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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcmsms 19301 A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( G  e. CMetSp  ->  G  e.  MetSp )
 
Theoremcmspropd 19302 Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  (
 ( dist `  K )  |`  ( B  X.  B ) )  =  (
 ( dist `  L )  |`  ( B  X.  B ) ) )   &    |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L ) )   =>    |-  ( ph  ->  ( K  e. CMetSp  <->  L  e. CMetSp ) )
 
Theoremcmsss 19303 The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  K  =  ( Ms  A )   &    |-  X  =  (
 Base `  M )   &    |-  J  =  ( TopOpen `  M )   =>    |-  (
 ( M  e. CMetSp  /\  A  C_  X )  ->  ( K  e. CMetSp  <->  A  e.  ( Clsd `  J ) ) )
 
Theoremlssbn 19304 A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  J  =  ( TopOpen `  W )   =>    |-  (
 ( W  e. Ban  /\  U  e.  S )  ->  ( X  e. Ban  <->  U  e.  ( Clsd `  J ) ) )
 
Theoremcmetcusp1OLD 19305 If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
 |-  X  =  ( Base `  F )   &    |-  D  =  ( ( dist `  F )  |`  ( X  X.  X ) )   &    |-  U  =  (UnifSt `  F )   =>    |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  ->  F  e. CUnifSp )
 
Theoremcmetcusp1 19306 If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
 |-  X  =  ( Base `  F )   &    |-  D  =  ( ( dist `  F )  |`  ( X  X.  X ) )   &    |-  U  =  (UnifSt `  F )   =>    |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) ) 
 ->  F  e. CUnifSp )
 
TheoremcmetcuspOLD 19307 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X ) )  ->  (toUnifSp `  (metUnifOLD
 `  D ) )  e. CUnifSp )
 
Theoremcmetcusp 19308 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X ) )  ->  (toUnifSp `  (metUnif `  D ) )  e. CUnifSp )
 
Theoremcncms 19309 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-fld  e. CMetSp
 
Theoremcnflduss 19310 The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  U  =  (UnifSt ` fld )   =>    |-  U  =  (metUnif `  ( abs  o.  -  ) )
 
Theoremcnfldcusp 19311 The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |-fld  e. CUnifSp
 
Theoremresscdrg 19312 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 19313 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 19314 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 19315 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 19316 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 19317 Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
 |-  ( W  e.  CHil  ->  W  e. Ban )
 
Theoremhlcph 19318 Every complex Hilbert space is a complex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( W  e.  CHil  ->  W  e.  CPreHil )
 
Theoremhlphl 19319 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 19320 Every complex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
 |-  ( W  e.  CHil  ->  W  e. CMetSp )
 
Theoremhlprlem 19321 Lemma for hlpr 19323. (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 19322 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 19323 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 19324 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 } ) )
 
11.5.6  Minimizing Vector Theorem
 
Theoremminveclem1 19325* Lemma for minvec 19337. 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 19326* Lemma for minvec 19337. 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 19327* Lemma for minvec 19337. 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 19328* Lemma for minvec 19337. 
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 19329* Lemma for minvec 19337. 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 19330* Lemma for minvec 19337. 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 19331* Lemma for minvec 19337. 
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 19332* Lemma for minvec 19337. 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 19333* Lemma for minvec 19337. 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 19334* Lemma for minvec 19337. Discharge the assumptions in minveclem4 19333. (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 19335* Lemma for minvec 19337. 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 19336* Lemma for minvec 19337. 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 19337* 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
 ) ) )
 
11.5.7  Projection Theorem
 
Theorempjthlem1 19338* Lemma for pjth 19340. (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 19339 Lemma for pjth 19340. (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 19340 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 19341 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 19342 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 19343 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 19344 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 19345* Lemma for pmltpc 19347. (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 19346* Lemma for pmltpc 19347. (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 19347* 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 19348* Lemma for ivth 19351. 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 19349* Lemma for ivth 19351. 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 19350* Lemma for ivth 19351, 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 19351* 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 19352* 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 19353* 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 19354* 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 19355* 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 19356* 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 19357* Combine ivthicc 19355 with evthicc 19356 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 19358* 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 19359 Extend class notation with the outer Lebesgue measure.
 class  vol *
 
Syntaxcvol 19360 Extend class notation with the Lebesgue measure.
 class  vol
 
Definitiondf-ovol 19361* 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 19362* 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 19363 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 19364* 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 19365* 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 19366 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 19367 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 19368 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 19369 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 19370* 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 19371* 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 19372* 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 19373* 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 19374 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 19375 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 19376* 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 19377 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR  ->  0  <_  ( vol * `
  A ) )
 
Theoremovolf 19378 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |- 
 vol * : ~P RR --> ( 0 [,]  +oo )
 
Theoremovollecl 19379 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 19380* Lemma for ovolss 19381. (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 19381 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 19382 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 19383 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 19384* Lemma for ovollb2 19385. (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 19385 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 19375). (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 19386 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 19387 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  QQ )  =  0
 
Theoremovolctb2 19388 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( ( A  C_  RR  /\  A  ~<_  NN )  ->  ( vol * `  A )  =  0
 )
 
Theoremovol0 19389 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
 |-  ( vol * `  (/) )  =  0
 
Theoremovolfi 19390 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 19391 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( A  e.  RR  ->  ( vol * `  { A } )  =  0 )
 
Theoremovolunlem1a 19392* Lemma for ovolun 19395. (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 19393* Lemma for ovolun 19395. (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 19394 Lemma for ovolun 19395. (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 19395 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 19401, 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 19396 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 19397* 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 19398* Lemma for ovoliun 19401. (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 19399* Lemma for ovoliun 19401. (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 19400* Lemma for ovoliun 19401. (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 ) )
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