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Theorem xrge0iifiso 26365
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 11378 . . 3  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 11118 . . 3  |-  <  Or  RR*
3 soss 4659 . . 3  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . 2  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 11378 . . 3  |-  ( 0 [,] +oo )  C_  RR*
6 cnvso 5376 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
72, 6mpbi 208 . . . 4  |-  `'  <  Or 
RR*
8 sopo 4658 . . . 4  |-  ( `'  <  Or  RR*  ->  `'  <  Po  RR* )
97, 8ax-mp 5 . . 3  |-  `'  <  Po 
RR*
10 poss 4643 . . 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 26363 . . . 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 5648 . . 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 9430 . . . . . . . 8  |-  0  e.  RR*
18 1re 9385 . . . . . . . . 9  |-  1  e.  RR
1918rexri 9436 . . . . . . . 8  |-  1  e.  RR*
20 0le1 9863 . . . . . . . 8  |-  0  <_  1
21 snunioc 11413 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
2217, 19, 20, 21mp3an 1314 . . . . . . 7  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
2322eleq2i 2507 . . . . . 6  |-  ( w  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
w  e.  ( 0 [,] 1 ) )
24 elun 3497 . . . . . 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 3891 . . . . . . 7  |-  ( w  e.  { 0 }  <-> 
w  =  0 )
27 elunitrn 26327 . . . . . . . . . . . 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 9386 . . . . . . . . . . . . . 14  |-  0  e.  RR
3130, 18elicc2i 11361 . . . . . . . . . . . . 13  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
3231simp3bi 1005 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  ->  z  <_  1 )
3332adantr 465 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  <_  1 )
34 elioc2 11358 . . . . . . . . . . . 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 1172 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  e.  ( 0 (,] 1 ) )
37 pnfxr 11092 . . . . . . . . . . . . . . 15  |- +oo  e.  RR*
38 0le0 10411 . . . . . . . . . . . . . . 15  |-  0  <_  0
39 ltpnf 11102 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR  ->  1  < +oo )
4018, 39ax-mp 5 . . . . . . . . . . . . . . 15  |-  1  < +oo
41 iocssioo 26061 . . . . . . . . . . . . . . 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 11373 . . . . . . . . . . . . . 14  |-  ( 0 (,) +oo )  = 
RR+
4442, 43sseqtri 3388 . . . . . . . . . . . . 13  |-  ( 0 (,] 1 )  C_  RR+
4544sseli 3352 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 (,] 1 )  ->  z  e.  RR+ )
46 relogcl 22027 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( log `  z )  e.  RR )
4746renegcld 9775 . . . . . . . . . . . . . 14  |-  ( z  e.  RR+  ->  -u ( log `  z )  e.  RR )
48 ltpnf 11102 . . . . . . . . . . . . . 14  |-  ( -u ( log `  z )  e.  RR  ->  -u ( log `  z )  < +oo )
4947, 48syl 16 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  -u ( log `  z )  < +oo )
50 brcnvg 5020 . . . . . . . . . . . . . 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 26364 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  ->  ( F `  z )  =  -u ( log `  z
) )
5553, 54breqtrrd 4318 . . . . . . . . . 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 4295 . . . . . . . . 9  |-  ( w  =  0  ->  (
w  <  z  <->  0  <  z ) )
59 fveq2 5691 . . . . . . . . . . 11  |-  ( w  =  0  ->  ( F `  w )  =  ( F ` 
0 ) )
60 0elunit 11403 . . . . . . . . . . . 12  |-  0  e.  ( 0 [,] 1
)
61 iftrue 3797 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  if ( x  =  0 , +oo ,  -u ( log `  x ) )  = +oo )
62 pnfex 11093 . . . . . . . . . . . . 13  |- +oo  e.  _V
6361, 12, 62fvmpt 5774 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  = +oo )
6460, 63ax-mp 5 . . . . . . . . . . 11  |-  ( F `
 0 )  = +oo
6559, 64syl6eq 2491 . . . . . . . . . 10  |-  ( w  =  0  ->  ( F `  w )  = +oo )
6665breq1d 4302 . . . . . . . . 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 3352 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,] 1 )  ->  w  e.  RR+ )
7473rpred 11027 . . . . . . . . . . . 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 11358 . . . . . . . . . . . . . 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 1004 . . . . . . . . . . . 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 9532 . . . . . . . . . 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 1172 . . . . . . . . 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 22072 . . . . . . . . . . . 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 22072 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( log `  z
)  e.  RR )
8986, 88ltnegd 9917 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( ( log `  w )  <  ( log `  z )  <->  -u ( log `  z )  <  -u ( log `  w ) ) )
90 logltb 22048 . . . . . . . . . . . 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 9608 . . . . . . . . . . . . 13  |-  -u ( log `  w )  e. 
_V
93 negex 9608 . . . . . . . . . . . . 13  |-  -u ( log `  z )  e. 
_V
9492, 93brcnv 5022 . . . . . . . . . . . 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 26364 . . . . . . . . . 10  |-  ( w  e.  ( 0 (,] 1 )  ->  ( F `  w )  =  -u ( log `  w
) )
9998, 54breqan12d 4307 . . . . . . . . 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 2782 . 2  |-  A. w  e.  ( 0 [,] 1
) A. z  e.  ( 0 [,] 1
) ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) )
107 soisoi 6019 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2715    u. cun 3326    C_ wss 3328   ifcif 3791   {csn 3877   class class class wbr 4292    e. cmpt 4350    Po wpo 4639    Or wor 4640   `'ccnv 4839   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418    Isom wiso 5419  (class class class)co 6091   RRcr 9281   0cc0 9282   1c1 9283   +oocpnf 9415   RR*cxr 9417    < clt 9418    <_ cle 9419   -ucneg 9596   RR+crp 10991   (,)cioo 11300   (,]cioc 11301   [,]cicc 11303   expce 13347   logclog 22006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008
This theorem is referenced by:  xrge0iifhmeo  26366
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