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Theorem i1fres 19550
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 19521 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21adantr 452 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F : RR --> RR )
3 ffn 5550 . . . . . . 7  |-  ( F : RR --> RR  ->  F  Fn  RR )
42, 3syl 16 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F  Fn  RR )
5 fnfvelrn 5826 . . . . . 6  |-  ( ( F  Fn  RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  ran  F
)
64, 5sylan 458 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  ( F `  x )  e.  ran  F )
7 i1f0rn 19527 . . . . . 6  |-  ( F  e.  dom  S.1  ->  0  e.  ran  F )
87ad2antrr 707 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  0  e.  ran  F )
9 ifcl 3735 . . . . 5  |-  ( ( ( F `  x
)  e.  ran  F  /\  0  e.  ran  F )  ->  if (
x  e.  A , 
( F `  x
) ,  0 )  e.  ran  F )
106, 8, 9syl2anc 643 . . . 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 5852 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> ran  F )
13 frn 5556 . . . 4  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
142, 13syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  C_  RR )
15 fss 5558 . . 3  |-  ( ( G : RR --> ran  F  /\  ran  F  C_  RR )  ->  G : RR --> RR )
1612, 14, 15syl2anc 643 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> RR )
17 i1frn 19522 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
1817adantr 452 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  e.  Fin )
19 frn 5556 . . . 4  |-  ( G : RR --> ran  F  ->  ran  G  C_  ran  F )
2012, 19syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  C_  ran  F )
21 ssfi 7288 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  C_  ran  F
)  ->  ran  G  e. 
Fin )
2218, 20, 21syl2anc 643 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  e.  Fin )
23 eleq1 2464 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
24 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
25 eqidd 2405 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  0  =  0 )
2623, 24, 25ifbieq12d 3721 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  =  if ( z  e.  A , 
( F `  z
) ,  0 ) )
27 fvex 5701 . . . . . . . . . . . . . 14  |-  ( F `
 z )  e. 
_V
28 c0ex 9041 . . . . . . . . . . . . . 14  |-  0  e.  _V
2927, 28ifex 3757 . . . . . . . . . . . . 13  |-  if ( z  e.  A , 
( F `  z
) ,  0 )  e.  _V
3026, 11, 29fvmpt 5765 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3130adantl 453 . . . . . . . . . . 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 2412 . . . . . . . . . 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 3888 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  ->  y  =/=  0
)
3433ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  y  =/=  0 )
3534necomd 2650 . . . . . . . . . . . . 13  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  0  =/=  y )
36 iffalse 3706 . . . . . . . . . . . . . 14  |-  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  0 )
3736neeq1d 2580 . . . . . . . . . . . . 13  |-  ( -.  z  e.  A  -> 
( if ( z  e.  A ,  ( F `  z ) ,  0 )  =/=  y  <->  0  =/=  y
) )
3835, 37syl5ibrcom 214 . . . . . . . . . . . 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 2629 . . . . . . . . . . 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 617 . . . . . . . . . 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 245 . . . . . . . . 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 3705 . . . . . . . . . . 11  |-  ( z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  ( F `
 z ) )
4342eqeq1d 2412 . . . . . . . . . 10  |-  ( z  e.  A  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( F `  z )  =  y ) )
4443pm5.32i 619 . . . . . . . . 9  |-  ( ( z  e.  A  /\  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y )  <-> 
( z  e.  A  /\  ( F `  z
)  =  y ) )
4541, 44syl6bb 253 . . . . . . . 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 623 . . . . . . 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 773 . . . . . . 7  |-  ( ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z
)  =  y ) )  <->  ( z  e.  A  /\  ( z  e.  RR  /\  ( F `  z )  =  y ) ) )
4846, 47syl6bb 253 . . . . . 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 5550 . . . . . . . . 9  |-  ( G : RR --> ran  F  ->  G  Fn  RR )
5012, 49syl 16 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  Fn  RR )
5150adantr 452 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  G  Fn  RR )
52 fniniseg 5810 . . . . . . 7  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
5351, 52syl 16 . . . . . 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 452 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  F  Fn  RR )
55 fniniseg 5810 . . . . . . . 8  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { y } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  y ) ) )
5654, 55syl 16 . . . . . . 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 685 . . . . . 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 277 . . . . 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 3490 . . . . 5  |-  ( z  e.  ( A  i^i  ( `' F " { y } ) )  <->  ( z  e.  A  /\  z  e.  ( `' F " { y } ) ) )
6058, 59syl6bbr 255 . . . 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 2402 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  =  ( A  i^i  ( `' F " { y } ) ) )
62 simplr 732 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  A  e.  dom  vol )
63 i1fima 19523 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
6463ad2antrr 707 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
65 inmbl 19389 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( `' F " { y } )  e.  dom  vol )  ->  ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol )
6662, 64, 65syl2anc 643 . . 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 2478 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  e.  dom  vol )
6861fveq2d 5691 . . . 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 19379 . . . . 5  |-  ( ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol  ->  ( vol `  ( A  i^i  ( `' F " { y } ) ) )  =  ( vol * `  ( A  i^i  ( `' F " { y } ) ) ) )
7066, 69syl 16 . . . 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 2436 . . 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 3522 . . . . 5  |-  ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } )
7372a1i 11 . . . 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 19380 . . . . 5  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( `' F " { y } ) 
C_  RR )
7564, 74syl 16 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  C_  RR )
76 mblvol 19379 . . . . . 6  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( vol `  ( `' F " { y } ) )  =  ( vol * `  ( `' F " { y } ) ) )
7764, 76syl 16 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol
* `  ( `' F " { y } ) ) )
78 i1fima2sn 19525 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' F " { y } ) )  e.  RR )
7978adantlr 696 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
8077, 79eqeltrrd 2479 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol * `
 ( `' F " { y } ) )  e.  RR )
81 ovolsscl 19335 . . . 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 1184 . . 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 2478 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  e.  RR )
8416, 22, 67, 83i1fd 19526 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    i^i cin 3279    C_ wss 3280   ifcif 3699   {csn 3774    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413   Fincfn 7068   RRcr 8945   0cc0 8946   vol *covol 19312   volcvol 19313   S.1citg1 19460
This theorem is referenced by:  i1fpos  19551  itg1climres  19559  itg2uba  19588  itg2splitlem  19593  itg2monolem1  19595
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-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-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-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-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  df-mbf 19465  df-itg1 19466
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