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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhlress 18801 The scalar field of a complex Hilbert space contains  RR. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CHil  ->  RR  C_  K )
 
Theoremhlpr 18802 The scalar field of a complex Hilbert space is either  RR or  CC. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CHil  ->  K  e.  { RR ,  CC } )
 
Theoremishl2 18803 A Hilbert space is a complete complex pre-Hilbert space over  RR or  CC. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CHil  <->  ( W  e. CMetSp  /\  W  e.  CPreHil  /\  K  e.  { RR ,  CC } ) )
 
11.4.6  Minimizing Vector Theorem
 
Theoremminveclem1 18804* Lemma for minvec 18816. The set of all distances from points of  Y to  A are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-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 ) ) )   =>    |-  ( ph  ->  ( R  C_  RR  /\  R  =/= 
 (/)  /\  A. w  e.  R  0  <_  w ) )
 
Theoremminveclem4c 18805* Lemma for minvec 18816. The infimum of the distances to  A is a real number. (Contributed by Mario Carneiro, 16-Jun-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 ,  `'  <  )   =>    |-  ( ph  ->  S  e.  RR )
 
Theoremminveclem2 18806* Lemma for minvec 18816. Any two points  K and 
L in  Y are close to each other if they are close to the infimum of distance to  A. (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  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  K  e.  Y )   &    |-  ( ph  ->  L  e.  Y )   &    |-  ( ph  ->  ( ( A D K ) ^
 2 )  <_  (
 ( S ^ 2
 )  +  B ) )   &    |-  ( ph  ->  ( ( A D L ) ^ 2 )  <_  ( ( S ^
 2 )  +  B ) )   =>    |-  ( ph  ->  (
 ( K D L ) ^ 2 )  <_  ( 4  x.  B ) )
 
Theoremminveclem3a 18807* Lemma for minvec 18816. 
D is a complete metric when restricted to  Y. (Contributed by Mario Carneiro, 7-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  ->  ( D  |`  ( Y  X.  Y ) )  e.  ( CMet `  Y )
 )
 
Theoremminveclem3b 18808* Lemma for minvec 18816. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed 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 ) )   &    |-  F  =  ran  ( r  e.  RR+  |->  { y  e.  Y  |  ( ( A D y ) ^ 2 )  <_  ( ( S ^
 2 )  +  r
 ) } )   =>    |-  ( ph  ->  F  e.  ( fBas `  Y ) )
 
Theoremminveclem3 18809* Lemma for minvec 18816. The filter formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-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 ) )   &    |-  F  =  ran  ( r  e.  RR+  |->  { y  e.  Y  |  ( ( A D y ) ^ 2 )  <_  ( ( S ^
 2 )  +  r
 ) } )   =>    |-  ( ph  ->  ( Y filGen F )  e.  (CauFil `  ( D  |`  ( Y  X.  Y ) ) ) )
 
Theoremminveclem4a 18810* Lemma for minvec 18816. 
F converges to a point 
P in  Y. (Contributed by Mario Carneiro, 7-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 ) )   &    |-  F  =  ran  ( r  e.  RR+  |->  { y  e.  Y  |  ( ( A D y ) ^ 2 )  <_  ( ( S ^
 2 )  +  r
 ) } )   &    |-  P  =  U. ( J  fLim  ( X filGen F ) )   =>    |-  ( ph  ->  P  e.  ( ( J  fLim  ( X filGen F ) )  i^i  Y ) )
 
Theoremminveclem4b 18811* Lemma for minvec 18816. The convergent point of the Cauchy sequence  F is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-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 ) )   &    |-  F  =  ran  ( r  e.  RR+  |->  { y  e.  Y  |  ( ( A D y ) ^ 2 )  <_  ( ( S ^
 2 )  +  r
 ) } )   &    |-  P  =  U. ( J  fLim  ( X filGen F ) )   =>    |-  ( ph  ->  P  e.  X )
 
Theoremminveclem4 18812* Lemma for minvec 18816. The convergent point of the Cauchy sequence  F attains the minimum distance, and so is closer to  A than any other point in  Y. (Contributed by Mario Carneiro, 7-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 ) )   &    |-  F  =  ran  ( r  e.  RR+  |->  { y  e.  Y  |  ( ( A D y ) ^ 2 )  <_  ( ( S ^
 2 )  +  r
 ) } )   &    |-  P  =  U. ( J  fLim  ( X filGen F ) )   &    |-  T  =  ( (
 ( ( ( A D P )  +  S )  /  2
 ) ^ 2 )  -  ( S ^
 2 ) )   =>    |-  ( ph  ->  E. x  e.  Y  A. y  e.  Y  ( N `  ( A  .-  x ) )  <_  ( N `  ( A 
 .-  y ) ) )
 
Theoremminveclem5 18813* Lemma for minvec 18816. Discharge the assumptions in minveclem4 18812. (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
 ) ) )
 
Theoremminveclem6 18814* Lemma for minvec 18816. Any minimal point is less than  S away from  A. (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  /\  x  e.  Y )  ->  (
 ( ( A D x ) ^ 2
 )  <_  ( ( S ^ 2 )  +  0 )  <->  A. y  e.  Y  ( N `  ( A 
 .-  x ) ) 
 <_  ( N `  ( A  .-  y ) ) ) )
 
Theoremminveclem7 18815* Lemma for minvec 18816. 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 18816* 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 18817* Lemma for pjth 18819. (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 18818 Lemma for pjth 18819. (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 18819 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 18820 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 18821 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 18822 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 18823 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 18824* Lemma for pmltpc 18826. (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 18825* Lemma for pmltpc 18826. (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 18826* 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 18827* Lemma for ivth 18830. 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 18828* Lemma for ivth 18830. 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 18829* Lemma for ivth 18830, 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 18830* 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 18831* 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 18832* 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 18833* 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 18834* 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 18835* 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 18836* Combine ivthicc 18834 with evthicc 18835 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 18837* 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 18838 Extend class notation with the outer Lebesgue measure.
 class  vol *
 
Syntaxcvol 18839 Extend class notation with the Lebesgue measure.
 class  vol
 
Definitiondf-ovol 18840* 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 18841* 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 18842 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 18843* 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 18844* 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 18845 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 18846 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 18847 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 18848 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 18849* 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 18850* 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 18851* 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 18852* 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 18853 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 18854 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 18855* 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 18856 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR  ->  0  <_  ( vol * `
  A ) )
 
Theoremovolf 18857 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |- 
 vol * : ~P RR --> ( 0 [,]  +oo )
 
Theoremovollecl 18858 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 18859* Lemma for ovolss 18860. (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 18860 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 18861 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 18862 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 18863* Lemma for ovollb2 18864. (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 18864 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 18854). (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 18865 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 18866 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  QQ )  =  0
 
Theoremovolctb2 18867 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  C_  RR  /\  A  ~<_  NN )  ->  ( vol * `  A )  =  0
 )
 
Theoremovol0 18868 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  (/) )  =  0
 
Theoremovolfi 18869 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 18870 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  { A } )  =  0 )
 
Theoremovolunlem1a 18871* Lemma for ovolun 18874. (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 18872* Lemma for ovolun 18874. (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 18873 Lemma for ovolun 18874. (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 18874 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 18880, 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 18875 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 18876* 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 18877* Lemma for ovoliun 18880. (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 18878* Lemma for ovoliun 18880. (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 18879* Lemma for ovoliun 18880. (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 18880* 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 18860, 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 18881* 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 18880.) (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 18882* 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 18883* 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 18884* Lemma for ovolshft 18886. (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 18885* Lemma for ovolshft 18886. (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 18886* 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 18887* 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 18888* Lemma for ovolsca 18890. (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 18889* Lemma for ovolshft 18886. (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 18890* 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 18891* 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 18892* Lemma for ovolicc2 18897. (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 18893* Lemma for ovolicc2 18897. (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 18894* Lemma for ovolicc2 18897. (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 18895* Lemma for ovolicc2 18897. (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 18896* Lemma for ovolicc2 18897. (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 18897* 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 18898 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 18899 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo )
 )  =  +oo )
 
Theoremovolre 18900 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( vol * `  RR )  =  +oo
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