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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremminveclem7 18801* Lemma for minvec 18802. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
 |-  X  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( norm `  U )   &    |-  ( ph  ->  U  e.  CPreHil )   &    |-  ( ph  ->  Y  e.  ( LSubSp `  U )
 )   &    |-  ( ph  ->  ( Us  Y )  e. CMetSp )   &    |-  ( ph  ->  A  e.  X )   &    |-  J  =  ( TopOpen `  U )   &    |-  R  =  ran  ( y  e.  Y  |->  ( N `  ( A 
 .-  y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   &    |-  D  =  ( ( dist `  U )  |`  ( X  X.  X ) )   =>    |-  ( ph  ->  E! x  e.  Y  A. y  e.  Y  ( N `  ( A  .-  x ) )  <_  ( N `  ( A  .-  y
 ) ) )
 
Theoremminvec 18802* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace  W that minimizes the distance to an arbitrary vector  A in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
 |-  X  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( norm `  U )   &    |-  ( ph  ->  U  e.  CPreHil )   &    |-  ( ph  ->  Y  e.  ( LSubSp `  U )
 )   &    |-  ( ph  ->  ( Us  Y )  e. CMetSp )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  E! x  e.  Y  A. y  e.  Y  ( N `  ( A  .-  x ) )  <_  ( N `  ( A  .-  y
 ) ) )
 
11.4.7  Projection Theorem
 
Theorempjthlem1 18803* Lemma for pjth 18805. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 norm `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .,  =  ( .i `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  CHil )   &    |-  ( ph  ->  U  e.  L )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  A. x  e.  U  ( N `  A )  <_  ( N `
  ( A  .-  x ) ) )   &    |-  T  =  ( ( A  .,  B )  /  ( ( B  .,  B )  +  1
 ) )   =>    |-  ( ph  ->  ( A  .,  B )  =  0 )
 
Theorempjthlem2 18804 Lemma for pjth 18805. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 norm `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .,  =  ( .i `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  CHil )   &    |-  ( ph  ->  U  e.  L )   &    |-  ( ph  ->  A  e.  V )   &    |-  J  =  ( TopOpen `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  O  =  ( ocv `  W )   &    |-  ( ph  ->  U  e.  ( Clsd `  J )
 )   =>    |-  ( ph  ->  A  e.  ( U  .(+)  ( O `
  U ) ) )
 
Theorempjth 18805 Projection Theorem: Any Hilbert space vector  A can be decomposed uniquely into a member  x of a closed subspace  H and a member  y of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.)
 |-  V  =  ( Base `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  O  =  ( ocv `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  L  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  CHil  /\  U  e.  L  /\  U  e.  ( Clsd `  J ) )  ->  ( U  .(+)  ( O `
  U ) )  =  V )
 
Theorempjth2 18806 Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)
 |-  J  =  ( TopOpen `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  K  =  ( proj `  W )   =>    |-  (
 ( W  e.  CHil  /\  U  e.  L  /\  U  e.  ( Clsd `  J ) )  ->  U  e.  dom  K )
 
Theoremcldcss 18807 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  L  =  ( LSubSp `  W )   &    |-  C  =  ( CSubSp `  W )   =>    |-  ( W  e.  CHil  ->  ( U  e.  C  <->  ( U  e.  L  /\  U  e.  ( Clsd `  J ) ) ) )
 
Theoremcldcss2 18808 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  L  =  ( LSubSp `  W )   &    |-  C  =  ( CSubSp `  W )   =>    |-  ( W  e.  CHil  ->  C  =  ( L  i^i  ( Clsd `  J ) ) )
 
Theoremhlhil 18809 Corollary of the Projection Theorem: A complex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.)
 |-  ( W  e.  CHil  ->  W  e.  Hil )
 
PART 12  BASIC REAL AND COMPLEX ANALYSIS
 
12.1  Continuity
 
12.1.1  Intermediate value theorem
 
Theorempmltpclem1 18810* Lemma for pmltpc 18812. (Contributed by Mario Carneiro, 1-Jul-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <  C )   &    |-  ( ph  ->  ( ( ( F `  A )  <  ( F `  B )  /\  ( F `
  C )  < 
 ( F `  B ) )  \/  (
 ( F `  B )  <  ( F `  A )  /\  ( F `
  B )  < 
 ( F `  C ) ) ) )   =>    |-  ( ph  ->  E. a  e.  S  E. b  e.  S  E. c  e.  S  ( a  < 
 b  /\  b  <  c 
 /\  ( ( ( F `  a )  <  ( F `  b )  /\  ( F `
  c )  < 
 ( F `  b
 ) )  \/  (
 ( F `  b
 )  <  ( F `  a )  /\  ( F `  b )  < 
 ( F `  c
 ) ) ) ) )
 
Theorempmltpclem2 18811* Lemma for pmltpc 18812. (Contributed by Mario Carneiro, 1-Jul-2014.)
 |-  ( ph  ->  F  e.  ( RR  ^pm  RR ) )   &    |-  ( ph  ->  A 
 C_  dom  F )   &    |-  ( ph  ->  U  e.  A )   &    |-  ( ph  ->  V  e.  A )   &    |-  ( ph  ->  W  e.  A )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  U  <_  V )   &    |-  ( ph  ->  W 
 <_  X )   &    |-  ( ph  ->  -.  ( F `  U )  <_  ( F `  V ) )   &    |-  ( ph  ->  -.  ( F `  X )  <_  ( F `  W ) )   =>    |-  ( ph  ->  E. a  e.  A  E. b  e.  A  E. c  e.  A  ( a  < 
 b  /\  b  <  c 
 /\  ( ( ( F `  a )  <  ( F `  b )  /\  ( F `
  c )  < 
 ( F `  b
 ) )  \/  (
 ( F `  b
 )  <  ( F `  a )  /\  ( F `  b )  < 
 ( F `  c
 ) ) ) ) )
 
Theorempmltpc 18812* Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)
 |-  ( ( F  e.  ( RR  ^pm  RR )  /\  A  C_  dom  F ) 
 ->  ( A. x  e.  A  A. y  e.  A  ( x  <_  y  ->  ( F `  x )  <_  ( F `
  y ) )  \/  A. x  e.  A  A. y  e.  A  ( x  <_  y  ->  ( F `  y )  <_  ( F `
  x ) )  \/  E. a  e.  A  E. b  e.  A  E. c  e.  A  ( a  < 
 b  /\  b  <  c 
 /\  ( ( ( F `  a )  <  ( F `  b )  /\  ( F `
  c )  < 
 ( F `  b
 ) )  \/  (
 ( F `  b
 )  <  ( F `  a )  /\  ( F `  b )  < 
 ( F `  c
 ) ) ) ) ) )
 
Theoremivthlem1 18813* Lemma for ivth 18816. The set  S of all 
x values with  ( F `  x ) less than  U is lower bounded by  A and upper bounded by  B. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }   =>    |-  ( ph  ->  ( A  e.  S  /\  A. z  e.  S  z 
 <_  B ) )
 
Theoremivthlem2 18814* Lemma for ivth 18816. Show that the supremum of  S cannot be less than  U. If it was, continuity of  F implies that there are points just above the supremum that are also less than  U, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }   &    |-  C  =  sup ( S ,  RR ,  <  )   =>    |-  ( ph  ->  -.  ( F `  C )  <  U )
 
Theoremivthlem3 18815* Lemma for ivth 18816, the intermediate value theorem. Show that  ( F `  C ) cannot be greater than  U, and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }   &    |-  C  =  sup ( S ,  RR ,  <  )   =>    |-  ( ph  ->  ( C  e.  ( A (,) B )  /\  ( F `  C )  =  U ) )
 
Theoremivth 18816* The intermediate value theorem, increasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
Theoremivth2 18817* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `  A ) ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
Theoremivthle 18818* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <_  U  /\  U  <_  ( F `  B ) ) )   =>    |-  ( ph  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
 
Theoremivthle2 18819* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  B )  <_  U  /\  U  <_  ( F `  A ) ) )   =>    |-  ( ph  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
 
Theoremivthicc 18820* The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  M  e.  ( A [,] B ) )   &    |-  ( ph  ->  N  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   =>    |-  ( ph  ->  (
 ( F `  M ) [,] ( F `  N ) )  C_  ran 
 F )
 
Theoremevthicc 18821* Specialization of the Extreme Value Theorem to a closed interval of  RR. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   =>    |-  ( ph  ->  ( E. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( F `
  y )  <_  ( F `  x ) 
 /\  E. z  e.  ( A [,] B ) A. w  e.  ( A [,] B ) ( F `
  z )  <_  ( F `  w ) ) )
 
Theoremevthicc2 18822* Combine ivthicc 18820 with evthicc 18821 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   =>    |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y
 ) )
 
Theoremcniccbdd 18823* A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  F  e.  ( ( A [,] B )
 -cn-> CC ) )  ->  E. x  e.  RR  A. y  e.  ( A [,] B ) ( abs `  ( F `  y ) )  <_  x )
 
12.2  Integrals
 
12.2.1  Lebesgue measure
 
Syntaxcovol 18824 Extend class notation with the outer Lebesgue measure.
 class  vol *
 
Syntaxcvol 18825 Extend class notation with the Lebesgue measure.
 class  vol
 
Definitiondf-ovol 18826* Define the outer Lebesgue measure for subsets of the reals. Here  f is a function from the natural numbers to pairs  <. a ,  b >. with  a  <_  b, and the outer volume of the set  x is the infimum over all such functions such that the union of the open intervals  ( a ,  b ) covers  x of the sum of  b  -  a. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |- 
 vol *  =  ( x  e.  ~P RR  |->  sup ( { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR )
 )  ^m  NN )
 ( x  C_  U. ran  ( (,)  o.  f ) 
 /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
 RR* ,  <  ) ) } ,  RR* ,  `'  <  ) )
 
Definitiondf-vol 18827* Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as  A  e.  dom  vol. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |- 
 vol  =  ( vol *  |`  { x  |  A. y  e.  ( `' vol * " RR )
 ( vol * `  y
 )  =  ( ( vol * `  (
 y  i^i  x )
 )  +  ( vol
 * `  ( y  \  x ) ) ) } )
 
Theoremovolfcl 18828 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( ( F : NN
 --> (  <_  i^i  ( RR  X.  RR ) ) 
 /\  N  e.  NN )  ->  ( ( 1st `  ( F `  N ) )  e.  RR  /\  ( 2nd `  ( F `  N ) )  e.  RR  /\  ( 1st `  ( F `  N ) )  <_  ( 2nd `  ( F `  N ) ) ) )
 
Theoremovolfioo 18829* Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) ) 
 ->  ( A  C_  U. ran  ( (,)  o.  F )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n ) )  <  z  /\  z  <  ( 2nd `  ( F `  n ) ) ) ) )
 
Theoremovolficc 18830* Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X. 
 RR ) ) ) 
 ->  ( A  C_  U. ran  ( [,]  o.  F )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n ) ) 
 <_  z  /\  z  <_  ( 2nd `  ( F `  n ) ) ) ) )
 
Theoremovolficcss 18831 Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( F : NN --> (  <_  i^i  ( RR  X. 
 RR ) )  ->  U. ran  ( [,]  o.  F )  C_  RR )
 
Theoremovolfsval 18832 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  G  =  ( ( abs  o.  -  )  o.  F )   =>    |-  ( ( F : NN
 --> (  <_  i^i  ( RR  X.  RR ) ) 
 /\  N  e.  NN )  ->  ( G `  N )  =  (
 ( 2nd `  ( F `  N ) )  -  ( 1st `  ( F `  N ) ) ) )
 
Theoremovolfsf 18833 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  G  =  ( ( abs  o.  -  )  o.  F )   =>    |-  ( F : NN --> (  <_  i^i  ( RR  X. 
 RR ) )  ->  G : NN --> ( 0 [,)  +oo ) )
 
Theoremovolsf 18834 Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  G  =  ( ( abs  o.  -  )  o.  F )   &    |-  S  =  seq  1 (  +  ,  G )   =>    |-  ( F : NN --> (  <_  i^i  ( RR  X. 
 RR ) )  ->  S : NN --> ( 0 [,)  +oo ) )
 
Theoremovolval 18835* The value of the outer measure. (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* ,  <  ) ) }   =>    |-  ( A  C_  RR  ->  ( vol * `  A )  =  sup ( M ,  RR* ,  `'  <  ) )
 
Theoremelovolm 18836* Elementhood in the set  M of approximations to the outer measure. (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* ,  <  ) ) }   =>    |-  ( B  e.  M  <->  E. f  e.  ( ( 
 <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
 ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq  1
 (  +  ,  (
 ( abs  o.  -  )  o.  f ) ) , 
 RR* ,  <  ) ) )
 
Theoremelovolmr 18837* Sufficient condition for elementhood in the set  M. (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* ,  <  ) ) }   &    |-  S  =  seq  1 (  +  ,  (
 ( abs  o.  -  )  o.  F ) )   =>    |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_  U.
 ran  ( (,)  o.  F ) )  ->  sup ( ran  S ,  RR*
 ,  <  )  e.  M )
 
Theoremovolmge0 18838* The set  M is composed of nonnegative extended real numbers. (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* ,  <  ) ) }   =>    |-  ( B  e.  M  ->  0  <_  B )
 
Theoremovolcl 18839 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR  ->  ( vol * `  A )  e.  RR* )
 
Theoremovollb 18840 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 18841* 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 18842 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR  ->  0  <_  ( vol * `
  A ) )
 
Theoremovolf 18843 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |- 
 vol * : ~P RR --> ( 0 [,]  +oo )
 
Theoremovollecl 18844 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 18845* Lemma for ovolss 18846. (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 18846 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 18847 If a set is contained in 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 18848 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 18849* Lemma for ovollb2 18850. (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 18850 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 18840). (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 18851 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 18852 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  QQ )  =  0
 
Theoremovolctb2 18853 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  C_  RR  /\  A  ~<_  NN )  ->  ( vol * `  A )  =  0
 )
 
Theoremovol0 18854 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  (/) )  =  0
 
Theoremovolfi 18855 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 18856 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  { A } )  =  0 )
 
Theoremovolunlem1a 18857* Lemma for ovolun 18860. (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 18858* Lemma for ovolun 18860. (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 18859 Lemma for ovolun 18860. (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 18860 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 18866, 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 18861 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 18862* 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 18863* Lemma for ovoliun 18866. (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 18864* Lemma for ovoliun 18866. (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 18865* Lemma for ovoliun 18866. (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 18866* 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 18846, 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 18867* 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 18866.) (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 18868* 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 18869* 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 18870* Lemma for ovolshft 18872. (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 18871* Lemma for ovolshft 18872. (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 18872* 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 18873* 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 18874* Lemma for ovolsca 18876. (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 18875* Lemma for ovolshft 18872. (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 18876* 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 18877* 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 18878* Lemma for ovolicc2 18883. (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 18879* Lemma for ovolicc2 18883. (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 18880* Lemma for ovolicc2 18883. (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 18881* Lemma for ovolicc2 18883. (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 18882* Lemma for ovolicc2 18883. (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 18883* 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 18884 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 18885 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo )
 )  =  +oo )
 
Theoremovolre 18886 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( vol * `  RR )  =  +oo
 
Theoremismbl 18887* 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 18888* From ovolun 18860, 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 18889 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol  =  ( vol *  |`  dom  vol )
 
Theoremvolf 18890 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |- 
 vol : dom  vol --> ( 0 [,]  +oo )
 
Theoremmblvol 18891 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 18892 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( A  e.  dom  vol 
 ->  A  C_  RR )
 
Theoremmblsplit 18893 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 18894 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 18895 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  ->  A  e.  dom 
 vol )
 
Theoremnulmbl2 18896* 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 18897 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 18898* 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 18899 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  (/)  e.  dom  vol
 
Theoremrembl 18900 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |- 
 RR  e.  dom  vol
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