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Theorem vitalilem5 19457
Description: Lemma for vitali 19458. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
vitali.2  |-  S  =  ( ( 0 [,] 1 ) /.  .~  )
vitali.3  |-  ( ph  ->  F  Fn  S )
vitali.4  |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
vitali.5  |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
vitali.6  |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F } )
vitali.7  |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \  dom  vol ) )
Assertion
Ref Expression
vitalilem5  |-  -.  ph
Distinct variable groups:    n, s, x, y, z, G    ph, n, x, z    z, S    x, T    n, F, s, x, y, z    .~ , n, s, x, y, z
Allowed substitution hints:    ph( y, s)    S( x, y, n, s)    T( y, z, n, s)

Proof of Theorem vitalilem5
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 0lt1 9506 . . . 4  |-  0  <  1
2 0re 9047 . . . . . 6  |-  0  e.  RR
3 1re 9046 . . . . . 6  |-  1  e.  RR
4 0le1 9507 . . . . . 6  |-  0  <_  1
5 ovolicc 19372 . . . . . 6  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  <_  1 )  ->  ( vol * `  ( 0 [,] 1 ) )  =  ( 1  -  0 ) )
62, 3, 4, 5mp3an 1279 . . . . 5  |-  ( vol
* `  ( 0 [,] 1 ) )  =  ( 1  -  0 )
7 ax-1cn 9004 . . . . . 6  |-  1  e.  CC
87subid1i 9328 . . . . 5  |-  ( 1  -  0 )  =  1
96, 8eqtri 2424 . . . 4  |-  ( vol
* `  ( 0 [,] 1 ) )  =  1
101, 9breqtrri 4197 . . 3  |-  0  <  ( vol * `  ( 0 [,] 1
) )
119, 3eqeltri 2474 . . . 4  |-  ( vol
* `  ( 0 [,] 1 ) )  e.  RR
122, 11ltnlei 9150 . . 3  |-  ( 0  <  ( vol * `  ( 0 [,] 1
) )  <->  -.  ( vol * `  ( 0 [,] 1 ) )  <_  0 )
1310, 12mpbi 200 . 2  |-  -.  ( vol * `  ( 0 [,] 1 ) )  <_  0
14 vitali.1 . . . . . 6  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
15 vitali.2 . . . . . 6  |-  S  =  ( ( 0 [,] 1 ) /.  .~  )
16 vitali.3 . . . . . 6  |-  ( ph  ->  F  Fn  S )
17 vitali.4 . . . . . 6  |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
18 vitali.5 . . . . . 6  |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
19 vitali.6 . . . . . 6  |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F } )
20 vitali.7 . . . . . 6  |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \  dom  vol ) )
2114, 15, 16, 17, 18, 19, 20vitalilem2 19454 . . . . 5  |-  ( ph  ->  ( ran  F  C_  ( 0 [,] 1
)  /\  ( 0 [,] 1 )  C_  U_ m  e.  NN  ( T `  m )  /\  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) ) )
2221simp2d 970 . . . 4  |-  ( ph  ->  ( 0 [,] 1
)  C_  U_ m  e.  NN  ( T `  m ) )
2314vitalilem1 19453 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  .~  Er  ( 0 [,] 1
)
24 erdm 6874 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (  .~  Er  ( 0 [,] 1
)  ->  dom  .~  =  ( 0 [,] 1
) )
2523, 24ax-mp 8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  dom  .~  =  ( 0 [,] 1 )
26 simpr 448 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  z  e.  S )  ->  z  e.  S )
2726, 15syl6eleq 2494 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  z  e.  S )  ->  z  e.  ( ( 0 [,] 1 ) /.  .~  ) )
28 elqsn0 6932 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( dom  .~  =  ( 0 [,] 1 )  /\  z  e.  ( ( 0 [,] 1
) /.  .~  )
)  ->  z  =/=  (/) )
2925, 27, 28sylancr 645 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  z  e.  S )  ->  z  =/=  (/) )
3023a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  .~  Er  ( 0 [,] 1 ) )
3130qsss 6924 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( 0 [,] 1 ) /.  .~  )  C_  ~P ( 0 [,] 1 ) )
3215, 31syl5eqss 3352 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  S  C_  ~P (
0 [,] 1 ) )
3332sselda 3308 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  z  e.  S )  ->  z  e.  ~P ( 0 [,] 1 ) )
3433elpwid 3768 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  z  e.  S )  ->  z  C_  ( 0 [,] 1
) )
3534sseld 3307 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  z  e.  S )  ->  (
( F `  z
)  e.  z  -> 
( F `  z
)  e.  ( 0 [,] 1 ) ) )
3629, 35embantd 52 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  z  e.  S )  ->  (
( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  ( F `  z )  e.  ( 0 [,] 1
) ) )
3736ralimdva 2744 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) ) )
3817, 37mpd 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) )
39 ffnfv 5853 . . . . . . . . . . . . . . . . . . 19  |-  ( F : S --> ( 0 [,] 1 )  <->  ( F  Fn  S  /\  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) ) )
4016, 38, 39sylanbrc 646 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : S --> ( 0 [,] 1 ) )
41 frn 5556 . . . . . . . . . . . . . . . . . 18  |-  ( F : S --> ( 0 [,] 1 )  ->  ran  F  C_  ( 0 [,] 1 ) )
4240, 41syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  F  C_  (
0 [,] 1 ) )
43 unitssre 10998 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,] 1 )  C_  RR
4442, 43syl6ss 3320 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ran  F  C_  RR )
45 reex 9037 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
4645elpw2 4324 . . . . . . . . . . . . . . . 16  |-  ( ran 
F  e.  ~P RR  <->  ran 
F  C_  RR )
4744, 46sylibr 204 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  e.  ~P RR )
4847anim1i 552 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  -> 
( ran  F  e.  ~P RR  /\  -.  ran  F  e.  dom  vol )
)
49 eldif 3290 . . . . . . . . . . . . . 14  |-  ( ran 
F  e.  ( ~P RR  \  dom  vol ) 
<->  ( ran  F  e. 
~P RR  /\  -.  ran  F  e.  dom  vol ) )
5048, 49sylibr 204 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  ->  ran  F  e.  ( ~P RR  \  dom  vol ) )
5150ex 424 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  ran  F  e.  dom  vol  ->  ran  F  e.  ( ~P RR  \  dom  vol ) ) )
5220, 51mt3d 119 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  e.  dom  vol )
5352adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ran  F  e.  dom  vol )
54 f1of 5633 . . . . . . . . . . . . 13  |-  ( G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  G : NN
--> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
5518, 54syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) ) )
56 inss1 3521 . . . . . . . . . . . . 13  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
57 qssre 10540 . . . . . . . . . . . . 13  |-  QQ  C_  RR
5856, 57sstri 3317 . . . . . . . . . . . 12  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
59 fss 5558 . . . . . . . . . . . 12  |-  ( ( G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) )  /\  ( QQ  i^i  ( -u 1 [,] 1 ) )  C_  RR )  ->  G : NN --> RR )
6055, 58, 59sylancl 644 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
6160ffvelrnda 5829 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
62 shftmbl 19386 . . . . . . . . . 10  |-  ( ( ran  F  e.  dom  vol 
/\  ( G `  n )  e.  RR )  ->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e. 
ran  F }  e.  dom  vol )
6353, 61, 62syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  e.  dom  vol )
6463, 19fmptd 5852 . . . . . . . 8  |-  ( ph  ->  T : NN --> dom  vol )
6564ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
6665ralrimiva 2749 . . . . . 6  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
67 iunmbl 19400 . . . . . 6  |-  ( A. m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
6866, 67syl 16 . . . . 5  |-  ( ph  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
69 mblss 19380 . . . . 5  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
7068, 69syl 16 . . . 4  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  RR )
71 ovolss 19334 . . . 4  |-  ( ( ( 0 [,] 1
)  C_  U_ m  e.  NN  ( T `  m )  /\  U_ m  e.  NN  ( T `  m )  C_  RR )  ->  ( vol * `  ( 0 [,] 1 ) )  <_  ( vol * `  U_ m  e.  NN  ( T `  m ) ) )
7222, 70, 71syl2anc 643 . . 3  |-  ( ph  ->  ( vol * `  ( 0 [,] 1
) )  <_  ( vol * `  U_ m  e.  NN  ( T `  m ) ) )
73 eqid 2404 . . . . . 6  |-  seq  1
(  +  ,  ( m  e.  NN  |->  ( vol * `  ( T `  m )
) ) )  =  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol * `  ( T `  m
) ) ) )
74 eqid 2404 . . . . . 6  |-  ( m  e.  NN  |->  ( vol
* `  ( T `  m ) ) )  =  ( m  e.  NN  |->  ( vol * `  ( T `  m
) ) )
75 mblss 19380 . . . . . . 7  |-  ( ( T `  m )  e.  dom  vol  ->  ( T `  m ) 
C_  RR )
7665, 75syl 16 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  C_  RR )
7714, 15, 16, 17, 18, 19, 20vitalilem4 19456 . . . . . . 7  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ( T `  m ) )  =  0 )
7877, 2syl6eqel 2492 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ( T `  m ) )  e.  RR )
7977mpteq2dva 4255 . . . . . . . . . 10  |-  ( ph  ->  ( m  e.  NN  |->  ( vol * `  ( T `  m )
) )  =  ( m  e.  NN  |->  0 ) )
80 fconstmpt 4880 . . . . . . . . . . 11  |-  ( NN 
X.  { 0 } )  =  ( m  e.  NN  |->  0 )
81 nnuz 10477 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
8281xpeq1i 4857 . . . . . . . . . . 11  |-  ( NN 
X.  { 0 } )  =  ( (
ZZ>= `  1 )  X. 
{ 0 } )
8380, 82eqtr3i 2426 . . . . . . . . . 10  |-  ( m  e.  NN  |->  0 )  =  ( ( ZZ>= ` 
1 )  X.  {
0 } )
8479, 83syl6eq 2452 . . . . . . . . 9  |-  ( ph  ->  ( m  e.  NN  |->  ( vol * `  ( T `  m )
) )  =  ( ( ZZ>= `  1 )  X.  { 0 } ) )
8584seqeq3d 11286 . . . . . . . 8  |-  ( ph  ->  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol * `  ( T `  m
) ) ) )  =  seq  1 (  +  ,  ( (
ZZ>= `  1 )  X. 
{ 0 } ) ) )
86 1z 10267 . . . . . . . . 9  |-  1  e.  ZZ
87 serclim0 12326 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  seq  1 (  +  , 
( ( ZZ>= `  1
)  X.  { 0 } ) )  ~~>  0 )
8886, 87ax-mp 8 . . . . . . . 8  |-  seq  1
(  +  ,  ( ( ZZ>= `  1 )  X.  { 0 } ) )  ~~>  0
8985, 88syl6eqbr 4209 . . . . . . 7  |-  ( ph  ->  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol * `  ( T `  m
) ) ) )  ~~>  0 )
90 seqex 11280 . . . . . . . 8  |-  seq  1
(  +  ,  ( m  e.  NN  |->  ( vol * `  ( T `  m )
) ) )  e. 
_V
91 c0ex 9041 . . . . . . . 8  |-  0  e.  _V
9290, 91breldm 5033 . . . . . . 7  |-  (  seq  1 (  +  , 
( m  e.  NN  |->  ( vol * `  ( T `  m )
) ) )  ~~>  0  ->  seq  1 (  +  , 
( m  e.  NN  |->  ( vol * `  ( T `  m )
) ) )  e. 
dom 
~~>  )
9389, 92syl 16 . . . . . 6  |-  ( ph  ->  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol * `  ( T `  m
) ) ) )  e.  dom  ~~>  )
9473, 74, 76, 78, 93ovoliun2 19355 . . . . 5  |-  ( ph  ->  ( vol * `  U_ m  e.  NN  ( T `  m )
)  <_  sum_ m  e.  NN  ( vol * `  ( T `  m
) ) )
9577sumeq2dv 12452 . . . . . 6  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  ( T `  m )
)  =  sum_ m  e.  NN  0 )
9681eqimssi 3362 . . . . . . . 8  |-  NN  C_  ( ZZ>= `  1 )
9796orci 380 . . . . . . 7  |-  ( NN  C_  ( ZZ>= `  1 )  \/  NN  e.  Fin )
98 sumz 12471 . . . . . . 7  |-  ( ( NN  C_  ( ZZ>= ` 
1 )  \/  NN  e.  Fin )  ->  sum_ m  e.  NN  0  =  0 )
9997, 98ax-mp 8 . . . . . 6  |-  sum_ m  e.  NN  0  =  0
10095, 99syl6eq 2452 . . . . 5  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  ( T `  m )
)  =  0 )
10194, 100breqtrd 4196 . . . 4  |-  ( ph  ->  ( vol * `  U_ m  e.  NN  ( T `  m )
)  <_  0 )
102 ovolge0 19330 . . . . 5  |-  ( U_ m  e.  NN  ( T `  m )  C_  RR  ->  0  <_  ( vol * `  U_ m  e.  NN  ( T `  m ) ) )
10370, 102syl 16 . . . 4  |-  ( ph  ->  0  <_  ( vol * `
 U_ m  e.  NN  ( T `  m ) ) )
104 ovolcl 19327 . . . . . 6  |-  ( U_ m  e.  NN  ( T `  m )  C_  RR  ->  ( vol * `
 U_ m  e.  NN  ( T `  m ) )  e.  RR* )
10570, 104syl 16 . . . . 5  |-  ( ph  ->  ( vol * `  U_ m  e.  NN  ( T `  m )
)  e.  RR* )
106 0xr 9087 . . . . 5  |-  0  e.  RR*
107 xrletri3 10701 . . . . 5  |-  ( ( ( vol * `  U_ m  e.  NN  ( T `  m )
)  e.  RR*  /\  0  e.  RR* )  ->  (
( vol * `  U_ m  e.  NN  ( T `  m )
)  =  0  <->  (
( vol * `  U_ m  e.  NN  ( T `  m )
)  <_  0  /\  0  <_  ( vol * `  U_ m  e.  NN  ( T `  m ) ) ) ) )
108105, 106, 107sylancl 644 . . . 4  |-  ( ph  ->  ( ( vol * `  U_ m  e.  NN  ( T `  m ) )  =  0  <->  (
( vol * `  U_ m  e.  NN  ( T `  m )
)  <_  0  /\  0  <_  ( vol * `  U_ m  e.  NN  ( T `  m ) ) ) ) )
109101, 103, 108mpbir2and 889 . . 3  |-  ( ph  ->  ( vol * `  U_ m  e.  NN  ( T `  m )
)  =  0 )
11072, 109breqtrd 4196 . 2  |-  ( ph  ->  ( vol * `  ( 0 [,] 1
) )  <_  0
)
11113, 110mto 169 1  |-  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U_ciun 4053   class class class wbr 4172   {copab 4225    e. cmpt 4226    X. cxp 4835   dom cdm 4837   ran crn 4838    Fn wfn 5408   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    Er wer 6861   /.cqs 6863   Fincfn 7068   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248   NNcn 9956   2c2 10005   ZZcz 10238   ZZ>=cuz 10444   QQcq 10530   [,]cicc 10875    seq cseq 11278    ~~> cli 12233   sum_csu 12434   vol *covol 19312   volcvol 19313
This theorem is referenced by:  vitali  19458
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-cc 8271  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
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-disj 4143  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-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  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-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-rest 13605  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cmp 17404  df-ovol 19314  df-vol 19315
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