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Theorem xrge0iifiso 27581
Description: The defined bijection from the closed unit interval and the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)
Hypothesis
Ref Expression
xrge0iifhmeo.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
Assertion
Ref Expression
xrge0iifiso  |-  F  Isom  <  ,  `'  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
Distinct variable group:    x, F

Proof of Theorem xrge0iifiso
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iccssxr 11607 . . 3  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 11347 . . 3  |-  <  Or  RR*
3 soss 4818 . . 3  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . 2  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 11607 . . 3  |-  ( 0 [,] +oo )  C_  RR*
6 cnvso 5546 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
72, 6mpbi 208 . . . 4  |-  `'  <  Or 
RR*
8 sopo 4817 . . . 4  |-  ( `'  <  Or  RR*  ->  `'  <  Po  RR* )
97, 8ax-mp 5 . . 3  |-  `'  <  Po 
RR*
10 poss 4802 . . 3  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( `'  <  Po 
RR*  ->  `'  <  Po  ( 0 [,] +oo ) ) )
115, 9, 10mp2 9 . 2  |-  `'  <  Po  ( 0 [,] +oo )
12 xrge0iifhmeo.1 . . . . 5  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
1312xrge0iifcnv 27579 . . . 4  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( z  e.  ( 0 [,] +oo )  |->  if ( z  = +oo ,  0 ,  ( exp `  -u z
) ) ) )
1413simpli 458 . . 3  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
15 f1ofo 5823 . . 3  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  ->  F : ( 0 [,] 1 )
-onto-> ( 0 [,] +oo ) )
1614, 15ax-mp 5 . 2  |-  F :
( 0 [,] 1
) -onto-> ( 0 [,] +oo )
17 0xr 9640 . . . . . . . 8  |-  0  e.  RR*
18 1re 9595 . . . . . . . . 9  |-  1  e.  RR
1918rexri 9646 . . . . . . . 8  |-  1  e.  RR*
20 0le1 10076 . . . . . . . 8  |-  0  <_  1
21 snunioc 11648 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
2217, 19, 20, 21mp3an 1324 . . . . . . 7  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
2322eleq2i 2545 . . . . . 6  |-  ( w  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
w  e.  ( 0 [,] 1 ) )
24 elun 3645 . . . . . 6  |-  ( w  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( w  e.  {
0 }  \/  w  e.  ( 0 (,] 1
) ) )
2523, 24bitr3i 251 . . . . 5  |-  ( w  e.  ( 0 [,] 1 )  <->  ( w  e.  { 0 }  \/  w  e.  ( 0 (,] 1 ) ) )
26 elsn 4041 . . . . . . 7  |-  ( w  e.  { 0 }  <-> 
w  =  0 )
27 elunitrn 27543 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
2827adantr 465 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  e.  RR )
29 simpr 461 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
0  <  z )
30 0re 9596 . . . . . . . . . . . . . 14  |-  0  e.  RR
3130, 18elicc2i 11590 . . . . . . . . . . . . 13  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
3231simp3bi 1013 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  ->  z  <_  1 )
3332adantr 465 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  <_  1 )
34 elioc2 11587 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
z  e.  ( 0 (,] 1 )  <->  ( z  e.  RR  /\  0  < 
z  /\  z  <_  1 ) ) )
3517, 18, 34mp2an 672 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  <->  ( z  e.  RR  /\  0  < 
z  /\  z  <_  1 ) )
3628, 29, 33, 35syl3anbrc 1180 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  e.  ( 0 (,] 1 ) )
37 pnfxr 11321 . . . . . . . . . . . . . . 15  |- +oo  e.  RR*
38 0le0 10625 . . . . . . . . . . . . . . 15  |-  0  <_  0
39 ltpnf 11331 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR  ->  1  < +oo )
4018, 39ax-mp 5 . . . . . . . . . . . . . . 15  |-  1  < +oo
41 iocssioo 11614 . . . . . . . . . . . . . . 15  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <_  0  /\  1  < +oo ) )  -> 
( 0 (,] 1
)  C_  ( 0 (,) +oo ) )
4217, 37, 38, 40, 41mp4an 673 . . . . . . . . . . . . . 14  |-  ( 0 (,] 1 )  C_  ( 0 (,) +oo )
43 ioorp 11602 . . . . . . . . . . . . . 14  |-  ( 0 (,) +oo )  = 
RR+
4442, 43sseqtri 3536 . . . . . . . . . . . . 13  |-  ( 0 (,] 1 )  C_  RR+
4544sseli 3500 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 (,] 1 )  ->  z  e.  RR+ )
46 relogcl 22719 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( log `  z )  e.  RR )
4746renegcld 9986 . . . . . . . . . . . . . 14  |-  ( z  e.  RR+  ->  -u ( log `  z )  e.  RR )
48 ltpnf 11331 . . . . . . . . . . . . . 14  |-  ( -u ( log `  z )  e.  RR  ->  -u ( log `  z )  < +oo )
4947, 48syl 16 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  -u ( log `  z )  < +oo )
50 brcnvg 5183 . . . . . . . . . . . . . 14  |-  ( ( +oo  e.  RR*  /\  -u ( log `  z )  e.  RR )  ->  ( +oo `'  <  -u ( log `  z
)  <->  -u ( log `  z
)  < +oo )
)
5137, 47, 50sylancr 663 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  ( +oo `'  <  -u ( log `  z
)  <->  -u ( log `  z
)  < +oo )
)
5249, 51mpbird 232 . . . . . . . . . . . 12  |-  ( z  e.  RR+  -> +oo `'  <  -u ( log `  z
) )
5345, 52syl 16 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  -> +oo `'  <  -u ( log `  z
) )
5412xrge0iifcv 27580 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  ->  ( F `  z )  =  -u ( log `  z
) )
5553, 54breqtrrd 4473 . . . . . . . . . 10  |-  ( z  e.  ( 0 (,] 1 )  -> +oo `'  <  ( F `  z
) )
5636, 55syl 16 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> +oo `'  <  ( F `  z ) )
5756ex 434 . . . . . . . 8  |-  ( z  e.  ( 0 [,] 1 )  ->  (
0  <  z  -> +oo `'  <  ( F `  z ) ) )
58 breq1 4450 . . . . . . . . 9  |-  ( w  =  0  ->  (
w  <  z  <->  0  <  z ) )
59 fveq2 5866 . . . . . . . . . . 11  |-  ( w  =  0  ->  ( F `  w )  =  ( F ` 
0 ) )
60 0elunit 11638 . . . . . . . . . . . 12  |-  0  e.  ( 0 [,] 1
)
61 iftrue 3945 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  if ( x  =  0 , +oo ,  -u ( log `  x ) )  = +oo )
62 pnfex 11322 . . . . . . . . . . . . 13  |- +oo  e.  _V
6361, 12, 62fvmpt 5950 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  = +oo )
6460, 63ax-mp 5 . . . . . . . . . . 11  |-  ( F `
 0 )  = +oo
6559, 64syl6eq 2524 . . . . . . . . . 10  |-  ( w  =  0  ->  ( F `  w )  = +oo )
6665breq1d 4457 . . . . . . . . 9  |-  ( w  =  0  ->  (
( F `  w
) `'  <  ( F `  z )  <-> +oo `'  <  ( F `  z ) ) )
6758, 66imbi12d 320 . . . . . . . 8  |-  ( w  =  0  ->  (
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
)  <->  ( 0  < 
z  -> +oo `'  <  ( F `  z ) ) ) )
6857, 67syl5ibr 221 . . . . . . 7  |-  ( w  =  0  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
6926, 68sylbi 195 . . . . . 6  |-  ( w  e.  { 0 }  ->  ( z  e.  ( 0 [,] 1
)  ->  ( w  <  z  ->  ( F `  w ) `'  <  ( F `  z ) ) ) )
70 simpll 753 . . . . . . . . 9  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  e.  ( 0 (,] 1
) )
7127ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  z  e.  RR )
7230a1i 11 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  0  e.  RR )
7344sseli 3500 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,] 1 )  ->  w  e.  RR+ )
7473rpred 11256 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 (,] 1 )  ->  w  e.  RR )
7574ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  e.  RR )
76 elioc2 11587 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
w  e.  ( 0 (,] 1 )  <->  ( w  e.  RR  /\  0  < 
w  /\  w  <_  1 ) ) )
7717, 18, 76mp2an 672 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,] 1 )  <->  ( w  e.  RR  /\  0  < 
w  /\  w  <_  1 ) )
7877simp2bi 1012 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 (,] 1 )  ->  0  <  w )
7978ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  0  <  w )
80 simpr 461 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  <  z )
8172, 75, 71, 79, 80lttrd 9742 . . . . . . . . . 10  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  0  <  z )
8232ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  z  <_  1 )
8371, 81, 82, 35syl3anbrc 1180 . . . . . . . . 9  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  z  e.  ( 0 (,] 1
) )
8470, 83jca 532 . . . . . . . 8  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  (
w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) ) )
8573adantr 465 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  w  e.  RR+ )
8685relogcld 22764 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( log `  w
)  e.  RR )
8745adantl 466 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  z  e.  RR+ )
8887relogcld 22764 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( log `  z
)  e.  RR )
8986, 88ltnegd 10130 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( ( log `  w )  <  ( log `  z )  <->  -u ( log `  z )  <  -u ( log `  w ) ) )
90 logltb 22740 . . . . . . . . . . . 12  |-  ( ( w  e.  RR+  /\  z  e.  RR+ )  ->  (
w  <  z  <->  ( log `  w )  <  ( log `  z ) ) )
9173, 45, 90syl2an 477 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  <->  ( log `  w
)  <  ( log `  z ) ) )
92 negex 9818 . . . . . . . . . . . . 13  |-  -u ( log `  w )  e. 
_V
93 negex 9818 . . . . . . . . . . . . 13  |-  -u ( log `  z )  e. 
_V
9492, 93brcnv 5185 . . . . . . . . . . . 12  |-  ( -u ( log `  w ) `'  <  -u ( log `  z
)  <->  -u ( log `  z
)  <  -u ( log `  w ) )
9594a1i 11 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( -u ( log `  w ) `'  <  -u ( log `  z
)  <->  -u ( log `  z
)  <  -u ( log `  w ) ) )
9689, 91, 953bitr4d 285 . . . . . . . . . 10  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  <->  -u ( log `  w
) `'  <  -u ( log `  z ) ) )
9796biimpd 207 . . . . . . . . 9  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  ->  -u ( log `  w ) `'  <  -u ( log `  z
) ) )
9812xrge0iifcv 27580 . . . . . . . . . 10  |-  ( w  e.  ( 0 (,] 1 )  ->  ( F `  w )  =  -u ( log `  w
) )
9998, 54breqan12d 4462 . . . . . . . . 9  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 w ) `'  <  ( F `  z )  <->  -u ( log `  w ) `'  <  -u ( log `  z
) ) )
10097, 99sylibrd 234 . . . . . . . 8  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) ) )
10184, 80, 100sylc 60 . . . . . . 7  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  ( F `  w ) `'  <  ( F `  z ) )
102101exp31 604 . . . . . 6  |-  ( w  e.  ( 0 (,] 1 )  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
10369, 102jaoi 379 . . . . 5  |-  ( ( w  e.  { 0 }  \/  w  e.  ( 0 (,] 1
) )  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
10425, 103sylbi 195 . . . 4  |-  ( w  e.  ( 0 [,] 1 )  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
105104imp 429 . . 3  |-  ( ( w  e.  ( 0 [,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  ->  ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) ) )
106105rgen2a 2891 . 2  |-  A. w  e.  ( 0 [,] 1
) A. z  e.  ( 0 [,] 1
) ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) )
107 soisoi 6212 . 2  |-  ( ( (  <  Or  (
0 [,] 1 )  /\  `'  <  Po  ( 0 [,] +oo ) )  /\  ( F : ( 0 [,] 1 ) -onto-> ( 0 [,] +oo )  /\  A. w  e.  ( 0 [,] 1 ) A. z  e.  ( 0 [,] 1 ) ( w  <  z  -> 
( F `  w
) `'  <  ( F `  z )
) ) )  ->  F  Isom  <  ,  `'  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo )
) )
1084, 11, 16, 106, 107mp4an 673 1  |-  F  Isom  <  ,  `'  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    u. cun 3474    C_ wss 3476   ifcif 3939   {csn 4027   class class class wbr 4447    |-> cmpt 4505    Po wpo 4798    Or wor 4799   `'ccnv 4998   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588    Isom wiso 5589  (class class class)co 6284   RRcr 9491   0cc0 9492   1c1 9493   +oocpnf 9625   RR*cxr 9627    < clt 9628    <_ cle 9629   -ucneg 9806   RR+crp 11220   (,)cioo 11529   (,]cioc 11530   [,]cicc 11532   expce 13659   logclog 22698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700
This theorem is referenced by:  xrge0iifhmeo  27582
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