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Theorem xrge0iifiso 24282
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 10957 . . 3  |-  ( 0 [,] 1 )  C_  RR*
2 xrltso 10698 . . 3  |-  <  Or  RR*
3 soss 4489 . . 3  |-  ( ( 0 [,] 1 ) 
C_  RR*  ->  (  <  Or 
RR*  ->  <  Or  (
0 [,] 1 ) ) )
41, 2, 3mp2 9 . 2  |-  <  Or  ( 0 [,] 1
)
5 iccssxr 10957 . . 3  |-  ( 0 [,]  +oo )  C_  RR*
6 cnvso 5378 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
72, 6mpbi 200 . . . 4  |-  `'  <  Or 
RR*
8 sopo 4488 . . . 4  |-  ( `'  <  Or  RR*  ->  `'  <  Po  RR* )
97, 8ax-mp 8 . . 3  |-  `'  <  Po 
RR*
10 poss 4473 . . 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 24280 . . . 4  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( z  e.  ( 0 [,]  +oo )  |->  if ( z  = 
+oo ,  0 , 
( exp `  -u z
) ) ) )
1413simpli 445 . . 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 8 . 2  |-  F :
( 0 [,] 1
) -onto-> ( 0 [,] 
+oo )
17 0xr 9095 . . . . . . . 8  |-  0  e.  RR*
18 1re 9054 . . . . . . . . 9  |-  1  e.  RR
1918rexri 9101 . . . . . . . 8  |-  1  e.  RR*
20 0le1 9515 . . . . . . . 8  |-  0  <_  1
21 snunioc 24098 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
2217, 19, 20, 21mp3an 1279 . . . . . . 7  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
2322eleq2i 2476 . . . . . 6  |-  ( w  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
w  e.  ( 0 [,] 1 ) )
24 elun 3456 . . . . . 6  |-  ( w  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( w  e.  {
0 }  \/  w  e.  ( 0 (,] 1
) ) )
2523, 24bitr3i 243 . . . . 5  |-  ( w  e.  ( 0 [,] 1 )  <->  ( w  e.  { 0 }  \/  w  e.  ( 0 (,] 1 ) ) )
26 elsn 3797 . . . . . . 7  |-  ( w  e.  { 0 }  <-> 
w  =  0 )
27 elunitrn 24256 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
2827adantr 452 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  e.  RR )
29 simpr 448 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
0  <  z )
30 0re 9055 . . . . . . . . . . . . . 14  |-  0  e.  RR
3130, 18elicc2i 10940 . . . . . . . . . . . . 13  |-  ( z  e.  ( 0 [,] 1 )  <->  ( z  e.  RR  /\  0  <_ 
z  /\  z  <_  1 ) )
3231simp3bi 974 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  ->  z  <_  1 )
3332adantr 452 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  <_  1 )
34 elioc2 10937 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
z  e.  ( 0 (,] 1 )  <->  ( z  e.  RR  /\  0  < 
z  /\  z  <_  1 ) ) )
3517, 18, 34mp2an 654 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  <->  ( z  e.  RR  /\  0  < 
z  /\  z  <_  1 ) )
3628, 29, 33, 35syl3anbrc 1138 . . . . . . . . . 10  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  -> 
z  e.  ( 0 (,] 1 ) )
37 pnfxr 10677 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
38 0le0 10045 . . . . . . . . . . . . . . 15  |-  0  <_  0
39 ltpnf 10685 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR  ->  1  <  +oo )
4018, 39ax-mp 8 . . . . . . . . . . . . . . 15  |-  1  <  +oo
41 iocssioo 24093 . . . . . . . . . . . . . . 15  |-  ( ( ( 0  e.  RR*  /\ 
+oo  e.  RR* )  /\  ( 0  <_  0  /\  1  <  +oo )
)  ->  ( 0 (,] 1 )  C_  ( 0 (,)  +oo ) )
4217, 37, 38, 40, 41mp4an 655 . . . . . . . . . . . . . 14  |-  ( 0 (,] 1 )  C_  ( 0 (,)  +oo )
43 ioorp 10952 . . . . . . . . . . . . . 14  |-  ( 0 (,)  +oo )  =  RR+
4442, 43sseqtri 3348 . . . . . . . . . . . . 13  |-  ( 0 (,] 1 )  C_  RR+
4544sseli 3312 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 (,] 1 )  ->  z  e.  RR+ )
46 relogcl 20434 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( log `  z )  e.  RR )
4746renegcld 9428 . . . . . . . . . . . . . 14  |-  ( z  e.  RR+  ->  -u ( log `  z )  e.  RR )
48 ltpnf 10685 . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  (  +oo `'  <  -u ( log `  z
)  <->  -u ( log `  z
)  <  +oo ) )
5249, 51mpbird 224 . . . . . . . . . . . 12  |-  ( z  e.  RR+  ->  +oo `'  <  -u ( log `  z
) )
5345, 52syl 16 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  ->  +oo `'  <  -u ( log `  z
) )
5412xrge0iifcv 24281 . . . . . . . . . . 11  |-  ( z  e.  ( 0 (,] 1 )  ->  ( F `  z )  =  -u ( log `  z
) )
5553, 54breqtrrd 4206 . . . . . . . . . 10  |-  ( z  e.  ( 0 (,] 1 )  ->  +oo `'  <  ( F `  z
) )
5636, 55syl 16 . . . . . . . . 9  |-  ( ( z  e.  ( 0 [,] 1 )  /\  0  <  z )  ->  +oo `'  <  ( F `  z ) )
5756ex 424 . . . . . . . 8  |-  ( z  e.  ( 0 [,] 1 )  ->  (
0  <  z  ->  +oo `'  <  ( F `  z ) ) )
58 breq1 4183 . . . . . . . . 9  |-  ( w  =  0  ->  (
w  <  z  <->  0  <  z ) )
59 fveq2 5695 . . . . . . . . . . 11  |-  ( w  =  0  ->  ( F `  w )  =  ( F ` 
0 ) )
60 0elunit 10979 . . . . . . . . . . . 12  |-  0  e.  ( 0 [,] 1
)
61 iftrue 3713 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  +oo )
6237elexi 2933 . . . . . . . . . . . . 13  |-  +oo  e.  _V
6361, 12, 62fvmpt 5773 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  =  +oo )
6460, 63ax-mp 8 . . . . . . . . . . 11  |-  ( F `
 0 )  = 
+oo
6559, 64syl6eq 2460 . . . . . . . . . 10  |-  ( w  =  0  ->  ( F `  w )  =  +oo )
6665breq1d 4190 . . . . . . . . 9  |-  ( w  =  0  ->  (
( F `  w
) `'  <  ( F `  z )  <->  +oo `'  <  ( F `  z ) ) )
6758, 66imbi12d 312 . . . . . . . 8  |-  ( w  =  0  ->  (
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
)  <->  ( 0  < 
z  ->  +oo `'  <  ( F `  z ) ) ) )
6857, 67syl5ibr 213 . . . . . . 7  |-  ( w  =  0  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
6926, 68sylbi 188 . . . . . 6  |-  ( w  e.  { 0 }  ->  ( z  e.  ( 0 [,] 1
)  ->  ( w  <  z  ->  ( F `  w ) `'  <  ( F `  z ) ) ) )
70 simpll 731 . . . . . . . . 9  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  e.  ( 0 (,] 1
) )
7127ad2antlr 708 . . . . . . . . . 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 3312 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,] 1 )  ->  w  e.  RR+ )
7473rpred 10612 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 (,] 1 )  ->  w  e.  RR )
7574ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  e.  RR )
76 elioc2 10937 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
w  e.  ( 0 (,] 1 )  <->  ( w  e.  RR  /\  0  < 
w  /\  w  <_  1 ) ) )
7717, 18, 76mp2an 654 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,] 1 )  <->  ( w  e.  RR  /\  0  < 
w  /\  w  <_  1 ) )
7877simp2bi 973 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 (,] 1 )  ->  0  <  w )
7978ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  0  <  w )
80 simpr 448 . . . . . . . . . . 11  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  w  <  z )
8172, 75, 71, 79, 80lttrd 9195 . . . . . . . . . 10  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  0  <  z )
8232ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  z  <_  1 )
8371, 81, 82, 35syl3anbrc 1138 . . . . . . . . 9  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  z  e.  ( 0 (,] 1
) )
8470, 83jca 519 . . . . . . . 8  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  (
w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) ) )
8573adantr 452 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  w  e.  RR+ )
8685relogcld 20479 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( log `  w
)  e.  RR )
8745adantl 453 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  z  e.  RR+ )
8887relogcld 20479 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( log `  z
)  e.  RR )
8986, 88ltnegd 9568 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( ( log `  w )  <  ( log `  z )  <->  -u ( log `  z )  <  -u ( log `  w ) ) )
90 logltb 20455 . . . . . . . . . . . 12  |-  ( ( w  e.  RR+  /\  z  e.  RR+ )  ->  (
w  <  z  <->  ( log `  w )  <  ( log `  z ) ) )
9173, 45, 90syl2an 464 . . . . . . . . . . 11  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  <->  ( log `  w
)  <  ( log `  z ) ) )
92 negex 9268 . . . . . . . . . . . . 13  |-  -u ( log `  w )  e. 
_V
93 negex 9268 . . . . . . . . . . . . 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 277 . . . . . . . . . 10  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  <->  -u ( log `  w
) `'  <  -u ( log `  z ) ) )
9796biimpd 199 . . . . . . . . 9  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  ->  -u ( log `  w ) `'  <  -u ( log `  z
) ) )
9812xrge0iifcv 24281 . . . . . . . . . 10  |-  ( w  e.  ( 0 (,] 1 )  ->  ( F `  w )  =  -u ( log `  w
) )
9998, 54breqan12d 4195 . . . . . . . . 9  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 w ) `'  <  ( F `  z )  <->  -u ( log `  w ) `'  <  -u ( log `  z
) ) )
10097, 99sylibrd 226 . . . . . . . 8  |-  ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 (,] 1 ) )  ->  ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) ) )
10184, 80, 100sylc 58 . . . . . . 7  |-  ( ( ( w  e.  ( 0 (,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  /\  w  < 
z )  ->  ( F `  w ) `'  <  ( F `  z ) )
102101exp31 588 . . . . . 6  |-  ( w  e.  ( 0 (,] 1 )  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
10369, 102jaoi 369 . . . . 5  |-  ( ( w  e.  { 0 }  \/  w  e.  ( 0 (,] 1
) )  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
10425, 103sylbi 188 . . . 4  |-  ( w  e.  ( 0 [,] 1 )  ->  (
z  e.  ( 0 [,] 1 )  -> 
( w  <  z  ->  ( F `  w
) `'  <  ( F `  z )
) ) )
105104imp 419 . . 3  |-  ( ( w  e.  ( 0 [,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  ->  ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) ) )
106105rgen2a 2740 . 2  |-  A. w  e.  ( 0 [,] 1
) A. z  e.  ( 0 [,] 1
) ( w  < 
z  ->  ( F `  w ) `'  <  ( F `  z ) )
107 soisoi 6015 . 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 655 1  |-  F  Isom  <  ,  `'  <  ( ( 0 [,] 1 ) ,  ( 0 [,] 
+oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674    u. cun 3286    C_ wss 3288   ifcif 3707   {csn 3782   class class class wbr 4180    e. cmpt 4234    Po wpo 4469    Or wor 4470   `'ccnv 4844   -onto->wfo 5419   -1-1-onto->wf1o 5420   ` cfv 5421    Isom wiso 5422  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955    +oocpnf 9081   RR*cxr 9083    < clt 9084    <_ cle 9085   -ucneg 9256   RR+crp 10576   (,)cioo 10880   (,]cioc 10881   [,]cicc 10883   expce 12627   logclog 20413
This theorem is referenced by:  xrge0iifhmeo  24283
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636  df-pi 12638  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-log 20415
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