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Theorem i1fres 19158
Description: The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside  A.) (Contributed by Mario Carneiro, 29-Jun-2014.)
Hypothesis
Ref Expression
i1fres.1  |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )
Assertion
Ref Expression
i1fres  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    G( x)

Proof of Theorem i1fres
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1ff 19129 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21adantr 451 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F : RR --> RR )
3 ffn 5469 . . . . . . 7  |-  ( F : RR --> RR  ->  F  Fn  RR )
42, 3syl 15 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F  Fn  RR )
5 fnfvelrn 5742 . . . . . 6  |-  ( ( F  Fn  RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  ran  F
)
64, 5sylan 457 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  ( F `  x )  e.  ran  F )
7 i1f0rn 19135 . . . . . 6  |-  ( F  e.  dom  S.1  ->  0  e.  ran  F )
87ad2antrr 706 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  0  e.  ran  F )
9 ifcl 3677 . . . . 5  |-  ( ( ( F `  x
)  e.  ran  F  /\  0  e.  ran  F )  ->  if (
x  e.  A , 
( F `  x
) ,  0 )  e.  ran  F )
106, 8, 9syl2anc 642 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  e.  ran  F
)
11 i1fres.1 . . . 4  |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )
1210, 11fmptd 5764 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> ran  F )
13 frn 5475 . . . 4  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
142, 13syl 15 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  C_  RR )
15 fss 5477 . . 3  |-  ( ( G : RR --> ran  F  /\  ran  F  C_  RR )  ->  G : RR --> RR )
1612, 14, 15syl2anc 642 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> RR )
17 i1frn 19130 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
1817adantr 451 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  e.  Fin )
19 frn 5475 . . . 4  |-  ( G : RR --> ran  F  ->  ran  G  C_  ran  F )
2012, 19syl 15 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  C_  ran  F )
21 ssfi 7168 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  C_  ran  F
)  ->  ran  G  e. 
Fin )
2218, 20, 21syl2anc 642 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  e.  Fin )
23 eleq1 2418 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
24 fveq2 5605 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
25 eqidd 2359 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  0  =  0 )
2623, 24, 25ifbieq12d 3663 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  =  if ( z  e.  A , 
( F `  z
) ,  0 ) )
27 fvex 5619 . . . . . . . . . . . . . 14  |-  ( F `
 z )  e. 
_V
28 c0ex 8919 . . . . . . . . . . . . . 14  |-  0  e.  _V
2927, 28ifex 3699 . . . . . . . . . . . . 13  |-  if ( z  e.  A , 
( F `  z
) ,  0 )  e.  _V
3026, 11, 29fvmpt 5682 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3130adantl 452 . . . . . . . . . . 11  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3231eqeq1d 2366 . . . . . . . . . 10  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  if (
z  e.  A , 
( F `  z
) ,  0 )  =  y ) )
33 eldifsni 3826 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  ->  y  =/=  0
)
3433ad2antlr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  y  =/=  0 )
3534necomd 2604 . . . . . . . . . . . . 13  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  0  =/=  y )
36 iffalse 3648 . . . . . . . . . . . . . 14  |-  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  0 )
3736neeq1d 2534 . . . . . . . . . . . . 13  |-  ( -.  z  e.  A  -> 
( if ( z  e.  A ,  ( F `  z ) ,  0 )  =/=  y  <->  0  =/=  y
) )
3835, 37syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `
 z ) ,  0 )  =/=  y
) )
3938necon4bd 2583 . . . . . . . . . . 11  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  -> 
z  e.  A ) )
4039pm4.71rd 616 . . . . . . . . . 10  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( z  e.  A  /\  if ( z  e.  A , 
( F `  z
) ,  0 )  =  y ) ) )
4132, 40bitrd 244 . . . . . . . . 9  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  ( z  e.  A  /\  if ( z  e.  A , 
( F `  z
) ,  0 )  =  y ) ) )
42 iftrue 3647 . . . . . . . . . . 11  |-  ( z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  ( F `
 z ) )
4342eqeq1d 2366 . . . . . . . . . 10  |-  ( z  e.  A  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( F `  z )  =  y ) )
4443pm5.32i 618 . . . . . . . . 9  |-  ( ( z  e.  A  /\  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y )  <-> 
( z  e.  A  /\  ( F `  z
)  =  y ) )
4541, 44syl6bb 252 . . . . . . . 8  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  ( z  e.  A  /\  ( F `  z )  =  y ) ) )
4645pm5.32da 622 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  RR  /\  ( G `  z )  =  y )  <->  ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z )  =  y ) ) ) )
47 an12 772 . . . . . . 7  |-  ( ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z
)  =  y ) )  <->  ( z  e.  A  /\  ( z  e.  RR  /\  ( F `  z )  =  y ) ) )
4846, 47syl6bb 252 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  RR  /\  ( G `  z )  =  y )  <->  ( z  e.  A  /\  (
z  e.  RR  /\  ( F `  z )  =  y ) ) ) )
49 ffn 5469 . . . . . . . . 9  |-  ( G : RR --> ran  F  ->  G  Fn  RR )
5012, 49syl 15 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  Fn  RR )
5150adantr 451 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  G  Fn  RR )
52 fniniseg 5726 . . . . . . 7  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
5351, 52syl 15 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
544adantr 451 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  F  Fn  RR )
55 fniniseg 5726 . . . . . . . 8  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { y } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  y ) ) )
5654, 55syl 15 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' F " { y } )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  y ) ) )
5756anbi2d 684 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  A  /\  z  e.  ( `' F " { y } ) )  <->  ( z  e.  A  /\  (
z  e.  RR  /\  ( F `  z )  =  y ) ) ) )
5848, 53, 573bitr4d 276 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  A  /\  z  e.  ( `' F " { y } ) ) ) )
59 elin 3434 . . . . 5  |-  ( z  e.  ( A  i^i  ( `' F " { y } ) )  <->  ( z  e.  A  /\  z  e.  ( `' F " { y } ) ) )
6058, 59syl6bbr 254 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
z  e.  ( A  i^i  ( `' F " { y } ) ) ) )
6160eqrdv 2356 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  =  ( A  i^i  ( `' F " { y } ) ) )
62 simplr 731 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  A  e.  dom  vol )
63 i1fima 19131 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
6463ad2antrr 706 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
65 inmbl 18997 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( `' F " { y } )  e.  dom  vol )  ->  ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol )
6662, 64, 65syl2anc 642 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( A  i^i  ( `' F " { y } ) )  e.  dom  vol )
6761, 66eqeltrd 2432 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  e.  dom  vol )
6861fveq2d 5609 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  =  ( vol `  ( A  i^i  ( `' F " { y } ) ) ) )
69 mblvol 18987 . . . . 5  |-  ( ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol  ->  ( vol `  ( A  i^i  ( `' F " { y } ) ) )  =  ( vol * `  ( A  i^i  ( `' F " { y } ) ) ) )
7066, 69syl 15 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( A  i^i  ( `' F " { y } ) ) )  =  ( vol * `  ( A  i^i  ( `' F " { y } ) ) ) )
7168, 70eqtrd 2390 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  =  ( vol
* `  ( A  i^i  ( `' F " { y } ) ) ) )
72 inss2 3466 . . . . 5  |-  ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } )
7372a1i 10 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } ) )
74 mblss 18988 . . . . 5  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( `' F " { y } ) 
C_  RR )
7564, 74syl 15 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  C_  RR )
76 mblvol 18987 . . . . . 6  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( vol `  ( `' F " { y } ) )  =  ( vol * `  ( `' F " { y } ) ) )
7764, 76syl 15 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol
* `  ( `' F " { y } ) ) )
78 i1fima2sn 19133 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' F " { y } ) )  e.  RR )
7978adantlr 695 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
8077, 79eqeltrrd 2433 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol * `
 ( `' F " { y } ) )  e.  RR )
81 ovolsscl 18943 . . . 4  |-  ( ( ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } )  /\  ( `' F " { y } )  C_  RR  /\  ( vol * `  ( `' F " { y } ) )  e.  RR )  ->  ( vol * `  ( A  i^i  ( `' F " { y } ) ) )  e.  RR )
8273, 75, 80, 81syl3anc 1182 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol * `
 ( A  i^i  ( `' F " { y } ) ) )  e.  RR )
8371, 82eqeltrd 2432 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  e.  RR )
8416, 22, 67, 83i1fd 19134 1  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225    i^i cin 3227    C_ wss 3228   ifcif 3641   {csn 3716    e. cmpt 4156   `'ccnv 4767   dom cdm 4768   ran crn 4769   "cima 4771    Fn wfn 5329   -->wf 5330   ` cfv 5334   Fincfn 6948   RRcr 8823   0cc0 8824   vol *covol 18920   volcvol 18921   S.1citg1 19068
This theorem is referenced by:  i1fpos  19159  itg1climres  19167  itg2uba  19196  itg2splitlem  19201  itg2monolem1  19203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-fl 11014  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-sum 12250  df-rest 13420  df-topgen 13437  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-top 16736  df-bases 16738  df-topon 16739  df-cmp 17214  df-ovol 18922  df-vol 18923  df-mbf 19073  df-itg1 19074
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