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Theorem xrge0iifcnv 24272
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 9087 . . . . . . 7  |-  0  e.  RR*
3 pnfxr 10669 . . . . . . 7  |-  +oo  e.  RR*
4 pnfge 10683 . . . . . . . 8  |-  ( 0  e.  RR*  ->  0  <_  +oo )
52, 4ax-mp 8 . . . . . . 7  |-  0  <_  +oo
6 ubicc2 10970 . . . . . . 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 24082 . . . . . 6  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
10 uncom 3451 . . . . . . . . . . . . . 14  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( ( 0 (,] 1 )  u.  { 0 } )
11 1re 9046 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
1211rexri 9093 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
13 0le1 9507 . . . . . . . . . . . . . . 15  |-  0  <_  1
14 snunioc 24090 . . . . . . . . . . . . . . 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 2426 . . . . . . . . . . . . 13  |-  ( ( 0 (,] 1 )  u.  { 0 } )  =  ( 0 [,] 1 )
1716eleq2i 2468 . . . . . . . . . . . 12  |-  ( x  e.  ( ( 0 (,] 1 )  u. 
{ 0 } )  <-> 
x  e.  ( 0 [,] 1 ) )
18 elun 3448 . . . . . . . . . . . 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 3798 . . . . . . . . . 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 10037 . . . . . . . . . . . . . 14  |-  0  <_  0
27 ltpnf 10677 . . . . . . . . . . . . . . 15  |-  ( 1  e.  RR  ->  1  <  +oo )
2811, 27ax-mp 8 . . . . . . . . . . . . . 14  |-  1  <  +oo
29 iocssioo 24085 . . . . . . . . . . . . . 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 10944 . . . . . . . . . . . . 13  |-  ( 0 (,)  +oo )  =  RR+
3230, 31sseqtri 3340 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR+
3332sseli 3304 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  x  e.  RR+ )
3433relogcld 20471 . . . . . . . . . 10  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  e.  RR )
3534renegcld 9420 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e.  RR )
3635rexrd 9090 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  e. 
RR* )
37 elioc1 10914 . . . . . . . . . . . . 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 10572 . . . . . . . . . . . . 13  |-  1  e.  RR+
4140a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 (,] 1 )  ->  1  e.  RR+ )
4233, 41logled 20475 . . . . . . . . . . 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 20433 . . . . . . . . . 10  |-  ( log `  1 )  =  0
4543, 44syl6breq 4211 . . . . . . . . 9  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  <_ 
0 )
4634le0neg1d 9554 . . . . . . . . 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 10677 . . . . . . . . 9  |-  ( -u ( log `  x )  e.  RR  ->  -u ( log `  x )  <  +oo )
4935, 48syl 16 . . . . . . . 8  |-  ( x  e.  ( 0 (,] 1 )  ->  -u ( log `  x )  <  +oo )
50 elico1 10915 . . . . . . . . 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 3306 . . . . 5  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  =  0
)  ->  -u ( log `  x )  e.  ( 0 [,]  +oo )
)
558, 54ifclda 3726 . . . 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 10971 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
5857a1i 11 . . . . 5  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  y  =  +oo )  ->  0  e.  ( 0 [,] 1
) )
59 iocssicc 24083 . . . . . 6  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
60 snunico 10980 . . . . . . . . . . . . . 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 2468 . . . . . . . . . . . 12  |-  ( y  e.  ( ( 0 [,)  +oo )  u.  {  +oo } )  <->  y  e.  ( 0 [,]  +oo ) )
63 elun 3448 . . . . . . . . . . . 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 3798 . . . . . . . . . 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 10670 . . . . . . . . . . . . . 14  |-  -oo  e.  RR*
72 0re 9047 . . . . . . . . . . . . . . 15  |-  0  e.  RR
73 mnflt 10678 . . . . . . . . . . . . . . 15  |-  ( 0  e.  RR  ->  -oo  <  0 )
7472, 73ax-mp 8 . . . . . . . . . . . . . 14  |-  -oo  <  0
75 xrleid 10699 . . . . . . . . . . . . . . 15  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
763, 75ax-mp 8 . . . . . . . . . . . . . 14  |-  +oo  <_  +oo
77 icossioo 24086 . . . . . . . . . . . . . 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 10941 . . . . . . . . . . . . 13  |-  (  -oo (,) 
+oo )  =  RR
8078, 79sseqtri 3340 . . . . . . . . . . . 12  |-  ( 0 [,)  +oo )  C_  RR
8180sseli 3304 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  RR )
8281renegcld 9420 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  -u y  e.  RR )
8382reefcld 12645 . . . . . . . . 9  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e.  RR )
8483rexrd 9090 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  e. 
RR* )
85 efgt0 12659 . . . . . . . . 9  |-  ( -u y  e.  RR  ->  0  <  ( exp `  -u y
) )
8682, 85syl 16 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  0  <  ( exp `  -u y
) )
87 elico1 10915 . . . . . . . . . . . . 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 9555 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  (
0  <_  y  <->  -u y  <_ 
0 ) )
9189, 90mpbid 202 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,) 
+oo )  ->  -u y  <_  0 )
92 efle 12674 . . . . . . . . . . 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 12648 . . . . . . . . 9  |-  ( exp `  0 )  =  1
9694, 95syl6breq 4211 . . . . . . . 8  |-  ( y  e.  ( 0 [,) 
+oo )  ->  ( exp `  -u y )  <_ 
1 )
97 elioc1 10914 . . . . . . . . 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 3306 . . . . 5  |-  ( ( y  e.  ( 0 [,]  +oo )  /\  -.  y  =  +oo )  -> 
( exp `  -u y
)  e.  ( 0 [,] 1 ) )
10258, 101ifclda 3726 . . . 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 2413 . . . . . 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 2413 . . . . . 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 3705 . . . . . . . 8  |-  ( x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  +oo )
110109eqeq2d 2415 . . . . . . 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 24091 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  -.  +oo 
e.  ( 0 [,) 
+oo ) )
11372, 3, 112mp2an 654 . . . . . . . . . 10  |-  -.  +oo  e.  ( 0 [,)  +oo )
114 df-nel 2570 . . . . . . . . . 10  |-  (  +oo  e/  ( 0 [,)  +oo ) 
<->  -.  +oo  e.  (
0 [,)  +oo ) )
115113, 114mpbir 201 . . . . . . . . 9  |-  +oo  e/  ( 0 [,)  +oo )
116 neleq1 2660 . . . . . . . . . 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 2660 . . . . . . . 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 2570 . . . . . . . 8  |-  ( if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e/  ( 0 [,)  +oo )  <->  -.  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  e.  ( 0 [,) 
+oo ) )
122 iffalse 3706 . . . . . . . . . . . . 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 2478 . . . . . . . . . . 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 2413 . . . . . . 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 2413 . . . . . . 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 9165 . . . . . . . . . . 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 2583 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,]  +oo )
)  /\  -.  y  =  +oo )  ->  -.  0  =  ( exp `  -u y ) )
140 eqeq1 2410 . . . . . . . . . 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 2406 . . . . . . . . . . 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 20432 . . . . . . . . . . 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 9070 . . . . . . . . . . 11  |-  ( x  e.  ( 0 (,] 1 )  ->  ( log `  x )  e.  CC )
15181recnd 9070 . . . . . . . . . . 11  |-  ( y  e.  ( 0 [,) 
+oo )  ->  y  e.  CC )
152 negcon2 9310 . . . . . . . . . . 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 3729 . . . . 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 3729 . . . 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 6254 . 2  |-  (  T. 
->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  (
y  e.  ( 0 [,]  +oo )  |->  if ( y  =  +oo , 
0 ,  ( exp `  -u y ) ) ) ) )
162161trud 1329 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 1322    = wceq 1649    e. wcel 1721    =/= wne 2567    e/ wnel 2568    u. cun 3278    C_ wss 3280   ifcif 3699   {csn 3774   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075    < clt 9076    <_ cle 9077   -ucneg 9248   RR+crp 10568   (,)cioo 10872   (,]cioc 10873   [,)cico 10874   [,]cicc 10875   expce 12619   logclog 20405
This theorem is referenced by:  xrge0iifiso  24274  xrge0iifmhm  24278  xrge0pluscn  24279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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