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Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcncms 18701 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-fld  e. CMetSp
 
Theoremresscdrg 18702 The real numbers are a subset of any complete subfield in the complexes. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  F  =  (flds  K )   =>    |-  ( ( K  e.  (SubRing ` fld )  /\  F  e.  DivRing  /\  F  e. CMetSp )  ->  RR  C_  K )
 
Theoremcncdrg 18703 The only complete subfields of the complexes are  RR and 
CC. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  F  =  (flds  K )   =>    |-  ( ( K  e.  (SubRing ` fld )  /\  F  e.  DivRing  /\  F  e. CMetSp )  ->  K  e.  { RR ,  CC } )
 
Theoremsrabn 18704 The subring algebra over a complete normed ring is a Banach space iff the subring is a closed division ring. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  A  =  ( ( subringAlg  `  W ) `  S )   &    |-  J  =  ( TopOpen `  W )   =>    |-  ( ( W  e. NrmRing  /\  W  e. CMetSp  /\  S  e.  (SubRing `  W ) ) 
 ->  ( A  e. Ban  <->  ( S  e.  ( Clsd `  J )  /\  ( Ws  S )  e.  DivRing ) ) )
 
Theoremrlmbn 18705 The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( R  e. NrmRing  /\  R  e.  DivRing  /\  R  e. CMetSp )  ->  (ringLMod `  R )  e. Ban )
 
Theoremishl 18706 The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
 |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
 
Theoremhlbn 18707 Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
 |-  ( W  e.  CHil  ->  W  e. Ban )
 
Theoremhlcph 18708 Every complex Hilbert space is a complex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( W  e.  CHil  ->  W  e.  CPreHil )
 
Theoremhlphl 18709 Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
 |-  ( W  e.  CHil  ->  W  e.  PreHil )
 
Theoremhlcms 18710 Every complex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
 |-  ( W  e.  CHil  ->  W  e. CMetSp )
 
Theoremhlprlem 18711 Lemma for hlpr 18713. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CHil  ->  ( K  e.  (SubRing ` fld ) 
 /\  (flds  K )  e.  DivRing  /\  (flds  K )  e. CMetSp )
 )
 
Theoremhlress 18712 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 18713 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 18714 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 } ) )
 
11.3.17  Minimizing Vector Theorem
 
Theoremminveclem1 18715* Lemma for minvec 18727. 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 18716* Lemma for minvec 18727. 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 18717* Lemma for minvec 18727. 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 18718* Lemma for minvec 18727. 
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 18719* Lemma for minvec 18727. 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 18720* Lemma for minvec 18727. 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 18721* Lemma for minvec 18727. 
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 18722* Lemma for minvec 18727. 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 18723* Lemma for minvec 18727. 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 18724* Lemma for minvec 18727. Discharge the assumptions in minveclem4 18723. (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 18725* Lemma for minvec 18727. 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 18726* Lemma for minvec 18727. 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 18727* 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
 ) ) )
 
11.3.18  Projection theorem
 
Theorempjthlem1 18728* Lemma for pjth 18730. (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 18729 Lemma for pjth 18730. (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 18730 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 18731 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 18732 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 18733 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 18734 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 18735* Lemma for pmltpc 18737. (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 18736* Lemma for pmltpc 18737. (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 18737* 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 18738* Lemma for ivth 18741. 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 18739* Lemma for ivth 18741. 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 18740* Lemma for ivth 18741, 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 18741* 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 18742* 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 18743* 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 18744* 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 18745* 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 18746* 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 18747* Combine ivthicc 18745 with evthicc 18746 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 18748* 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 18749 Extend class notation with the outer Lebesgue measure.
 class  vol *
 
Syntaxcvol 18750 Extend class notation with the Lebesgue measure.
 class  vol
 
Definitiondf-ovol 18751* 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 18752* 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 18753 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 18754* 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 18755* 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 18756 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 18757 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 18758 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 18759 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 18760* 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 18761* 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 18762* 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 18763* 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 18764 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 18765 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 18766* 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 18767 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR  ->  0  <_  ( vol * `
  A ) )
 
Theoremovolf 18768 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |- 
 vol * : ~P RR --> ( 0 [,]  +oo )
 
Theoremovollecl 18769 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 18770* Lemma for ovolss 18771. (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 18771 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 18772 If a set is contained in a another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( ( A  C_  B  /\  B  C_  RR  /\  ( vol * `  B )  e.  RR )  ->  ( vol * `  A )  e.  RR )
 
Theoremovolssnul 18773 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 18774* Lemma for ovollb2 18775. (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 18775 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 18765). (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 18776 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 18777 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  QQ )  =  0
 
Theoremovolctb2 18778 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  C_  RR  /\  A  ~<_  NN )  ->  ( vol * `  A )  =  0
 )
 
Theoremovol0 18779 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  (/) )  =  0
 
Theoremovolfi 18780 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 18781 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  { A } )  =  0 )
 
Theoremovolunlem1a 18782* Lemma for ovolun 18785. (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 18783* Lemma for ovolun 18785. (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 18784 Lemma for ovolun 18785. (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 18785 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 18791, 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 18786 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 18787* 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 18788* Lemma for ovoliun 18791. (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 18789* Lemma for ovoliun 18791. (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 18790* Lemma for ovoliun 18791. (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 18791* 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 18771, 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 18792* 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 18791.) (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 18793* 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 18794* 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 18795* Lemma for ovolshft 18797. (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 18796* Lemma for ovolshft 18797. (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 18797* 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 18798* 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 18799* Lemma for ovolsca 18801. (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 18800* Lemma for ovolshft 18797. (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 ) )
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