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Theorem xrge0iifcnv 24311
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 9123 . . . . . . 7  |-  0  e.  RR*
3 pnfxr 10705 . . . . . . 7  |-  +oo  e.  RR*
4 pnfge 10719 . . . . . . . 8  |-  ( 0  e.  RR*  ->  0  <_  +oo )
52, 4ax-mp 8 . . . . . . 7  |-  0  <_  +oo
6 ubicc2 11006 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <_  +oo )  ->  +oo  e.  ( 0 [,]  +oo ) )
72, 3, 5, 6mp3an 1279 . . . . . 6  |-  +oo  e.  ( 0 [,]  +oo )
87a1i 11 . . . . 5  |-  ( ( x  e.  ( 0 [,] 1 )  /\  x  =  0 )  ->  +oo  e.  (
0 [,]  +oo ) )
9 icossicc 24121 . . . . . 6  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
10 uncom 3483 . . . . . . . . . . . . . 14  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( ( 0 (,] 1 )  u.  { 0 } )
11 1re 9082 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
1211rexri 9129 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
13 0le1 9543 . . . . . . . . . . . . . . 15  |-  0  <_  1
14 snunioc 24129 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
152, 12, 13, 14mp3an 1279 . . . . . . . . . . . . . 14  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
1610, 15eqtr3i 2457 . . . . . . . . . . . . 13  |-  ( ( 0 (,] 1 )  u.  { 0 } )  =  ( 0 [,] 1 )
1716eleq2i 2499 . . . . . . . . . . . 12  |-  ( x  e.  ( ( 0 (,] 1 )  u. 
{ 0 } )  <-> 
x  e.  ( 0 [,] 1 ) )
18 elun 3480 . . . . . . . . . . . 12  |-  ( x  e.  ( ( 0 (,] 1 )  u. 
{ 0 } )  <-> 
( x  e.  ( 0 (,] 1 )  \/  x  e.  {
0 } ) )
1917, 18bitr3i 243 . . . . . . . . . . 11  |-  ( x  e.  ( 0 [,] 1 )  <->  ( x  e.  ( 0 (,] 1
)  \/  x  e. 
{ 0 } ) )
20 pm2.53 363 . . . . . . . . . . 11  |-  ( ( x  e.  ( 0 (,] 1 )  \/  x  e.  { 0 } )  ->  ( -.  x  e.  (
0 (,] 1 )  ->  x  e.  {
0 } ) )
2119, 20sylbi 188 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  e.  (
0 (,] 1 )  ->  x  e.  {
0 } ) )
22 elsni 3830 . . . . . . . . . 10  |-  ( x  e.  { 0 }  ->  x  =  0 )
2321, 22syl6 31 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  e.  (
0 (,] 1 )  ->  x  =  0 ) )
2423con1d 118 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  =  0  ->  x  e.  ( 0 (,] 1 ) ) )
2524imp 419 . . . . . . 7  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  x  e.  ( 0 (,] 1
) )
26 0le0 10073 . . . . . . . . . . . . . 14  |-  0  <_  0
27 ltpnf 10713 . . . . . . . . . . . . . . 15  |-  ( 1  e.  RR  ->  1  <  +oo )
2811, 27ax-mp 8 . . . . . . . . . . . . . 14  |-  1  <  +oo
29 iocssioo 24124 . . . . . . . . . . . . . 14  |-  ( ( ( 0  e.  RR*  /\ 
+oo  e.  RR* )  /\  ( 0  <_  0  /\  1  <  +oo )
)  ->  ( 0 (,] 1 )  C_  ( 0 (,)  +oo ) )
302, 3, 26, 28, 29mp4an 655 . . . . . . . . . . . . 13  |-  ( 0 (,] 1 )  C_  ( 0 (,)  +oo )
31 ioorp 10980 . . . . . . . . . . . . 13  |-  ( 0 (,)  +oo )  =  RR+
3230, 31sseqtri 3372 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR+
3332sseli 3336 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  x  e.  RR+ )
3433relogcld 20510 . . . . . . . . . 10  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  e.  RR )
3534renegcld 9456 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e.  RR )
3635rexrd 9126 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e. 
RR* )
37 elioc1 10950 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
x  e.  ( 0 (,] 1 )  <->  ( x  e.  RR*  /\  0  < 
x  /\  x  <_  1 ) ) )
382, 12, 37mp2an 654 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 (,] 1 )  <->  ( x  e.  RR*  /\  0  < 
x  /\  x  <_  1 ) )
3938simp3bi 974 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  x  <_  1 )
40 1rp 10608 . . . . . . . . . . . . 13  |-  1  e.  RR+
4140a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 (,] 1 )  ->  1  e.  RR+ )
4233, 41logled 20514 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  (
x  <_  1  <->  ( log `  x )  <_  ( log `  1 ) ) )
4339, 42mpbid 202 . . . . . . . . . 10  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  <_ 
( log `  1
) )
44 log1 20472 . . . . . . . . . 10  |-  ( log `  1 )  =  0
4543, 44syl6breq 4243 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  <_ 
0 )
4634le0neg1d 9590 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  (
( log `  x
)  <_  0  <->  0  <_  -u ( log `  x ) ) )
4745, 46mpbid 202 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  0  <_ 
-u ( log `  x
) )
48 ltpnf 10713 . . . . . . . . 9  |-  ( -u ( log `  x )  e.  RR  ->  -u ( log `  x )  <  +oo )
4935, 48syl 16 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  <  +oo )
50 elico1 10951 . . . . . . . . 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 ) ) )
512, 3, 50mp2an 654 . . . . . . . 8  |-  ( -u ( log `  x )  e.  ( 0 [,) 
+oo )  <->  ( -u ( log `  x )  e. 
RR*  /\  0  <_  -u ( log `  x )  /\  -u ( log `  x
)  <  +oo ) )
5236, 47, 49, 51syl3anbrc 1138 . . . . . . 7  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e.  ( 0 [,)  +oo ) )
5325, 52syl 16 . . . . . 6  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  -u ( log `  x )  e.  ( 0 [,)  +oo )
)
549, 53sseldi 3338 . . . . 5  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  -u ( log `  x )  e.  ( 0 [,]  +oo )
)
558, 54ifclda 3758 . . . 4  |-  ( x  e.  ( 0 [,] 1 )  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,] 
+oo ) )
5655adantl 453 . . 3  |-  ( (  T.  /\  x  e.  ( 0 [,] 1
) )  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,] 
+oo ) )
57 0elunit 11007 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
5857a1i 11 . . . . 5  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  y  =  +oo )  ->  0  e.  ( 0 [,] 1
) )
59 iocssicc 24122 . . . . . 6  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
60 snunico 11016 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <_  +oo )  ->  ( ( 0 [,)  +oo )  u.  {  +oo } )  =  ( 0 [,] 
+oo ) )
612, 3, 5, 60mp3an 1279 . . . . . . . . . . . . 13  |-  ( ( 0 [,)  +oo )  u.  {  +oo } )  =  ( 0 [,] 
+oo )
6261eleq2i 2499 . . . . . . . . . . . 12  |-  ( y  e.  ( ( 0 [,)  +oo )  u.  {  +oo } )  <->  y  e.  ( 0 [,]  +oo ) )
63 elun 3480 . . . . . . . . . . . 12  |-  ( y  e.  ( ( 0 [,)  +oo )  u.  {  +oo } )  <->  ( y  e.  ( 0 [,)  +oo )  \/  y  e.  { 
+oo } ) )
6462, 63bitr3i 243 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,] 
+oo )  <->  ( y  e.  ( 0 [,)  +oo )  \/  y  e.  { 
+oo } ) )
65 pm2.53 363 . . . . . . . . . . 11  |-  ( ( y  e.  ( 0 [,)  +oo )  \/  y  e.  {  +oo } )  ->  ( -.  y  e.  ( 0 [,)  +oo )  ->  y  e.  {  +oo } ) )
6664, 65sylbi 188 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,] 
+oo )  ->  ( -.  y  e.  (
0 [,)  +oo )  -> 
y  e.  {  +oo } ) )
67 elsni 3830 . . . . . . . . . 10  |-  ( y  e.  {  +oo }  ->  y  =  +oo )
6866, 67syl6 31 . . . . . . . . 9  |-  ( y  e.  ( 0 [,] 
+oo )  ->  ( -.  y  e.  (
0 [,)  +oo )  -> 
y  =  +oo )
)
6968con1d 118 . . . . . . . 8  |-  ( y  e.  ( 0 [,] 
+oo )  ->  ( -.  y  =  +oo  ->  y  e.  ( 0 [,)  +oo ) ) )
7069imp 419 . . . . . . 7  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
y  e.  ( 0 [,)  +oo ) )
71 mnfxr 10706 . . . . . . . . . . . . . 14  |-  -oo  e.  RR*
72 0re 9083 . . . . . . . . . . . . . . 15  |-  0  e.  RR
73 mnflt 10714 . . . . . . . . . . . . . . 15  |-  ( 0  e.  RR  ->  -oo  <  0 )
7472, 73ax-mp 8 . . . . . . . . . . . . . 14  |-  -oo  <  0
75 xrleid 10735 . . . . . . . . . . . . . . 15  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
763, 75ax-mp 8 . . . . . . . . . . . . . 14  |-  +oo  <_  +oo
77 icossioo 24125 . . . . . . . . . . . . . 14  |-  ( ( (  -oo  e.  RR*  /\ 
+oo  e.  RR* )  /\  (  -oo  <  0  /\  +oo 
<_  +oo ) )  -> 
( 0 [,)  +oo )  C_  (  -oo (,)  +oo ) )
7871, 3, 74, 76, 77mp4an 655 . . . . . . . . . . . . 13  |-  ( 0 [,)  +oo )  C_  (  -oo (,)  +oo )
79 ioomax 10977 . . . . . . . . . . . . 13  |-  (  -oo (,) 
+oo )  =  RR
8078, 79sseqtri 3372 . . . . . . . . . . . 12  |-  ( 0 [,)  +oo )  C_  RR
8180sseli 3336 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  RR )
8281renegcld 9456 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  -u y  e.  RR )
8382reefcld 12682 . . . . . . . . 9  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e.  RR )
8483rexrd 9126 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e. 
RR* )
85 efgt0 12696 . . . . . . . . 9  |-  ( -u y  e.  RR  ->  0  <  ( exp `  -u y
) )
8682, 85syl 16 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  0  <  ( exp `  -u y
) )
87 elico1 10951 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR* )  ->  (
y  e.  ( 0 [,)  +oo )  <->  ( y  e.  RR*  /\  0  <_ 
y  /\  y  <  +oo ) ) )
882, 3, 87mp2an 654 . . . . . . . . . . . 12  |-  ( y  e.  ( 0 [,) 
+oo )  <->  ( y  e.  RR*  /\  0  <_ 
y  /\  y  <  +oo ) )
8988simp2bi 973 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  0  <_  y )
9081le0neg2d 9591 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  (
0  <_  y  <->  -u y  <_ 
0 ) )
9189, 90mpbid 202 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  -u y  <_  0 )
92 efle 12711 . . . . . . . . . . 11  |-  ( (
-u y  e.  RR  /\  0  e.  RR )  ->  ( -u y  <_  0  <->  ( exp `  -u y
)  <_  ( exp `  0 ) ) )
9382, 72, 92sylancl 644 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( -u y  <_  0  <->  ( exp `  -u y )  <_  ( exp `  0 ) ) )
9491, 93mpbid 202 . . . . . . . . 9  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  <_ 
( exp `  0
) )
95 ef0 12685 . . . . . . . . 9  |-  ( exp `  0 )  =  1
9694, 95syl6breq 4243 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  <_ 
1 )
97 elioc1 10950 . . . . . . . . 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
) ) )
982, 12, 97mp2an 654 . . . . . . . 8  |-  ( ( exp `  -u y
)  e.  ( 0 (,] 1 )  <->  ( ( exp `  -u y )  e. 
RR*  /\  0  <  ( exp `  -u y
)  /\  ( exp `  -u y )  <_  1
) )
9984, 86, 96, 98syl3anbrc 1138 . . . . . . 7  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e.  ( 0 (,] 1
) )
10070, 99syl 16 . . . . . 6  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
( exp `  -u y
)  e.  ( 0 (,] 1 ) )
10159, 100sseldi 3338 . . . . 5  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
( exp `  -u y
)  e.  ( 0 [,] 1 ) )
10258, 101ifclda 3758 . . . 4  |-  ( y  e.  ( 0 [,] 
+oo )  ->  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) )  e.  ( 0 [,] 1 ) )
103102adantl 453 . . 3  |-  ( (  T.  /\  y  e.  ( 0 [,]  +oo ) )  ->  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) )  e.  ( 0 [,] 1 ) )
104 eqeq2 2444 . . . . . 6  |-  ( 0  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y
) )  ->  (
x  =  0  <->  x  =  if ( y  = 
+oo ,  0 , 
( exp `  -u y
) ) ) )
105104bibi1d 311 . . . . 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 ) ) ) ) )
106 eqeq2 2444 . . . . . 6  |-  ( ( exp `  -u y
)  =  if ( y  =  +oo , 
0 ,  ( exp `  -u y ) )  ->  ( x  =  ( exp `  -u y
)  <->  x  =  if ( y  =  +oo ,  0 ,  ( exp `  -u y ) ) ) )
107106bibi1d 311 . . . . 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 ) ) ) ) )
108 simpr 448 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  y  = 
+oo )
109 iftrue 3737 . . . . . . . 8  |-  ( x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  +oo )
110109eqeq2d 2446 . . . . . . 7  |-  ( x  =  0  ->  (
y  =  if ( x  =  0 , 
+oo ,  -u ( log `  x ) )  <->  y  =  +oo ) )
111108, 110syl5ibrcom 214 . . . . . 6  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( x  =  0  ->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) ) )
112 ubico 24130 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  -.  +oo 
e.  ( 0 [,) 
+oo ) )
11372, 3, 112mp2an 654 . . . . . . . . . 10  |-  -.  +oo  e.  ( 0 [,)  +oo )
114 df-nel 2601 . . . . . . . . . 10  |-  (  +oo  e/  ( 0 [,)  +oo ) 
<->  -.  +oo  e.  (
0 [,)  +oo ) )
115113, 114mpbir 201 . . . . . . . . 9  |-  +oo  e/  ( 0 [,)  +oo )
116 neleq1 2691 . . . . . . . . . 10  |-  ( y  =  +oo  ->  (
y  e/  ( 0 [,)  +oo )  <->  +oo  e/  (
0 [,)  +oo ) ) )
117116adantl 453 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( y  e/  ( 0 [,) 
+oo )  <->  +oo  e/  (
0 [,)  +oo ) ) )
118115, 117mpbiri 225 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  y  e/  ( 0 [,)  +oo ) )
119 neleq1 2691 . . . . . . . 8  |-  ( y  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  (
y  e/  ( 0 [,)  +oo )  <->  if (
x  =  0 , 
+oo ,  -u ( log `  x ) )  e/  ( 0 [,)  +oo ) ) )
120118, 119syl5ibcom 212 . . . . . . 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 ) ) )
121 df-nel 2601 . . . . . . . 8  |-  ( if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e/  ( 0 [,)  +oo )  <->  -.  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,) 
+oo ) )
122 iffalse 3738 . . . . . . . . . . . . 13  |-  ( -.  x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  -u ( log `  x ) )
123122adantl 453 . . . . . . . . . . . 12  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  if (
x  =  0 , 
+oo ,  -u ( log `  x ) )  = 
-u ( log `  x
) )
124123, 53eqeltrd 2509 . . . . . . . . . . 11  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  if (
x  =  0 , 
+oo ,  -u ( log `  x ) )  e.  ( 0 [,)  +oo ) )
125124ex 424 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 1 )  ->  ( -.  x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) )  e.  ( 0 [,)  +oo )
) )
126125ad2antrr 707 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( -.  x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,)  +oo ) ) )
127126con1d 118 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( -.  if ( x  =  0 ,  +oo ,  -u ( log `  x
) )  e.  ( 0 [,)  +oo )  ->  x  =  0 ) )
128121, 127syl5bi 209 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e/  ( 0 [,)  +oo )  ->  x  =  0 ) )
129120, 128syld 42 . . . . . 6  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( y  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  x  =  0 ) )
130111, 129impbid 184 . . . . 5  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  y  =  +oo )  ->  ( x  =  0  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
131 eqeq2 2444 . . . . . . 7  |-  (  +oo  =  if ( x  =  0 ,  +oo ,  -u ( log `  x
) )  ->  (
y  =  +oo  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
132131bibi2d 310 . . . . . 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 ) ) ) ) )
133 eqeq2 2444 . . . . . . 7  |-  ( -u ( log `  x )  =  if ( x  =  0 ,  +oo , 
-u ( log `  x
) )  ->  (
y  =  -u ( log `  x )  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
134133bibi2d 310 . . . . . 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 ) ) ) ) )
13572a1i 11 . . . . . . . . . . . 12  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
0  e.  RR )
13670, 86syl 16 . . . . . . . . . . . 12  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
0  <  ( exp `  -u y ) )
137135, 136ltned 9201 . . . . . . . . . . 11  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
0  =/=  ( exp `  -u y ) )
138137adantll 695 . . . . . . . . . 10  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  0  =/=  ( exp `  -u y
) )
139138neneqd 2614 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  -.  0  =  ( exp `  -u y ) )
140 eqeq1 2441 . . . . . . . . . 10  |-  ( x  =  0  ->  (
x  =  ( exp `  -u y )  <->  0  =  ( exp `  -u y
) ) )
141140notbid 286 . . . . . . . . 9  |-  ( x  =  0  ->  ( -.  x  =  ( exp `  -u y )  <->  -.  0  =  ( exp `  -u y
) ) )
142139, 141syl5ibrcom 214 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  (
x  =  0  ->  -.  x  =  ( exp `  -u y ) ) )
143142imp 419 . . . . . . 7  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  x  =  0 )  ->  -.  x  =  ( exp `  -u y
) )
144 simplr 732 . . . . . . 7  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  x  =  0 )  ->  -.  y  =  +oo )
145143, 1442falsed 341 . . . . . 6  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  x  =  0 )  ->  ( x  =  ( exp `  -u y
)  <->  y  =  +oo ) )
146 eqcom 2437 . . . . . . . . . . 11  |-  ( x  =  ( exp `  -u y
)  <->  ( exp `  -u y
)  =  x )
147146a1i 11 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( x  =  ( exp `  -u y
)  <->  ( exp `  -u y
)  =  x ) )
148 relogeftb 20471 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  -u y  e.  RR )  ->  (
( log `  x
)  =  -u y  <->  ( exp `  -u y
)  =  x ) )
14933, 82, 148syl2an 464 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( ( log `  x
)  =  -u y  <->  ( exp `  -u y
)  =  x ) )
15034recnd 9106 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  e.  CC )
15181recnd 9106 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  CC )
152 negcon2 9346 . . . . . . . . . . 11  |-  ( ( ( log `  x
)  e.  CC  /\  y  e.  CC )  ->  ( ( log `  x
)  =  -u y  <->  y  =  -u ( log `  x
) ) )
153150, 151, 152syl2an 464 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( ( log `  x
)  =  -u y  <->  y  =  -u ( log `  x
) ) )
154147, 149, 1533bitr2d 273 . . . . . . . . 9  |-  ( ( x  e.  ( 0 (,] 1 )  /\  y  e.  ( 0 [,)  +oo ) )  -> 
( x  =  ( exp `  -u y
)  <->  y  =  -u ( log `  x ) ) )
15525, 70, 154syl2an 464 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0 )  /\  (
y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo ) )  ->  ( x  =  ( exp `  -u y
)  <->  y  =  -u ( log `  x ) ) )
156155an4s 800 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  ( -.  x  =  0  /\  -.  y  =  +oo ) )  ->  (
x  =  ( exp `  -u y )  <->  y  =  -u ( log `  x
) ) )
157156anass1rs 783 . . . . . 6  |-  ( ( ( ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,]  +oo ) )  /\  -.  y  =  +oo )  /\  -.  x  =  0
)  ->  ( x  =  ( exp `  -u y
)  <->  y  =  -u ( log `  x ) ) )
158132, 134, 145, 157ifbothda 3761 . . . . 5  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  (
x  =  ( exp `  -u y )  <->  y  =  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) ) )
159105, 107, 130, 158ifbothda 3761 . . . 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 ) ) ) )
160159adantl 453 . . 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 ) ) ) )
1611, 56, 103, 160f1ocnv2d 6287 . 2  |-  (  T. 
->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  (
y  e.  ( 0 [,]  +oo )  |->  if ( y  =  +oo , 
0 ,  ( exp `  -u y ) ) ) ) )
162161trud 1332 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    T. wtru 1325    = wceq 1652    e. wcel 1725    =/= wne 2598    e/ wnel 2599    u. cun 3310    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   -ucneg 9284   RR+crp 10604   (,)cioo 10908   (,]cioc 10909   [,)cico 10910   [,]cicc 10911   expce 12656   logclog 20444
This theorem is referenced by:  xrge0iifiso  24313  xrge0iifmhm  24317  xrge0pluscn  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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446
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