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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovollb 18801 The outer volume is a lower bound on the sum of all interval coverings of  A. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U.
 ran  ( (,)  o.  F ) )  ->  ( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  )
 )
 
Theoremovolgelb 18802* The outer volume is the greatest lower bound on the sum of all interval coverings of  A. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  g ) )   =>    |-  ( ( A 
 C_  RR  /\  ( vol
 * `  A )  e.  RR  /\  B  e.  RR+ )  ->  E. g  e.  ( (  <_  i^i  ( RR  X.  RR )
 )  ^m  NN )
 ( A  C_  U. ran  ( (,)  o.  g ) 
 /\  sup ( ran  S ,  RR* ,  <  )  <_  ( ( vol * `  A )  +  B ) ) )
 
Theoremovolge0 18803 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR  ->  0  <_  ( vol * `
  A ) )
 
Theoremovolf 18804 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |- 
 vol * : ~P RR --> ( 0 [,]  +oo )
 
Theoremovollecl 18805 If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  C_  RR  /\  B  e.  RR  /\  ( vol * `  A )  <_  B ) 
 ->  ( vol * `  A )  e.  RR )
 
Theoremovolsslem 18806* Lemma for ovolss 18807. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR )
 )  ^m  NN )
 ( A  C_  U. ran  ( (,)  o.  f ) 
 /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
 RR* ,  <  ) ) }   &    |-  N  =  {
 y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
 ran  ( (,)  o.  f )  /\  y  = 
 sup ( ran  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  f ) ) , 
 RR* ,  <  ) ) }   =>    |-  ( ( A  C_  B  /\  B  C_  RR )  ->  ( vol * `  A )  <_  ( vol * `  B ) )
 
Theoremovolss 18807 The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( ( A  C_  B  /\  B  C_  RR )  ->  ( vol * `  A )  <_  ( vol * `  B ) )
 
Theoremovolsscl 18808 If a set is contained in a another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  C_  B  /\  B  C_  RR  /\  ( vol * `  B )  e.  RR )  ->  ( vol * `  A )  e.  RR )
 
Theoremovolssnul 18809 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |-  ( ( A  C_  B  /\  B  C_  RR  /\  ( vol * `  B )  =  0
 )  ->  ( vol * `
  A )  =  0 )
 
Theoremovollb2lem 18810* Lemma for ovollb2 18811. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  G  =  ( n  e.  NN  |->  <.
 ( ( 1st `  ( F `  n ) )  -  ( ( B 
 /  2 )  /  ( 2 ^ n ) ) ) ,  ( ( 2nd `  ( F `  n ) )  +  ( ( B 
 /  2 )  /  ( 2 ^ n ) ) ) >. )   &    |-  T  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  A  C_  U. ran  ( [,]  o.  F ) )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  e.  RR )   =>    |-  ( ph  ->  ( vol * `  A ) 
 <_  ( sup ( ran 
 S ,  RR* ,  <  )  +  B ) )
 
Theoremovollb2 18811 It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 18801). (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U.
 ran  ( [,]  o.  F ) )  ->  ( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  )
 )
 
Theoremovolctb 18812 The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
 |-  ( ( A  C_  RR  /\  A  ~~  NN )  ->  ( vol * `  A )  =  0 )
 
Theoremovolq 18813 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  QQ )  =  0
 
Theoremovolctb2 18814 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  C_  RR  /\  A  ~<_  NN )  ->  ( vol * `  A )  =  0
 )
 
Theoremovol0 18815 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  (/) )  =  0
 
Theoremovolfi 18816 A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ( A  e.  Fin  /\  A  C_  RR )  ->  ( vol * `  A )  =  0
 )
 
Theoremovolsn 18817 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  { A } )  =  0 )
 
Theoremovolunlem1a 18818* Lemma for ovolun 18821. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ph  ->  ( A  C_  RR  /\  ( vol * `  A )  e.  RR ) )   &    |-  ( ph  ->  ( B  C_ 
 RR  /\  ( vol * `
  B )  e. 
 RR ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F  e.  (
 (  <_  i^i  ( RR 
 X.  RR ) )  ^m  NN ) )   &    |-  ( ph  ->  A 
 C_  U. ran  ( (,) 
 o.  F ) )   &    |-  ( ph  ->  sup ( ran 
 S ,  RR* ,  <  ) 
 <_  ( ( vol * `  A )  +  ( C  /  2 ) ) )   &    |-  ( ph  ->  G  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )
 )   &    |-  ( ph  ->  B  C_ 
 U. ran  ( (,)  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol * `  B )  +  ( C  /  2 ) ) )   &    |-  H  =  ( n  e.  NN  |->  if ( ( n  / 
 2 )  e.  NN ,  ( G `  ( n  /  2 ) ) ,  ( F `  ( ( n  +  1 )  /  2
 ) ) ) )   =>    |-  ( ( ph  /\  k  e.  NN )  ->  ( U `  k )  <_  ( ( ( vol
 * `  A )  +  ( vol * `  B ) )  +  C ) )
 
Theoremovolunlem1 18819* Lemma for ovolun 18821. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ph  ->  ( A  C_  RR  /\  ( vol * `  A )  e.  RR ) )   &    |-  ( ph  ->  ( B  C_ 
 RR  /\  ( vol * `
  B )  e. 
 RR ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F  e.  (
 (  <_  i^i  ( RR 
 X.  RR ) )  ^m  NN ) )   &    |-  ( ph  ->  A 
 C_  U. ran  ( (,) 
 o.  F ) )   &    |-  ( ph  ->  sup ( ran 
 S ,  RR* ,  <  ) 
 <_  ( ( vol * `  A )  +  ( C  /  2 ) ) )   &    |-  ( ph  ->  G  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )
 )   &    |-  ( ph  ->  B  C_ 
 U. ran  ( (,)  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol * `  B )  +  ( C  /  2 ) ) )   &    |-  H  =  ( n  e.  NN  |->  if ( ( n  / 
 2 )  e.  NN ,  ( G `  ( n  /  2 ) ) ,  ( F `  ( ( n  +  1 )  /  2
 ) ) ) )   =>    |-  ( ph  ->  ( vol * `
  ( A  u.  B ) )  <_  ( ( ( vol
 * `  A )  +  ( vol * `  B ) )  +  C ) )
 
Theoremovolunlem2 18820 Lemma for ovolun 18821. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ph  ->  ( A  C_  RR  /\  ( vol * `  A )  e.  RR ) )   &    |-  ( ph  ->  ( B  C_ 
 RR  /\  ( vol * `
  B )  e. 
 RR ) )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( vol * `  ( A  u.  B ) ) 
 <_  ( ( ( vol
 * `  A )  +  ( vol * `  B ) )  +  C ) )
 
Theoremovolun 18821 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 18827, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ( ( A 
 C_  RR  /\  ( vol
 * `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol * `  B )  e.  RR ) ) 
 ->  ( vol * `  ( A  u.  B ) )  <_  ( ( vol * `  A )  +  ( vol * `
  B ) ) )
 
Theoremovolunnul 18822 Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0
 )  ->  ( vol * `
  ( A  u.  B ) )  =  ( vol * `  A ) )
 
Theoremovolfiniun 18823* The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( A  e.  Fin  /\  A. k  e.  A  ( B  C_  RR  /\  ( vol * `  B )  e.  RR )
 )  ->  ( vol * `
  U_ k  e.  A  B )  <_  sum_ k  e.  A  ( vol * `  B ) )
 
Theoremovoliunlem1 18824* Lemma for ovoliun 18827. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  T  =  seq  1
 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol
 * `  A )
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( vol * `  A )  e.  RR )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  ( F `  n ) ) )   &    |-  U  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  H ) )   &    |-  H  =  ( k  e.  NN  |->  ( ( F `  ( 1st `  ( J `  k ) ) ) `
  ( 2nd `  ( J `  k ) ) ) )   &    |-  ( ph  ->  J : NN -1-1-onto-> ( NN  X.  NN ) )   &    |-  ( ph  ->  F : NN --> ( ( 
 <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  A  C_  U. ran  ( (,)  o.  ( F `  n ) ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  sup ( ran  S ,  RR* ,  <  ) 
 <_  ( ( vol * `  A )  +  ( B  /  ( 2 ^ n ) ) ) )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  L  e.  ZZ )   &    |-  ( ph  ->  A. w  e.  ( 1 ... K ) ( 1st `  ( J `  w ) ) 
 <_  L )   =>    |-  ( ph  ->  ( U `  K )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B ) )
 
Theoremovoliunlem2 18825* Lemma for ovoliun 18827. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  T  =  seq  1
 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol
 * `  A )
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( vol * `  A )  e.  RR )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  ( F `  n ) ) )   &    |-  U  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  H ) )   &    |-  H  =  ( k  e.  NN  |->  ( ( F `  ( 1st `  ( J `  k ) ) ) `
  ( 2nd `  ( J `  k ) ) ) )   &    |-  ( ph  ->  J : NN -1-1-onto-> ( NN  X.  NN ) )   &    |-  ( ph  ->  F : NN --> ( ( 
 <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  A  C_  U. ran  ( (,)  o.  ( F `  n ) ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  sup ( ran  S ,  RR* ,  <  ) 
 <_  ( ( vol * `  A )  +  ( B  /  ( 2 ^ n ) ) ) )   =>    |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B ) )
 
Theoremovoliunlem3 18826* Lemma for ovoliun 18827. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  T  =  seq  1
 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol
 * `  A )
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( vol * `  A )  e.  RR )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B ) )
 
Theoremovoliun 18827* The Lebesgue outer measure function is countably sub-additive. (Many books allow  +oo as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 18807, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  T  =  seq  1
 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol
 * `  A )
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( vol * `  A )  e.  RR )   =>    |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR*
 ,  <  ) )
 
Theoremovoliun2 18828* The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 18827.) (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  T  =  seq  1
 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol
 * `  A )
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( vol * `  A )  e.  RR )   &    |-  ( ph  ->  T  e.  dom  ~~>  )   =>    |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A )  <_  sum_ n  e.  NN  ( vol * `  A ) )
 
Theoremovoliunnul 18829* A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ( A  ~<_  NN  /\  A. n  e.  A  ( B  C_  RR  /\  ( vol * `  B )  =  0 ) ) 
 ->  ( vol * `  U_ n  e.  A  B )  =  0 )
 
Theoremshft2rab 18830* If  B is a shift of  A by  C, then  A is a shift of  B by  -u C. (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )   =>    |-  ( ph  ->  A  =  {
 y  e.  RR  |  ( y  -  -u C )  e.  B }
 )
 
Theoremovolshftlem1 18831* Lemma for ovolshft 18833. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )   &    |-  M  =  { y  e.  RR*  | 
 E. f  e.  (
 (  <_  i^i  ( RR 
 X.  RR ) )  ^m  NN ) ( B  C_  U.
 ran  ( (,)  o.  f )  /\  y  = 
 sup ( ran  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  f ) ) , 
 RR* ,  <  ) ) }   &    |-  S  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  G  =  ( n  e.  NN  |->  <.
 ( ( 1st `  ( F `  n ) )  +  C ) ,  ( ( 2nd `  ( F `  n ) )  +  C ) >. )   &    |-  ( ph  ->  F : NN
 --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  A 
 C_  U. ran  ( (,) 
 o.  F ) )   =>    |-  ( ph  ->  sup ( ran 
 S ,  RR* ,  <  )  e.  M )
 
Theoremovolshftlem2 18832* Lemma for ovolshft 18833. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )   &    |-  M  =  { y  e.  RR*  | 
 E. f  e.  (
 (  <_  i^i  ( RR 
 X.  RR ) )  ^m  NN ) ( B  C_  U.
 ran  ( (,)  o.  f )  /\  y  = 
 sup ( ran  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  f ) ) , 
 RR* ,  <  ) ) }   =>    |-  ( ph  ->  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR )
 )  ^m  NN )
 ( A  C_  U. ran  ( (,)  o.  g ) 
 /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  g ) ) , 
 RR* ,  <  ) ) }  C_  M )
 
Theoremovolshft 18833* The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )   =>    |-  ( ph  ->  ( vol * `  A )  =  ( vol * `  B ) )
 
Theoremsca2rab 18834* If  B is a scale of  A by  C, then  A is a scale of  B by  1  /  C. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   =>    |-  ( ph  ->  A  =  { y  e. 
 RR  |  ( ( 1  /  C )  x.  y )  e.  B } )
 
Theoremovolscalem1 18835* Lemma for ovolsca 18837. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   &    |-  ( ph  ->  ( vol * `  A )  e.  RR )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  G  =  ( n  e.  NN  |->  <.
 ( ( 1st `  ( F `  n ) ) 
 /  C ) ,  ( ( 2nd `  ( F `  n ) ) 
 /  C ) >. )   &    |-  ( ph  ->  F : NN
 --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  A 
 C_  U. ran  ( (,) 
 o.  F ) )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol * `  A )  +  ( C  x.  R ) ) )   =>    |-  ( ph  ->  ( vol * `  B ) 
 <_  ( ( ( vol
 * `  A )  /  C )  +  R ) )
 
Theoremovolscalem2 18836* Lemma for ovolshft 18833. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   &    |-  ( ph  ->  ( vol * `  A )  e.  RR )   =>    |-  ( ph  ->  ( vol * `  B ) 
 <_  ( ( vol * `  A )  /  C ) )
 
Theoremovolsca 18837* The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  B  =  { x  e. 
 RR  |  ( C  x.  x )  e.  A } )   &    |-  ( ph  ->  ( vol * `  A )  e.  RR )   =>    |-  ( ph  ->  ( vol * `  B )  =  ( ( vol
 * `  A )  /  C ) )
 
Theoremovolicc1 18838* The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  G  =  ( n  e.  NN  |->  if ( n  =  1 ,  <. A ,  B >. , 
 <. 0 ,  0 >.
 ) )   =>    |-  ( ph  ->  ( vol * `  ( A [,] B ) ) 
 <_  ( B  -  A ) )
 
Theoremovolicc2lem1 18839* Lemma for ovolicc2 18844. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   =>    |-  ( ( ph  /\  X  e.  U ) 
 ->  ( P  e.  X  <->  ( P  e.  RR  /\  ( 1st `  ( F `  ( G `  X ) ) )  <  P  /\  P  <  ( 2nd `  ( F `  ( G `  X ) ) ) ) ) )
 
Theoremovolicc2lem2 18840* Lemma for ovolicc2 18844. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   &    |-  ( ph  ->  H : T --> T )   &    |-  ( ( ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) ) 
 <_  B ,  ( 2nd `  ( F `  ( G `  t ) ) ) ,  B )  e.  ( H `  t ) )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  C  e.  T )   &    |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )   &    |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }   =>    |-  (
 ( ph  /\  ( N  e.  NN  /\  -.  N  e.  W )
 )  ->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  <_  B )
 
Theoremovolicc2lem3 18841* Lemma for ovolicc2 18844. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   &    |-  ( ph  ->  H : T --> T )   &    |-  ( ( ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) ) 
 <_  B ,  ( 2nd `  ( F `  ( G `  t ) ) ) ,  B )  e.  ( H `  t ) )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  C  e.  T )   &    |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )   &    |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }   =>    |-  (
 ( ph  /\  ( N  e.  { n  e. 
 NN  |  A. m  e.  W  n  <_  m }  /\  P  e.  { n  e.  NN  |  A. m  e.  W  n  <_  m } ) ) 
 ->  ( N  =  P  <->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  =  ( 2nd `  ( F `  ( G `  ( K `  P ) ) ) ) ) )
 
Theoremovolicc2lem4 18842* Lemma for ovolicc2 18844. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   &    |-  ( ph  ->  H : T --> T )   &    |-  ( ( ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) ) 
 <_  B ,  ( 2nd `  ( F `  ( G `  t ) ) ) ,  B )  e.  ( H `  t ) )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  C  e.  T )   &    |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )   &    |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }   &    |-  M  =  sup ( W ,  RR ,  `'  <  )   =>    |-  ( ph  ->  ( B  -  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
 
Theoremovolicc2lem5 18843* Lemma for ovolicc2 18844. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  U  e.  ( ~P ran  ( (,) 
 o.  F )  i^i 
 Fin ) )   &    |-  ( ph  ->  ( A [,] B )  C_  U. U )   &    |-  ( ph  ->  G : U
 --> NN )   &    |-  ( ( ph  /\  t  e.  U ) 
 ->  ( ( (,)  o.  F ) `  ( G `  t ) )  =  t )   &    |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B ) )  =/=  (/) }   =>    |-  ( ph  ->  ( B  -  A )  <_  sup ( ran  S ,  RR*
 ,  <  ) )
 
Theoremovolicc2 18844* The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR )
 )  ^m  NN )
 ( ( A [,] B )  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran 
 seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) ,  RR* ,  <  ) ) }   =>    |-  ( ph  ->  ( B  -  A )  <_  ( vol * `  ( A [,] B ) ) )
 
Theoremovolicc 18845 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol * `  ( A [,] B ) )  =  ( B  -  A ) )
 
Theoremovolicopnf 18846 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo )
 )  =  +oo )
 
Theoremovolre 18847 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( vol * `  RR )  =  +oo
 
Theoremismbl 18848* The predicate " A is Lebesgue-measurable". A set is measurable if it splits every other set  x in a "nice" way, that is, if the measure of the pieces  x  i^i  A and  x  \  A sum up to the measure of 
x (assuming that the measure of 
x is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR (
 ( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol * `  ( x  \  A ) ) ) ) ) )
 
Theoremismbl2 18849* From ovolun 18821, it suffices to show that the measure of  x is at least the sum of the measures of  x  i^i  A and  x  \  A. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR (
 ( vol * `  x )  e.  RR  ->  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
 * `  ( x  \  A ) ) ) 
 <_  ( vol * `  x ) ) ) )
 
Theoremvolres 18850 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol  =  ( vol *  |`  dom  vol )
 
Theoremvolf 18851 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol : dom  vol --> ( 0 [,]  +oo )
 
Theoremmblvol 18852 The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  ( vol `  A )  =  ( vol * `
  A ) )
 
Theoremmblss 18853 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  A  C_  RR )
 
Theoremmblsplit 18854 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  C_  RR  /\  ( vol * `  B )  e.  RR )  ->  ( vol * `  B )  =  ( ( vol * `  ( B  i^i  A ) )  +  ( vol
 * `  ( B  \  A ) ) ) )
 
Theoremcmmbl 18855 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  ( RR  \  A )  e.  dom  vol )
 
Theoremnulmbl 18856 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  ->  A  e.  dom 
 vol )
 
Theoremnulmbl2 18857* A set of outer measure zero is measurable. The term "outer measure zero" here is slightly different from "nullset/negligible set"; a nullset has  vol * ( A )  =  0 while "outer measure zero" means that for any  x there is a  y containing  A with volume less than  x. Assuming AC, these notions are equivalent (because the intersection of all such  y is a nullset) but in ZF this is a strictly weaker notion. Proposition 563Gb of [Fremlin5] p. 193. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A. x  e.  RR+  E. y  e.  dom  vol ( A  C_  y  /\  ( vol * `  y )  <_  x ) 
 ->  A  e.  dom  vol )
 
Theoremunmbl 18858 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  u.  B )  e.  dom  vol )
 
Theoremshftmbl 18859* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  RR )  ->  { x  e. 
 RR  |  ( x  -  B )  e.  A }  e.  dom  vol )
 
Theorem0mbl 18860 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  (/)  e.  dom  vol
 
Theoremrembl 18861 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |- 
 RR  e.  dom  vol
 
Theoreminmbl 18862 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  i^i  B )  e.  dom  vol )
 
Theoremdifmbl 18863 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  dom  vol )  ->  ( A  \  B )  e.  dom  vol )
 
Theoremfiniunmbl 18864* A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ( A  e.  Fin  /\  A. k  e.  A  B  e.  dom  vol )  -> 
 U_ k  e.  A  B  e.  dom  vol )
 
Theoremvolun 18865 The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( ( A  e.  dom  vol  /\  B  e.  dom  vol  /\  ( A  i^i  B )  =  (/) )  /\  ( ( vol `  A )  e.  RR  /\  ( vol `  B )  e.  RR ) )  ->  ( vol `  ( A  u.  B ) )  =  (
 ( vol `  A )  +  ( vol `  B ) ) )
 
Theoremvolinun 18866 Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ( ( A  e.  dom  vol  /\  B  e.  dom  vol )  /\  (
 ( vol `  A )  e.  RR  /\  ( vol `  B )  e.  RR ) )  ->  ( ( vol `  A )  +  ( vol `  B ) )  =  (
 ( vol `  ( A  i^i  B ) )  +  ( vol `  ( A  u.  B ) ) ) )
 
Theoremvolfiniun 18867* The volume of a disjoint finite union of measurable sets is the sum of the measures. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( ( A  e.  Fin  /\  A. k  e.  A  ( B  e.  dom  vol  /\  ( vol `  B )  e.  RR )  /\ Disj  k  e.  A B )  ->  ( vol `  U_ k  e.  A  B )  = 
 sum_ k  e.  A  ( vol `  B )
 )
 
Theoremiundisj 18868* Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( n  =  k 
 ->  A  =  B )   =>    |-  U_ n  e.  NN  A  =  U_ n  e.  NN  ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisj2 18869* A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( n  =  k 
 ->  A  =  B )   =>    |- Disj  n  e.  NN ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremvoliunlem1 18870* Lemma for voliun 18874. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ph  ->  F : NN --> dom  vol )   &    |-  ( ph  -> Disj  i  e.  NN ( F `  i ) )   &    |-  H  =  ( n  e.  NN  |->  ( vol * `  ( E  i^i  ( F `  n ) ) ) )   &    |-  ( ph  ->  E 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  E )  e.  RR )   =>    |-  (
 ( ph  /\  k  e. 
 NN )  ->  (
 (  seq  1 (  +  ,  H ) `  k )  +  ( vol * `  ( E 
 \  U. ran  F ) ) )  <_  ( vol * `  E ) )
 
Theoremvoliunlem2 18871* Lemma for voliun 18874. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ph  ->  F : NN --> dom  vol )   &    |-  ( ph  -> Disj  i  e.  NN ( F `  i ) )   &    |-  H  =  ( n  e.  NN  |->  ( vol * `  ( x  i^i  ( F `  n ) ) ) )   =>    |-  ( ph  ->  U. ran  F  e.  dom  vol )
 
Theoremvoliunlem3 18872* Lemma for voliun 18874. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  ( ph  ->  F : NN --> dom  vol )   &    |-  ( ph  -> Disj  i  e.  NN ( F `  i ) )   &    |-  H  =  ( n  e.  NN  |->  ( vol * `  ( x  i^i  ( F `  n ) ) ) )   &    |-  S  =  seq  1 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol `  ( F `  n ) ) )   &    |-  ( ph  ->  A. i  e.  NN  ( vol `  ( F `  i ) )  e. 
 RR )   =>    |-  ( ph  ->  ( vol `  U. ran  F )  =  sup ( ran 
 S ,  RR* ,  <  ) )
 
Theoremiunmbl 18873 The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( A. n  e. 
 NN  A  e.  dom  vol 
 ->  U_ n  e.  NN  A  e.  dom  vol )
 
Theoremvoliun 18874 The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  S  =  seq  1
 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol `  A ) )   =>    |-  ( ( A. n  e.  NN  ( A  e.  dom  vol  /\  ( vol `  A )  e.  RR )  /\ Disj  n  e. 
 NN A )  ->  ( vol `  U_ n  e. 
 NN  A )  = 
 sup ( ran  S ,  RR* ,  <  )
 )
 
Theoremvolsuplem 18875* Lemma for volsup 18876. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  ( ( A. n  e.  NN  ( F `  n )  C_  ( F `
  ( n  +  1 ) )  /\  ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) ) ) 
 ->  ( F `  A )  C_  ( F `  B ) )
 
Theoremvolsup 18876* The volume of the limit of an increasing sequence of measurable sets is the limit of the volumes. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( ( F : NN
 --> dom  vol  /\  A. n  e.  NN  ( F `  n )  C_  ( F `
  ( n  +  1 ) ) ) 
 ->  ( vol `  U. ran  F )  =  sup (
 ( vol " ran  F ) ,  RR* ,  <  ) )
 
Theoremiunmbl2 18877* The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  ~<_  NN  /\  A. n  e.  A  B  e.  dom  vol )  ->  U_ n  e.  A  B  e.  dom  vol )
 
Theoremioombl1lem1 18878* Lemma for ioombl1 18882. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  B  =  ( A (,)  +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol * `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol * `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ph  ->  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  H : NN
 --> (  <_  i^i  ( RR  X.  RR ) ) ) )
 
Theoremioombl1lem2 18879* Lemma for ioombl1 18882. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  B  =  ( A (,)  +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol * `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol * `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  e.  RR )
 
Theoremioombl1lem3 18880* Lemma for ioombl1 18882. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  B  =  ( A (,)  +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol * `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol * `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ( ph  /\  n  e.  NN )  ->  ( ( ( ( abs  o.  -  )  o.  G ) `  n )  +  ( (
 ( abs  o.  -  )  o.  H ) `  n ) )  =  (
 ( ( abs  o.  -  )  o.  F ) `
  n ) )
 
Theoremioombl1lem4 18881* Lemma for ioombl1 18882. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  B  =  ( A (,)  +oo )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  E  C_  RR )   &    |-  ( ph  ->  ( vol * `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  S  =  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  T  =  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )   &    |-  U  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  H ) )   &    |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  F ) )   &    |-  ( ph  ->  sup ( ran  S ,  RR*
 ,  <  )  <_  ( ( vol * `  E )  +  C ) )   &    |-  P  =  ( 1st `  ( F `  n ) )   &    |-  Q  =  ( 2nd `  ( F `  n ) )   &    |-  G  =  ( n  e.  NN  |->  <. if ( if ( P  <_  A ,  A ,  P ) 
 <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) ,  Q >. )   &    |-  H  =  ( n  e.  NN  |->  <. P ,  if ( if ( P  <_  A ,  A ,  P )  <_  Q ,  if ( P  <_  A ,  A ,  P ) ,  Q ) >. )   =>    |-  ( ph  ->  ( ( vol * `  ( E  i^i  B ) )  +  ( vol
 * `  ( E  \  B ) ) ) 
 <_  ( ( vol * `  E )  +  C ) )
 
Theoremioombl1 18882 An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
 |-  ( A  e.  RR*  ->  ( A (,)  +oo )  e.  dom  vol )
 
Theoremicombl1 18883 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  dom  vol )
 
Theoremicombl 18884 A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  ( A [,) B )  e.  dom  vol )
 
Theoremioombl 18885 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( A (,) B )  e.  dom  vol
 
Theoremiccmbl 18886 A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B )  e.  dom  vol )
 
Theoremiccvolcl 18887 A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( vol `  ( A [,] B ) )  e.  RR )
 
Theoremovolioo 18888 The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol * `  ( A (,) B ) )  =  ( B  -  A ) )
 
Theoremovolfs2 18889 Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  G  =  ( ( abs  o.  -  )  o.  F )   =>    |-  ( F : NN --> (  <_  i^i  ( RR  X. 
 RR ) )  ->  G  =  ( ( vol *  o.  (,) )  o.  F ) )
 
Theoremioorcl2 18890 An open interval with finite volume has real endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ( ( A (,) B )  =/=  (/)  /\  ( vol * `  ( A (,) B ) )  e.  RR )  ->  ( A  e.  RR  /\  B  e.  RR ) )
 
Theoremioorf 18891 Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  F : ran  (,) --> ( 
 <_  i^i  ( RR*  X.  RR* ) )
 
Theoremioorval 18892* Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( A  e.  ran  (,) 
 ->  ( F `  A )  =  if ( A  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
 >. ) )
 
Theoremioorinv2 18893* The function  F is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( ( A (,) B )  =/=  (/)  ->  ( F `  ( A (,) B ) )  =  <. A ,  B >. )
 
Theoremioorinv 18894* The function  F is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( A  e.  ran  (,) 
 ->  ( (,) `  ( F `  A ) )  =  A )
 
Theoremioorcl 18895* The function  F does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  )
 >. ) )   =>    |-  ( ( A  e.  ran 
 (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
 
Theoremuniiccdif 18896 A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   =>    |-  ( ph  ->  ( U. ran  ( (,)  o.  F )  C_  U. ran  ( [,]  o.  F ) 
 /\  ( vol * `  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) ) )  =  0 ) )
 
Theoremuniioovol 18897* An disjoint union of open intervals has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 18874.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ph  ->  ( vol * `  U. ran  ( (,)  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  ) )
 
Theoremuniiccvol 18898* An almost-disjoint union of closed intervals (disjoint interiors) has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 18874.) (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ph  ->  ( vol * `  U. ran  ( [,]  o.  F ) )  =  sup ( ran  S ,  RR* ,  <  ) )
 
Theoremuniioombllem1 18899* Lemma for uniioombl 18907. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol * `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol * `  E )  +  C ) )   =>    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
 
Theoremuniioombllem2a 18900* Lemma for uniioombl 18907. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR )
 ) )   &    |-  ( ph  -> Disj  x  e.  NN ( (,) `  ( F `  x ) ) )   &    |-  S  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   &    |-  A  =  U. ran  ( (,) 
 o.  F )   &    |-  ( ph  ->  ( vol * `  E )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )   &    |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )   &    |-  T  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  G ) )   &    |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol * `  E )  +  C ) )   =>    |-  ( ( ( ph  /\  J  e.  NN )  /\  z  e.  NN )  ->  ( ( (,) `  ( F `  z
 ) )  i^i  ( (,) `  ( G `  J ) ) )  e.  ran  (,) )
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