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Theorem xrge0iifiso 26285
Description: The defined bijection from the closed unit interval and the extended non-negative 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 11374 . . 3  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 11114 . . 3  |-  <  Or  RR*
3 soss 4655 . . 3  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . 2  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 11374 . . 3  |-  ( 0 [,] +oo )  C_  RR*
6 cnvso 5373 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
72, 6mpbi 208 . . . 4  |-  `'  <  Or 
RR*
8 sopo 4654 . . . 4  |-  ( `'  <  Or  RR*  ->  `'  <  Po  RR* )
97, 8ax-mp 5 . . 3  |-  `'  <  Po 
RR*
10 poss 4639 . . 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 26283 . . . 4  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( z  e.  ( 0 [,] +oo )  |->  if ( z  = +oo ,  0 ,  ( exp `  -u z
) ) ) )
1413simpli 455 . . 3  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
15 f1ofo 5645 . . 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 9426 . . . . . . . 8  |-  0  e.  RR*
18 1re 9381 . . . . . . . . 9  |-  1  e.  RR
1918rexri 9432 . . . . . . . 8  |-  1  e.  RR*
20 0le1 9859 . . . . . . . 8  |-  0  <_  1
21 snunioc 11409 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
2217, 19, 20, 21mp3an 1309 . . . . . . 7  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
2322eleq2i 2505 . . . . . 6  |-  ( w  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
w  e.  ( 0 [,] 1 ) )
24 elun 3494 . . . . . 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 3888 . . . . . . 7  |-  ( w  e.  { 0 }  <-> 
w  =  0 )
27 elunitrn 26247 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
2827adantr 462 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  e.  RR )
29 simpr 458 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
0  <  z )
30 0re 9382 . . . . . . . . . . . . . 14  |-  0  e.  RR
3130, 18elicc2i 11357 . . . . . . . . . . . . 13  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
3231simp3bi 1000 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  ->  z  <_  1 )
3332adantr 462 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  <_  1 )
34 elioc2 11354 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
z  e.  ( 0 (,] 1 )  <->  ( z  e.  RR  /\  0  < 
z  /\  z  <_  1 ) ) )
3517, 18, 34mp2an 667 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  <->  ( z  e.  RR  /\  0  < 
z  /\  z  <_  1 ) )
3628, 29, 33, 35syl3anbrc 1167 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  e.  ( 0 (,] 1 ) )
37 pnfxr 11088 . . . . . . . . . . . . . . 15  |- +oo  e.  RR*
38 0le0 10407 . . . . . . . . . . . . . . 15  |-  0  <_  0
39 ltpnf 11098 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR  ->  1  < +oo )
4018, 39ax-mp 5 . . . . . . . . . . . . . . 15  |-  1  < +oo
41 iocssioo 25978 . . . . . . . . . . . . . . 15  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <_  0  /\  1  < +oo ) )  -> 
( 0 (,] 1
)  C_  ( 0 (,) +oo ) )
4217, 37, 38, 40, 41mp4an 668 . . . . . . . . . . . . . 14  |-  ( 0 (,] 1 )  C_  ( 0 (,) +oo )
43 ioorp 11369 . . . . . . . . . . . . . 14  |-  ( 0 (,) +oo )  = 
RR+
4442, 43sseqtri 3385 . . . . . . . . . . . . 13  |-  ( 0 (,] 1 )  C_  RR+
4544sseli 3349 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 (,] 1 )  ->  z  e.  RR+ )
46 relogcl 21970 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( log `  z )  e.  RR )
4746renegcld 9771 . . . . . . . . . . . . . 14  |-  ( z  e.  RR+  ->  -u ( log `  z )  e.  RR )
48 ltpnf 11098 . . . . . . . . . . . . . 14  |-  ( -u ( log `  z )  e.  RR  ->  -u ( log `  z )  < +oo )
4947, 48syl 16 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  -u ( log `  z )  < +oo )
50 brcnvg 5016 . . . . . . . . . . . . . 14  |-  ( ( +oo  e.  RR*  /\  -u ( log `  z )  e.  RR )  ->  ( +oo `'  <  -u ( log `  z
)  <->  -u ( log `  z
)  < +oo )
)
5137, 47, 50sylancr 658 . . . . . . . . . . . . 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 26284 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  ->  ( F `  z )  =  -u ( log `  z
) )
5553, 54breqtrrd 4315 . . . . . . . . . 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 4292 . . . . . . . . 9  |-  ( w  =  0  ->  (
w  <  z  <->  0  <  z ) )
59 fveq2 5688 . . . . . . . . . . 11  |-  ( w  =  0  ->  ( F `  w )  =  ( F ` 
0 ) )
60 0elunit 11399 . . . . . . . . . . . 12  |-  0  e.  ( 0 [,] 1
)
61 iftrue 3794 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  if ( x  =  0 , +oo ,  -u ( log `  x ) )  = +oo )
62 pnfex 11089 . . . . . . . . . . . . 13  |- +oo  e.  _V
6361, 12, 62fvmpt 5771 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  = +oo )
6460, 63ax-mp 5 . . . . . . . . . . 11  |-  ( F `
 0 )  = +oo
6559, 64syl6eq 2489 . . . . . . . . . 10  |-  ( w  =  0  ->  ( F `  w )  = +oo )
6665breq1d 4299 . . . . . . . . 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 748 . . . . . . . . 9  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  e.  ( 0 (,] 1
) )
7127ad2antlr 721 . . . . . . . . . 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 3349 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,] 1 )  ->  w  e.  RR+ )
7473rpred 11023 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 (,] 1 )  ->  w  e.  RR )
7574ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  e.  RR )
76 elioc2 11354 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
w  e.  ( 0 (,] 1 )  <->  ( w  e.  RR  /\  0  < 
w  /\  w  <_  1 ) ) )
7717, 18, 76mp2an 667 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,] 1 )  <->  ( w  e.  RR  /\  0  < 
w  /\  w  <_  1 ) )
7877simp2bi 999 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 (,] 1 )  ->  0  <  w )
7978ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  0  <  w )
80 simpr 458 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  <  z )
8172, 75, 71, 79, 80lttrd 9528 . . . . . . . . . 10  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  0  <  z )
8232ad2antlr 721 . . . . . . . . . 10  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  z  <_  1 )
8371, 81, 82, 35syl3anbrc 1167 . . . . . . . . 9  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  z  e.  ( 0 (,] 1
) )
8470, 83jca 529 . . . . . . . 8  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  (
w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) ) )
8573adantr 462 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  w  e.  RR+ )
8685relogcld 22015 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( log `  w
)  e.  RR )
8745adantl 463 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  z  e.  RR+ )
8887relogcld 22015 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( log `  z
)  e.  RR )
8986, 88ltnegd 9913 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( ( log `  w )  <  ( log `  z )  <->  -u ( log `  z )  <  -u ( log `  w ) ) )
90 logltb 21991 . . . . . . . . . . . 12  |-  ( ( w  e.  RR+  /\  z  e.  RR+ )  ->  (
w  <  z  <->  ( log `  w )  <  ( log `  z ) ) )
9173, 45, 90syl2an 474 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  <->  ( log `  w
)  <  ( log `  z ) ) )
92 negex 9604 . . . . . . . . . . . . 13  |-  -u ( log `  w )  e. 
_V
93 negex 9604 . . . . . . . . . . . . 13  |-  -u ( log `  z )  e. 
_V
9492, 93brcnv 5018 . . . . . . . . . . . 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 26284 . . . . . . . . . 10  |-  ( w  e.  ( 0 (,] 1 )  ->  ( F `  w )  =  -u ( log `  w
) )
9998, 54breqan12d 4304 . . . . . . . . 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 601 . . . . . 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 2780 . 2  |-  A. w  e.  ( 0 [,] 1
) A. z  e.  ( 0 [,] 1
) ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) )
107 soisoi 6016 . 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 668 1  |-  F  Isom  <  ,  `'  <  ( ( 0 [,] 1 ) ,  ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713    u. cun 3323    C_ wss 3325   ifcif 3788   {csn 3874   class class class wbr 4289    e. cmpt 4347    Po wpo 4635    Or wor 4636   `'ccnv 4835   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415    Isom wiso 5416  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279   +oocpnf 9411   RR*cxr 9413    < clt 9414    <_ cle 9415   -ucneg 9592   RR+crp 10987   (,)cioo 11296   (,]cioc 11297   [,]cicc 11299   expce 13343   logclog 21949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-limc 21241  df-dv 21242  df-log 21951
This theorem is referenced by:  xrge0iifhmeo  26286
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