260 . . . . . 6  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
10 ressxr 8878 . . . . . . . . . . . . . . . 16  |-  RR  C_  RR*
11 1re 8839 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
1210, 11sselii 3179 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
13 0le1 9299 . . . . . . . . . . . . . . 15  |-  0  <_  1
14 snunioc 23269 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
152, 12, 13, 14mp3an 1277 . . . . . . . . . . . . . 14  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
16 uncom 3321 . . . . . . . . . . . . . . 15  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( ( 0 (,] 1 )  u.  { 0 } )
1716eqeq1i 2292 . . . . . . . . . . . . . 14  |-  ( ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 )  <-> 
( ( 0 (,] 1 )  u.  {
0 } )  =  ( 0 [,] 1
) )
1815, 17mpbi 199 . . . . . . . . . . . . 13  |-  ( ( 0 (,] 1 )  u.  { 0 } )  =  ( 0 [,] 1 )
1918eleq2i 2349 . . . . . . . . . . . 12  |-  ( x  e.  ( ( 0 (,] 1 )  u. 
{ 0 } )  <-> 
x  e.  ( 0 [,] 1 ) )
20 elun 3318 . . . . . . . . . . . 12  |-  ( x  e.  ( ( 0 (,] 1 )  u. 
{ 0 } )  <-> 
( x  e.  ( 0 (,] 1 )  \/  x  e.  {
0 } ) )
2119, 20bitr3i 242 . . . . . . . . . . 11  |-  ( x  e.  ( 0 [,] 1 )  <->  ( x  e.  ( 0 (,] 1
)  \/  x  e. 
{ 0 } ) )
22 pm2.53 362 . . . . . . . . . . 11  |-  ( ( x  e.  ( 0 (,] 1 )  \/  x  e.  { 0 } )  ->  ( -.  x  e.  (
0 (,] 1 )  ->  x  e.  {
0 } ) )
2321, 22sylbi 187 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  e.  (
0 (,] 1 )  ->  x  e.  {
0 } ) )
24 elsni 3666 . . . . . . . . . 10  |-  ( x  e.  { 0 }  ->  x  =  0 )
2523, 24syl6 29 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  e.  (
0 (,] 1 )  ->  x  =  0 ) )
2625con1d 116 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  =  0  ->  x  e.  ( 0 (,] 1 ) ) )
2726imp 418 . . . . . . 7  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  x  e.  ( 0 (,] 1
) )
28 xrleid 10486 . . . . . . . . . . . . . . 15  |-  ( 0  e.  RR*  ->  0  <_ 
0 )
292, 28ax-mp 8 . . . . . . . . . . . . . 14  |-  0  <_  0
30 ltpnf 10465 . . . . . . . . . . . . . . 15  |-  ( 1  e.  RR  ->  1  <  +oo )
3111, 30ax-mp 8 . . . . . . . . . . . . . 14  |-  1  <  +oo
32 iocssioo 23263 . . . . . . . . . . . . . 14  |-  ( ( ( 0  e.  RR*  /\ 
+oo  e.  RR* )  /\  ( 0  <_  0  /\  1  <  +oo )
)  ->  ( 0 (,] 1 )  C_  ( 0 (,)  +oo ) )
332, 3, 29, 31, 32mp4an 654 . . . . . . . . . . . . 13  |-  ( 0 (,] 1 )  C_  ( 0 (,)  +oo )
34 ioorp 10729 . . . . . . . . . . . . 13  |-  ( 0 (,)  +oo )  =  RR+
3533, 34sseqtri 3212 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR+
3635sseli 3178 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  x  e.  RR+ )
37 relogcl 19934 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
3836, 37syl 15 . . . . . . . . . 10  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  e.  RR )
3938renegcld 9212 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e.  RR )
4010, 39sseldi 3180 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e. 
RR* )
41 elioc1 10700 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
x  e.  ( 0 (,] 1 )  <->  ( x  e.  RR*  /\  0  < 
x  /\  x  <_  1 ) ) )
422, 12, 41mp2an 653 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 (,] 1 )  <->  ( x  e.  RR*  /\  0  < 
x  /\  x  <_  1 ) )
4342simp3bi 972 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  x  <_  1 )
44 1rp 10360 . . . . . . . . . . . . 13  |-  1  e.  RR+
4544a1i 10 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 (,] 1 )  ->  1  e.  RR+ )
4636, 45logled 19980 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  (
x  <_  1  <->  ( log `  x )  <_  ( log `  1 ) ) )
4743, 46mpbid 201 . . . . . . . . . 10  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  <_ 
( log `  1
) )
48 log1 19941 . . . . . . . . . . 11  |-  ( log `  1 )  =  0
4948breq2i 4033 . . . . . . . . . 10  |-  ( ( log `  x )  <_  ( log `  1
)  <->  ( log `  x
)  <_  0 )
5047, 49sylib 188 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  <_ 
0 )
51 le0neg1 9284 . . . . . . . . . 10  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  <_  0  <->  0  <_  -u ( log `  x ) ) )
5238, 51syl 15 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  (
( log `  x
)  <_  0  <->  0  <_  -u ( log `  x ) ) )
5350, 52mpbid 201 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  0  <_ 
-u ( log `  x
) )
54 ltpnf 10465 . . . . . . . . 9  |-  ( -u ( log `  x )  e.  RR  ->  -u ( log `  x )  <  +oo )
5539, 54syl 15 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  <  +oo )
56 elico1 10701 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR* )  ->  ( -u ( log `  x
)  e.  ( 0 [,)  +oo )  <->  ( -u ( log `  x )  e. 
RR*  /\  0  <_  -u ( log `  x )  /\  -u ( log `  x
)  <  +oo ) ) )
572, 3, 56mp2an 653 . . . . . . . 8  |-  ( -u ( log `  x )  e.  ( 0 [,) 
+oo )  <->  ( -u ( log `  x )  e. 
RR*  /\  0  <_  -u ( log `  x )  /\  -u ( log `  x
)  <  +oo ) )
5840, 53, 55, 57syl3anbrc 1136 . . . . . . 7  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e.  ( 0 [,)  +oo ) )
5927, 58syl 15 . . . . . 6  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  -u ( log `  x )  e.  ( 0 [,)  +oo )
)
609, 59sseldi 3180 . . . . 5  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  -u ( log `  x )  e.  ( 0 [,]  +oo )
)
618, 60ifclda 3594 . . . 4  |-  ( x  e.  ( 0 [,] 1 )  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,] 
+oo ) )
6261adantl 452 . . 3  |-  ( (  T.  /\  x  e.  ( 0 [,] 1
) )  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,] 
+oo ) )
63 0elunit 10756 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
6463a1i 10 . . . . 5  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  y  =  +oo )  ->  0  e.  ( 0 [,] 1
) )
65 ssun2 3341 . . . . . . 7  |-  ( 0 (,] 1 )  C_  ( { 0 }  u.  ( 0 (,] 1
) )
6665, 15sseqtri 3212 . . . . . 6  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
67 snunico 10765 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <_  +oo )  ->  ( ( 0 [,)  +oo )  u.  {  +oo } )  =  ( 0 [,] 
+oo ) )
682, 3, 5, 67mp3an 1277 . . . . . . . . . . . . 13  |-  ( ( 0 [,)  +oo )  u.  {  +oo } )  =  ( 0 [,] 
+oo )
6968eleq2i 2349 . . . . . . . . . . . 12  |-  ( y  e.  ( ( 0 [,)  +oo )  u.  {  +oo } )  <->  y  e.  ( 0 [,]  +oo ) )
70 elun 3318 . . . . . . . . . . . 12  |-  ( y  e.  ( ( 0 [,)  +oo )  u.  {  +oo } )  <->  ( y  e.  ( 0 [,)  +oo )  \/  y  e.  { 
+oo } ) )
7169, 70bitr3i 242 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,] 
+oo )  <->  ( y  e.  ( 0 [,)  +oo )  \/  y  e.  { 
+oo } ) )
72 pm2.53 362 . . . . . . . . . . 11  |-  ( ( y  e.  ( 0 [,)  +oo )  \/  y  e.  {  +oo } )  ->  ( -.  y  e.  ( 0 [,)  +oo )  ->  y  e.  {  +oo } ) )
7371, 72sylbi 187 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,] 
+oo )  ->  ( -.  y  e.  (
0 [,)  +oo )  -> 
y  e.  {  +oo } ) )
74 elsni 3666 . . . . . . . . . 10  |-  ( y  e.  {  +oo }  ->  y  =  +oo )
7573, 74syl6 29 . . . . . . . . 9  |-  ( y  e.  ( 0 [,] 
+oo )  ->  ( -.  y  e.  (
0 [,)  +oo )  -> 
y  =  +oo )
)
7675con1d 116 . . . . . . . 8  |-  ( y  e.  ( 0 [,] 
+oo )  ->  ( -.  y  =  +oo  ->  y  e.  ( 0 [,)  +oo ) ) )
7776imp 418 . . . . . . 7  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
y  e.  ( 0 [,)  +oo ) )
78 mnfxr 10458 . . . . . . . . . . . . . 14  |-  -oo  e.  RR*
79 0re 8840 . . . . . . . . . . . . . . 15  |-  0  e.  RR
80 mnflt 10466 . . . . . . . . . . . . . . 15  |-  ( 0  e.  RR  ->  -oo  <  0 )
8179, 80ax-mp 8 . . . . . . . . . . . . . 14  |-  -oo  <  0
82 xrleid 10486 . . . . . . . . . . . . . . 15  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
833, 82ax-mp 8 . . . . . . . . . . . . . 14  |-  +oo  <_  +oo
84 icossioo 23264 . . . . . . . . . . . . . 14  |-  ( ( (  -oo  e.  RR*  /\ 
+oo  e.  RR* )  /\  (  -oo  <  0  /\  +oo 
<_  +oo ) )  -> 
( 0 [,)  +oo )  C_  (  -oo (,)  +oo ) )
8578, 3, 81, 83, 84mp4an 654 . . . . . . . . . . . . 13  |-  ( 0 [,)  +oo )  C_  (  -oo (,)  +oo )
86 ioomax 10726 . . . . . . . . . . . . 13  |-  (  -oo (,) 
+oo )  =  RR
8785, 86sseqtri 3212 . . . . . . . . . . . 12  |-  ( 0 [,)  +oo )  C_  RR
8887sseli 3178 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  RR )
8988renegcld 9212 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  -u y  e.  RR )
90 reefcl 12370 . . . . . . . . . 10  |-  ( -u y  e.  RR  ->  ( exp `  -u y
)  e.  RR )
9189, 90syl 15 . . . . . . . . 9  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e.  RR )
9210, 91sseldi 3180 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e. 
RR* )
93 efgt0 12385 . . . . . . . . 9  |-  ( -u y  e.  RR  ->  0  <  ( exp `  -u y
) )
9489, 93syl 15 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  0  <  ( exp `  -u y
) )
95 elico1 10701 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR* )  ->  (
y  e.  ( 0 [,)  +oo )  <->  ( y  e.  RR*  /\  0  <_ 
y  /\  y  <  +oo ) ) )
962, 3, 95mp2an 653 . . . . . . . . . . . 12  |-  ( y  e.  ( 0 [,) 
+oo )  <->  ( y  e.  RR*  /\  0  <_ 
y  /\  y  <  +oo ) )
9796simp2bi 971 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  0  <_  y )
98 le0neg2 9285 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  (
0  <_  y  <->  -u y  <_ 
0 ) )
9988, 98syl 15 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  (
0  <_  y  <->  -u y  <_ 
0 ) )
10097, 99mpbid 201 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  -u y  <_  0 )
101 efle 12400 . . . . . . . . . . 11  |-  ( (
-u y  e.  RR  /\  0  e.  RR )  ->  ( -u y  <_  0  <->  ( exp `  -u y
)  <_  ( exp `  0 ) ) )
10289, 79, 101sylancl 643 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( -u y  <_  0  <->  ( exp `  -u y )  <_  ( exp `  0 ) ) )
103100, 102mpbid 201 . . . . . . . . 9  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  <_ 
( exp `  0
) )
104 ef0 12374 . . . . . . . . 9  |-  ( exp `  0 )  =  1
105103, 104syl6breq 4064 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  <_ 
1 )
106 elioc1 10700 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
( exp `  -u y
)  e.  ( 0 (,] 1 )  <->  ( ( exp `  -u y )  e. 
RR*  /\  0  <  ( exp `  -u y
)  /\  ( exp `  -u y )  <_  1
) ) )
1072, 12, 106mp2an 653 . . . . . . . 8  |-  ( ( exp `  -u y
)  e.  ( 0 (,] 1 )  <->  ( ( exp `  -u y )  e. 
RR*  /\  0  <  ( exp `  -u y
)  /\  ( exp `  -u y )  <_  1
) )
10892, 94, 105, 107syl3anbrc 1136 . . . . . . 7  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e.  ( 0 (,] 1
) )
10977, 108syl 15 . . . . . 6  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
( exp `  -u y
)  e.  ( 0 (,] 1 ) )
11066, 109sseldi 3180 . . . . 5  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
( exp `  -u y
)  e.  ( 0 [,] 1 ) )
11164, 110ifclda 3594 . . . 4  |-  ( y  e.  ( 0 [,] 
+oo )  ->  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) )  e.  ( 0 [,] 1 ) )
112111adantl 452 . . 3  |-  ( (  T.  /\  y  e.  ( 0 [,]  +oo ) )  ->  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) )  e.  ( 0 [,] 1 ) )
113 eqeq2 2294 . . . . . 6  |-  ( 0  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y
) )  ->  (
x  =  0  <->  x  =  if ( y  = 
+oo ,  0 , 
( exp `  -u y
) ) ) )
114113bibi1d 310 . . . . 5  |-  ( 0  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y
) )  ->  (
( x  =  0  <-> 
y  =  if ( x  =  0 , 
+oo ,  -u ( log `  x ) ) )  <-> 
( x  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) )  <-> 
y  =  if ( x  =  0 , 
+oo ,  -u ( log `  x ) ) ) ) )
115 eqeq2 2294 . . . . . 6  |-  ( ( exp `  -u y
)  =  if ( y  =  +oo , 
0 ,  ( exp `  -u y ) )  ->  ( x  =  ( exp `  -u y
)  <->  x  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) ) ) )
116115bibi1d 310 . . . . 5  |-  ( ( exp `  -u y
)  =  if ( y  =  +oo , 
0 ,  ( exp `  -u y ) )  ->  ( ( x  =  ( exp `  -u y
)  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )  <->  ( x  =  if ( y  = 
+oo ,  0 , 
( exp `  -u y
) )  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) ) )
117 simpr 447 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  y  = 
+oo )
118 iftrue 3573 . . . . . . . 8  |-  ( x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  +oo )
119118eqeq2d 2296 . . . . . . 7  |-  ( x  =  0  ->  (
y  =  if ( x  =  0 , 
+oo ,  -u ( log `  x ) )  <->  y  =  +oo ) )
120117, 119syl5ibrcom 213 . . . . . 6  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( x  =  0  ->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) ) )
121 ubico 23270 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  -.  +oo 
e.  ( 0 [,) 
+oo ) )
12279, 3, 121mp2an 653 . . . . . . . . . 10  |-  -.  +oo  e.  ( 0 [,)  +oo )
123 df-nel 2451 . . . . . . . . . 10  |-  (  +oo  e/  ( 0 [,)  +oo ) 
<->  -.  +oo  e.  (
0 [,)  +oo ) )
124122, 123mpbir 200 . . . . . . . . 9  |-  +oo  e/  ( 0 [,)  +oo )
125 neleq1 2539 . . . . . . . . . 10  |-  ( y  =  +oo  ->  (
y  e/  ( 0 [,)  +oo )  <->  +oo  e/  (
0 [,)  +oo ) ) )
126125adantl 452 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( y  e/  ( 0 [,) 
+oo )  <->  +oo  e/  (
0 [,)  +oo ) ) )
127124, 126mpbiri 224 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  y  e/  ( 0 [,)  +oo ) )
128 neleq1 2539 . . . . . . . 8  |-  ( y  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  (
y  e/  ( 0 [,)  +oo )  <->  if (
x  =  0 , 
+oo ,  -u ( log `  x ) )  e/  ( 0 [,)  +oo ) ) )
129127, 128syl5ibcom 211 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( y  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e/  ( 0 [,) 
+oo ) ) )
130 df-nel 2451 . . . . . . . 8  |-  ( if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e/  ( 0 [,)  +oo )  <->  -.  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,) 
+oo ) )
131 iffalse 3574 . . . . . . . . . . . . 13  |-  ( -.  x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  -u ( log `  x ) )
132131adantl 452 . . . . . . . . . . . 12  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  if (
x  =  0 , 
+oo ,  -u ( log `  x ) )  = 
-u ( log `  x
) )
133132, 59eqeltrd 2359 . . . . . . . . . . 11  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  if (
x  =  0 , 
+oo ,  -u ( log `  x ) )  e.  ( 0 [,)  +oo ) )
134133ex 423 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) )  e.  ( 0 [,)  +oo )
) )
135134ad2antrr 706 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( -.  x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,)  +oo ) ) )
136135con1d 116 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( -.  if ( x  =  0 ,  +oo ,  -u ( log `  x
) )  e.  ( 0 [,)  +oo )  ->  x  =  0 ) )
137130, 136syl5bi 208 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e/  ( 0 [,)  +oo )  ->  x  =  0 ) )
138129, 137syld 40 . . . . . 6  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( y  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  x  =  0 ) )
139120, 138impbid 183 . . . . 5  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( x  =  0  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
140 eqeq2 2294 . . . . . . 7  |-  (  +oo  =  if ( x  =  0 ,  +oo ,  -u ( log `  x
) )  ->  (
y  =  +oo  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
141140bibi2d 309 . . . . . 6  |-  (  +oo  =  if ( x  =  0 ,  +oo ,  -u ( log `  x
) )  ->  (
( x  =  ( exp `  -u y
)  <->  y  =  +oo ) 
<->  ( x  =  ( exp `  -u y
)  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) ) )
142 eqeq2 2294 . . . . . . 7  |-  ( -u ( log `  x )  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  (
y  =  -u ( log `  x )  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
143142bibi2d 309 . . . . . 6  |-  ( -u ( log `  x )  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  (
( x  =  ( exp `  -u y
)  <->  y  =  -u ( log `  x ) )  <->  ( x  =  ( exp `  -u y
)  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) ) )
14479a1i 10 . . . . . . . . . . . 12  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
0  e.  RR )
14577, 94syl 15 . . . . . . . . . . . 12  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
0  <  ( exp `  -u y ) )
146144, 145ltned 8957 . . . . . . . . . . 11  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
0  =/=  ( exp `  -u y ) )
147146adantll 694 . . . . . . . . . 10  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  0  =/=  ( exp `  -u y
) )
148147neneqd 2464 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  -.  0  =  ( exp `  -u y ) )
149 eqeq1 2291 . . . . . . . . . 10  |-  ( x  =  0  ->  (
x  =  ( exp `  -u y )  <->  0  =  ( exp `  -u y
) ) )
150149notbid 285 . . . . . . . . 9  |-  ( x  =  0  ->  ( -.  x  =  ( exp `  -u y )  <->  -.  0  =  ( exp `  -u y
) ) )
151148, 150syl5ibrcom 213 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  (
x  =  0  ->  -.  x  =  ( exp `  -u y ) ) )
152151imp 418 . . . . . . 7  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  x  =  0 )  ->  -.  x  =  ( exp `  -u y
) )
153 simplr 731 . . . . . . 7  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  x  =  0 )  ->  -.  y  =  +oo )
154152, 1532falsed 340 . . . . . 6  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  x  =  0 )  ->  ( x  =  ( exp `  -u y
)  <->  y  =  +oo ) )
155 eqcom 2287 . . . . . . . . . . 11  |-  ( x  =  ( exp `  -u y
)  <->  ( exp `  -u y
)  =  x )
156155a1i 10 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( x  =  ( exp `  -u y
)  <->  ( exp `  -u y
)  =  x ) )
157 relogeftb 19940 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  -u y  e.  RR )  ->  (
( log `  x
)  =  -u y  <->  ( exp `  -u y
)  =  x ) )
15836, 89, 157syl2an 463 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( ( log `  x
)  =  -u y  <->  ( exp `  -u y
)  =  x ) )
15938recnd 8863 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  e.  CC )
16088recnd 8863 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  CC )
161 negcon2 9102 . . . . . . . . . . 11  |-  ( ( ( log `  x
)  e.  CC  /\  y  e.  CC )  ->  ( ( log `  x
)  =  -u y  <->  y  =  -u ( log `  x
) ) )
162159, 160, 161syl2an 463 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( ( log `  x
)  =  -u y  <->  y  =  -u ( log `  x
) ) )
163156, 158, 1623bitr2d 272 . . . . . . . . 9  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( x  =  ( exp `  -u y
)  <->  y  =  -u ( log `  x ) ) )
16427, 77, 163syl2an 463 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0 )  /\  (
y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo ) )  ->  ( x  =  ( exp `  -u y
)  <->  y  =  -u ( log `  x ) ) )
165164an4s 799 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  ( -.  x  =  0  /\  -.  y  =  +oo ) )  ->  (
x  =  ( exp `  -u y )  <->  y  =  -u ( log `  x
) ) )
166165anass1rs 782 . . . . . 6  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  -.  x  =  0
)  ->  ( x  =  ( exp `  -u y
)  <->  y  =  -u ( log `  x ) ) )
167141, 143, 154, 166ifbothda 3597 . . . . 5  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  (
x  =  ( exp `  -u y )  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
168114, 116, 139, 167ifbothda 3597 . . . 4  |-  ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo ) )  -> 
( x  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) )  <-> 
y  =  if ( x  =  0 , 
+oo ,  -u ( log `  x ) ) ) )
169168adantl 452 . . 3  |-  ( (  T.  /\  ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo ) ) )  -> 
( x  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) )  <-> 
y  =  if ( x  =  0 , 
+oo ,  -u ( log `  x ) ) ) )
1701, 62, 112, 169f1ocnv2d 6070 . 2  |-  (  T. 
->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  (
y  e.  ( 0 [,]  +oo )  |->  if ( y  =  +oo , 
0 ,  ( exp `  -u y ) ) ) ) )
171170trud 1314 1  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( y  e.  ( 0 [,]  +oo )  |->  if ( y  = 
+oo ,  0 , 
( exp `  -u y
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1625    e. wcel 1686    =/= wne 2448    e/ wnel 2449    u. cun 3152    C_ wss 3154   ifcif 3567   {csn 3642   class class class wbr 4025    e. cmpt 4079   `'ccnv 4690   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868    < clt 8869    <_ cle 8870   -ucneg 9040   RR+crp 10356   (,)cioo 10658   (,]cioc 10659   [,)cico 10660   [,]cicc 10661   expce 12345   logclog 19914
This theorem is referenced by:  xrge0iifiso  23319  xrge0iifmhm  23323  xrge0pluscn  23324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916
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