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Theorem List for Metamath Proof Explorer - 37701-37800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremetransclem35 37701*  P does not divide the P-1 -th derivative of  F applied to  0. This is case 2 of the proof in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n )  ^m  (
 0 ... M ) )  |  sum_ j  e.  (
 0 ... M ) ( c `  j )  =  n } )   &    |-  D  =  ( j  e.  (
 0 ... M )  |->  if ( j  =  0 ,  ( P  -  1 ) ,  0 ) )   =>    |-  ( ph  ->  (
 ( ( RR  Dn F ) `  ( P  -  1 ) ) `
  0 )  =  ( ( ! `  ( P  -  1
 ) )  x.  ( prod_ j  e.  ( 1
 ... M ) -u j ^ P ) ) )
 
Theoremetransclem36 37702* The  N-th derivative of  F applied to  J is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  X  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  H  =  ( j  e.  ( 0
 ... M )  |->  ( x  e.  X  |->  ( ( x  -  j
 ) ^ if (
 j  =  0 ,  ( P  -  1
 ) ,  P ) ) ) )   &    |-  ( ph  ->  J  e.  X )   &    |-  ( ph  ->  J  e.  ZZ )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n )  ^m  (
 0 ... M ) )  |  sum_ j  e.  (
 0 ... M ) ( c `  j )  =  n } )   =>    |-  ( ph  ->  ( ( ( S  Dn F ) `  N ) `
  J )  e. 
 ZZ )
 
Theoremetransclem37 37703*  ( P  -  1 ) factorial divides the  N-th derivative of  F applied to  J. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  X  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  H  =  ( j  e.  ( 0
 ... M )  |->  ( x  e.  X  |->  ( ( x  -  j
 ) ^ if (
 j  =  0 ,  ( P  -  1
 ) ,  P ) ) ) )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n ) 
 ^m  ( 0 ...
 M ) )  | 
 sum_ j  e.  (
 0 ... M ) ( c `  j )  =  n } )   &    |-  ( ph  ->  J  e.  (
 0 ... M ) )   &    |-  ( ph  ->  J  e.  X )   =>    |-  ( ph  ->  ( ! `  ( P  -  1 ) )  ||  ( ( ( S  Dn F ) `
  N ) `  J ) )
 
Theoremetransclem38 37704*  P divides the I -th derivative of  F applied to  J. if it is not the case that  I  =  P  - 
1 and  J  =  0. This is case 1 and the second part of case 2 proven in in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  ( ph  ->  I  e.  NN0 )   &    |-  ( ph  ->  J  e.  ( 0 ...
 M ) )   &    |-  ( ph  ->  -.  ( I  =  ( P  -  1
 )  /\  J  =  0 ) )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n ) 
 ^m  ( 0 ...
 M ) )  | 
 sum_ j  e.  (
 0 ... M ) ( c `  j )  =  n } )   =>    |-  ( ph  ->  P  ||  (
 ( ( ( RR 
 Dn F ) `
  I ) `  J )  /  ( ! `  ( P  -  1 ) ) ) )
 
Theoremetransclem39 37705*  G is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  G  =  ( x  e.  RR  |->  sum_ i  e.  ( 0 ...
 R ) ( ( ( RR  Dn F ) `  i
 ) `  x )
 )   =>    |-  ( ph  ->  G : RR --> CC )
 
Theoremetransclem40 37706* The  N-th derivative of  F is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  X  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ k  e.  (
 1 ... M ) ( ( x  -  k
 ) ^ P ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( S  Dn F ) `  N )  e.  ( X -cn->
 CC ) )
 
Theoremetransclem41 37707*  P does not divide the P-1 -th derivative of  F applied to  0. This is the first part of case 2: proven in in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( ! `  M )  <  P )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^
 ( P  -  1
 ) )  x.  prod_ j  e.  ( 1 ...
 M ) ( ( x  -  j ) ^ P ) ) )   =>    |-  ( ph  ->  -.  P  ||  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
  0 )  /  ( ! `  ( P  -  1 ) ) ) )
 
Theoremetransclem42 37708* The  N-th derivative of  F applied to  J is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  X  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  J  e.  X )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  (
 ( ( S  Dn F ) `  N ) `  J )  e. 
 ZZ )
 
Theoremetransclem43 37709*  G is a continuous function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  X  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  G  =  ( x  e.  X  |->  sum_ i  e.  ( 0 ...
 R ) ( ( ( S  Dn F ) `  i
 ) `  x )
 )   =>    |-  ( ph  ->  G  e.  ( X -cn-> CC )
 )
 
Theoremetransclem44 37710* The given finite sum is nonzero. This is the claim proved after equation (7) in [Juillerat] p. 12 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  A : NN0 --> ZZ )   &    |-  ( ph  ->  ( A `  0 )  =/=  0 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( abs `  ( A `  0 ) )  <  P )   &    |-  ( ph  ->  ( ! `  M )  <  P )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^
 ( P  -  1
 ) )  x.  prod_ j  e.  ( 1 ...
 M ) ( ( x  -  j ) ^ P ) ) )   &    |-  K  =  (
 sum_ k  e.  (
 ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR 
 Dn F ) `
  ( 2nd `  k
 ) ) `  ( 1st `  k ) ) )  /  ( ! `
  ( P  -  1 ) ) )   =>    |-  ( ph  ->  K  =/=  0 )
 
Theoremetransclem45 37711*  K is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  ( ph  ->  A : NN0 --> ZZ )   &    |-  K  =  ( sum_ k  e.  (
 ( 0 ... M )  X.  ( 0 ...
 R ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k )
 ) )  /  ( ! `  ( P  -  1 ) ) )   =>    |-  ( ph  ->  K  e.  ZZ )
 
Theoremetransclem46 37712* This is the proof for equation *(7) in [Juillerat] p. 12. The proven equality will lead to a contradiction, because the left-hand side goes to  0 for large  P, but the right-hand side is a non-zero integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  Q  e.  (
 (Poly `  ZZ )  \  { 0p }
 ) )   &    |-  ( ph  ->  ( Q `  _e )  =  0 )   &    |-  A  =  (coeff `  Q )   &    |-  M  =  (deg `  Q )   &    |-  ( ph  ->  RR  C_  RR )   &    |-  ( ph  ->  RR  e.  { RR ,  CC } )   &    |-  ( ph  ->  RR  e.  (
 ( TopOpen ` fld )t  RR ) )   &    |-  ( ph  ->  P  e.  NN )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  L  =  sum_ j  e.  ( 0 ...
 M ) ( ( ( A `  j
 )  x.  ( _e 
 ^c  j ) )  x.  S. (
 0 (,) j ) ( ( _e  ^c  -u x )  x.  ( F `  x ) )  _d x )   &    |-  R  =  ( ( M  x.  P )  +  ( P  -  1 ) )   &    |-  G  =  ( x  e.  RR  |->  sum_ i  e.  (
 0 ... R ) ( ( ( RR  Dn F ) `  i
 ) `  x )
 )   &    |-  O  =  ( x  e.  ( 0 [,] j )  |->  -u (
 ( _e  ^c  -u x )  x.  ( G `  x ) ) )   =>    |-  ( ph  ->  ( L  /  ( ! `  ( P  -  1
 ) ) )  =  ( -u sum_ k  e.  (
 ( 0 ... M )  X.  ( 0 ...
 R ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k )
 ) )  /  ( ! `  ( P  -  1 ) ) ) )
 
Theoremetransclem47 37713*  _e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  Q  e.  (
 (Poly `  ZZ )  \  { 0p }
 ) )   &    |-  ( ph  ->  ( Q `  _e )  =  0 )   &    |-  A  =  (coeff `  Q )   &    |-  ( ph  ->  ( A `  0 )  =/=  0
 )   &    |-  M  =  (deg `  Q )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( abs `  ( A `  0 ) )  <  P )   &    |-  ( ph  ->  ( ! `  M )  <  P )   &    |-  ( ph  ->  ( sum_ j  e.  ( 0 ... M ) ( ( abs `  ( ( A `  j )  x.  ( _e  ^c  j ) ) )  x.  ( M  x.  ( M ^
 ( M  +  1 ) ) ) )  x.  ( ( ( M ^ ( M  +  1 ) ) ^ ( P  -  1 ) )  /  ( ! `  ( P  -  1 ) ) ) )  <  1
 )   &    |-  F  =  ( x  e.  RR  |->  ( ( x ^ ( P  -  1 ) )  x.  prod_ j  e.  (
 1 ... M ) ( ( x  -  j
 ) ^ P ) ) )   &    |-  L  =  sum_ j  e.  ( 0 ...
 M ) ( ( ( A `  j
 )  x.  ( _e 
 ^c  j ) )  x.  S. (
 0 (,) j ) ( ( _e  ^c  -u x )  x.  ( F `  x ) )  _d x )   &    |-  K  =  ( L  /  ( ! `  ( P  -  1 ) ) )   =>    |-  ( ph  ->  E. k  e.  ZZ  ( k  =/=  0  /\  ( abs `  k )  <  1
 ) )
 
Theoremetransclem48 37714*  _e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. In this lemma, a large enough prime  p is chosen: it will be used by subsequent lemmas. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  Q  e.  (
 (Poly `  ZZ )  \  { 0p }
 ) )   &    |-  ( ph  ->  ( Q `  _e )  =  0 )   &    |-  A  =  (coeff `  Q )   &    |-  ( ph  ->  ( A `  0 )  =/=  0
 )   &    |-  M  =  (deg `  Q )   &    |-  C  =  sum_ j  e.  ( 0 ...
 M ) ( ( abs `  ( ( A `  j )  x.  ( _e  ^c 
 j ) ) )  x.  ( M  x.  ( M ^ ( M  +  1 ) ) ) )   &    |-  S  =  ( n  e.  NN0  |->  ( C  x.  ( ( ( M ^ ( M  +  1 ) ) ^ n )  /  ( ! `  n ) ) ) )   &    |-  I  =  sup ( { i  e.  NN0  |  A. n  e.  ( ZZ>= `  i )
 ( abs `  ( S `  n ) )  < 
 1 } ,  RR ,  `'  <  )   &    |-  T  =  sup ( { ( abs `  ( A `  0 ) ) ,  ( ! `  M ) ,  I } ,  RR* ,  <  )   =>    |-  ( ph  ->  E. k  e.  ZZ  ( k  =/=  0  /\  ( abs `  k )  <  1
 ) )
 
Theoremetransc 37715  _e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  _e  e.  ( CC  \  AA )
 
21.30.18  Basic measure theory
 
21.30.18.1  σ-Algebras

Proofs for most of the theorems in section 111 of [Fremlin1]

 
Syntaxcsalg 37716 Extend class notation with the class of all sigma algebra.
 class SAlg
 
Definitiondf-salg 37717* Define the class of sigma-algebras. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- SAlg  =  { x  |  ( (/)  e.  x  /\  A. y  e.  x  ( U. x  \  y
 )  e.  x  /\  A. y  e.  ~P  x ( y  ~<_  om  ->  U. y  e.  x ) ) }
 
Syntaxcsalon 37718 Extend class notation with the class of sigma-algebras on a set.
 class SalOn
 
Definitiondf-salon 37719* Define the set of sigma-algebra on a given set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- SalOn  =  ( x  e.  _V  |->  { s  e. SAlg  |  U. s  =  x } )
 
Syntaxcsalgen 37720 Extend class notation with the class of sigma-algebra generator.
 class SalGen
 
Definitiondf-salgen 37721* Define the set of sigma-algebra generated by a given set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- SalGen  =  ( x  e.  _V  |->  |^| { s  e. SAlg  |  U. s  =  U. x } )
 
Theoremissal 37722* Express the predicate " S is a sigma algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( S  e.  V  ->  ( S  e. SAlg  <->  ( (/)  e.  S  /\  A. y  e.  S  ( U. S  \  y
 )  e.  S  /\  A. y  e.  ~P  S ( y  ~<_  om  ->  U. y  e.  S ) ) ) )
 
Theorempwsal 37723 The power set of a given set is a sigma-algebra (the so called discrete sigma-algebra). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( X  e.  V  ->  ~P X  e. SAlg )
 
Theoremsalunicl 37724 SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  S  e. SAlg )   &    |-  ( ph  ->  T  e.  ~P S )   &    |-  ( ph  ->  T  ~<_ 
 om )   =>    |-  ( ph  ->  U. T  e.  S )
 
Theoremsaluncl 37725 The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( S  e. SAlg  /\  E  e.  S  /\  F  e.  S )  ->  ( E  u.  F )  e.  S )
 
Theoremprsal 37726 The pair of the empty set and the whole base is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( X  e.  V  ->  { (/) ,  X }  e. SAlg )
 
Theoremsaldifcl 37727 The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( S  e. SAlg  /\  E  e.  S )  ->  ( U. S  \  E )  e.  S )
 
Theorem0sal 37728 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( S  e. SAlg  ->  (/)  e.  S )
 
Theoremsalgenval 37729* The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( X  e.  V  ->  (SalGen `  X )  =  |^| { s  e. SAlg  |  U. s  =  U. X } )
 
Theoremsaliuncl 37730* SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  S  e. SAlg )   &    |-  ( ph  ->  K  ~<_  om )   &    |-  (
 ( ph  /\  k  e.  K )  ->  E  e.  S )   =>    |-  ( ph  ->  U_ k  e.  K  E  e.  S )
 
Theoremsalincl 37731 The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( S  e. SAlg  /\  E  e.  S  /\  F  e.  S )  ->  ( E  i^i  F )  e.  S )
 
Theoremsaluni 37732 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( S  e. SAlg  ->  U. S  e.  S )
 
Theoremsaliincl 37733* SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  S  e. SAlg )   &    |-  ( ph  ->  K  ~<_  om )   &    |-  ( ph  ->  K  =/=  (/) )   &    |-  (
 ( ph  /\  k  e.  K )  ->  E  e.  S )   =>    |-  ( ph  ->  |^|_ k  e.  K  E  e.  S )
 
Theoremsaldifcl2 37734 The difference of two elements of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( S  e. SAlg  /\  E  e.  S  /\  F  e.  S )  ->  ( E 
 \  F )  e.  S )
 
Theoremintsaluni 37735* The union of an arbitrary intersection of sigma-algebras on the same set  X, is  X. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  G  C_ SAlg )   &    |-  ( ph  ->  G  =/=  (/) )   &    |-  (
 ( ph  /\  s  e.  G )  ->  U. s  =  X )   =>    |-  ( ph  ->  U. |^| G  =  X )
 
Theoremintsal 37736* The arbitrary intersection of sigma-algebra (on the same set  X) is a sigma-algebra ( on the same set  X, see intsaluni 37735) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  G  C_ SAlg )   &    |-  ( ph  ->  G  =/=  (/) )   &    |-  (
 ( ph  /\  s  e.  G )  ->  U. s  =  X )   =>    |-  ( ph  ->  |^| G  e. SAlg )
 
Theoremsalgencl 37737 SalGen actually generates a sigma-algebra (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( X  e.  V  ->  (SalGen `  X )  e. SAlg )
 
21.30.18.2  Sum of nonnegative extended reals
 
Syntaxcsumge0 37738 Extend class notation to include the sum of nonnegative extended reals.
 class Σ^
 
Definitiondf-sumge0 37739* Define the arbitrary sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) $.
 |- Σ^ 
 =  ( x  e. 
 _V  |->  if ( +oo  e.  ran 
 x , +oo ,  sup ( ran  ( y  e.  ( ~P dom  x  i^i  Fin )  |->  sum_ w  e.  y  ( x `  w ) ) , 
 RR* ,  <  ) ) )
 
Theoremsge0rnre 37740* When Σ^ is applied to nonnegative real numbers the range used in its definition is a subset of the reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )   =>    |-  ( ph  ->  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  C_  RR )
 
Theoremfge0icoicc 37741 If  F maps to nonnegative reals, then  F maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )   =>    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
 
Theoremsge0val 37742* The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( X  e.  V  /\  F : X --> ( 0 [,] +oo ) )  ->  (Σ^ `  F )  =  if ( +oo  e.  ran  F , +oo ,  sup ( ran  ( y  e.  ( ~P X  i^i  Fin )  |-> 
 sum_ w  e.  y  ( F `  w ) ) ,  RR* ,  <  ) ) )
 
Theoremfge0npnf 37743 If  F maps to nonnegative reals, then +oo is not in its range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )   =>    |-  ( ph  ->  -. +oo  e.  ran  F )
 
Theoremsge0rnn0 37744* The range used in the definition of Σ^ is not empty. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
 sum_ y  e.  x  ( F `  y ) )  =/=  (/)
 
Theoremsge0vald 37745* The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  =  if ( +oo  e.  ran  F , +oo ,  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
 sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) ) )
 
Theoremfge0iccico 37746 A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  -. +oo  e.  ran  F )   =>    |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )
 
Theoremgsumge0cl 37747 Closure of group sum, for finitely supported nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  G  =  ( RR*ss  ( 0 [,] +oo ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  F finSupp 
 0 )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  e.  (
 0 [,] +oo ) )
 
Theoremsge0reval 37748* Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |-> 
 sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
 
Theoremsge0pnfval 37749 If a term in the sum of nonnegative extended reals is +oo, then the value of the sum is +oo. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  -> +oo 
 e.  ran  F )   =>    |-  ( ph  ->  (Σ^ `  F )  = +oo )
 
Theoremfge0iccre 37750 A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  -. +oo  e.  ran  F )   =>    |-  ( ph  ->  F : X --> RR )
 
Theoremsge0z 37751* Any nonnegative extended sum of zero is zero. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  0 ) )  =  0 )
 
Theoremsge00 37752 The sum of nonnegative extended reals is zero when applied to the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (Σ^ `  (/) )  =  0
 
Theoremfsumlesge0 37753* Every finite subsum of nonnegative reals is less than or equal to the extended sum over the whole (possibly infinite) domain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  ( ph  ->  Y  e.  Fin )   =>    |-  ( ph  ->  sum_ x  e.  Y  ( F `  x )  <_  (Σ^ `  F ) )
 
Theoremsge0revalmpt 37754* Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  =  sup ( ran  ( y  e.  ( ~P A  i^i  Fin )  |-> 
 sum_ x  e.  y  B ) ,  RR* ,  <  ) )
 
Theoremsge0sn 37755 A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : { A } --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  =  ( F `  A ) )
 
Theoremsge0tsms 37756 Σ^ applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  G  =  ( RR*ss  ( 0 [,] +oo ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  e.  ( G tsums  F ) )
 
Theoremsge0cl 37757 The arbitrary sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  e.  ( 0 [,] +oo ) )
 
Theoremsge0f1o 37758* Re-index a nonnegative extended sum using a bijection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  F/ n ph   &    |-  (
 k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ( ph  /\  n  e.  C ) 
 ->  ( F `  n )  =  G )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  =  (Σ^ `  ( n  e.  C  |->  D ) ) )
 
Theoremsge0snmpt 37759* A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  ( 0 [,] +oo ) )   &    |-  ( k  =  A  ->  B  =  C )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  { A }  |->  B ) )  =  C )
 
Theoremsge0ge0 37760 The sum of nonnegative extended reals is nonnegative (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  0  <_  (Σ^ `  F ) )
 
Theoremsge0xrcl 37761 The arbitrary sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  e.  RR* )
 
Theoremsge0repnf 37762 The of nonnegative extended reals is a real number if and only if it is not +oo. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (
 (Σ^ `  F )  e.  RR  <->  -.  (Σ^ `  F )  = +oo ) )
 
Theoremsge0fsum 37763* The arbitrary sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +oo (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  F : X --> ( 0 [,) +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  =  sum_ x  e.  X  ( F `  x ) )
 
Theoremsge0rern 37764 If the sum of nonnegative extended reals is not +oo then no terms is +oo. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  F )  e.  RR )   =>    |-  ( ph  ->  -. +oo  e.  ran  F )
 
Theoremsge0supre 37765* If the arbitrary sum of nonnegative extended reals is real, then it is the supremum (in the real numbers) of finite subsums. Similar to sge0sup 37767, but here we can use  sup with respect to  RR instead of  RR* (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  F )  e.  RR )   =>    |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR ,  <  ) )
 
Theoremsge0fsummpt 37766* The arbitrary sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +oo (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  =  sum_ k  e.  A  B )
 
Theoremsge0sup 37767* The arbitrary sum of nonnegative extended reals is the supremum of finite subsums (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  =  sup ( ran  ( x  e.  ( ~P X  i^i  Fin )  |->  (Σ^ `  ( F  |`  x ) ) ) ,  RR* ,  <  ) )
 
Theoremsge0less 37768 A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  ( F  |`  Y ) ) 
 <_  (Σ^ `  F ) )
 
Theoremsge0rnbnd 37769* The range used in the definition of Σ^ is bounded, when the whole sum is a real number. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  F )  e.  RR )   =>    |-  ( ph  ->  E. z  e.  RR  A. w  e.  ran  ( x  e.  ( ~P X  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) w 
 <_  z )
 
Theoremsge0pr 37770* Sum of a pair of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  E  e.  ( 0 [,] +oo ) )   &    |-  ( k  =  A  ->  C  =  D )   &    |-  ( k  =  B  ->  C  =  E )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  { A ,  B }  |->  C ) )  =  ( D +e E ) )
 
Theoremsge0gerp 37771* The arbitrary sum of nonnegative extended reals is larger or equal to a given extended real number if this number can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ( ph  /\  x  e.  RR+ )  ->  E. z  e.  ( ~P X  i^i  Fin ) A  <_  ( (Σ^ `  ( F  |`  z ) ) +e x ) )   =>    |-  ( ph  ->  A  <_  (Σ^ `  F ) )
 
Theoremsge0pnffigt 37772* If the sum of nonnegative extended reals is +oo, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  F )  = +oo )   &    |-  ( ph  ->  Y  e.  RR )   =>    |-  ( ph  ->  E. x  e.  ( ~P X  i^i  Fin ) Y  <  (Σ^ `  ( F  |`  x ) ) )
 
Theoremsge0ssre 37773 If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  F )  e.  RR )   =>    |-  ( ph  ->  (Σ^ `  ( F  |`  Y ) )  e.  RR )
 
Theoremsge0lefi 37774* A sum of nonnegative extended reals is smaller than a given extended real if and only if every finite subsum is smaller than it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  (
 (Σ^ `  F )  <_  A  <->  A. x  e.  ( ~P X  i^i  Fin )
 (Σ^ `  ( F  |`  x ) )  <_  A )
 )
 
Theoremsge0lessmpt 37775* A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  C 
 C_  A )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  C  |->  B ) )  <_  (Σ^ `  ( x  e.  A  |->  B ) ) )
 
Theoremsge0ltfirp 37776* If the sum of nonnegative extended reals is real, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  Y  e.  RR+ )   &    |-  ( ph  ->  (Σ^ `  F )  e.  RR )   =>    |-  ( ph  ->  E. x  e.  ( ~P X  i^i  Fin )
 (Σ^ `  F )  <  (
 (Σ^ `  ( F  |`  x ) )  +  Y ) )
 
Theoremsge0prle 37777* The sum of a pair of nonnegative extended reals is less than or equal their extended addition. When it is a distinct pair, than equality holds, see sge0pr 37770 (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  E  e.  ( 0 [,] +oo ) )   &    |-  ( k  =  A  ->  C  =  D )   &    |-  ( k  =  B  ->  C  =  E )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  { A ,  B }  |->  C ) )  <_  ( D +e E ) )
 
Theoremsge0gerpmpt 37778* The arbitrary sum of nonnegative extended reals is larger or equal to a given extended real number if this number can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ( ph  /\  y  e.  RR+ )  ->  E. z  e.  ( ~P A  i^i  Fin ) C  <_  ( (Σ^ `  ( x  e.  z  |->  B ) ) +e y ) )   =>    |-  ( ph  ->  C  <_  (Σ^ `  ( x  e.  A  |->  B ) ) )
 
Theoremsge0resrnlem 37779 The sum of nonnegative extended reals restricted to the range of a function is less or equal to the sum of the composition of the two functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : B --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  X  e.  ~P A )   &    |-  ( ph  ->  ( G  |`  X ) : X -1-1-onto-> ran  G )   =>    |-  ( ph  ->  (Σ^ `  ( F  |`  ran  G )
 )  <_  (Σ^ `  ( F  o.  G ) ) )
 
Theoremsge0resrn 37780 The sum of nonnegative extended reals restricted to the range of a function is less or equal to the sum of the composition of the two functions (well order hypothesis allows to avoid using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : B --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  R  We  A )   =>    |-  ( ph  ->  (Σ^ `  ( F  |`  ran  G )
 )  <_  (Σ^ `  ( F  o.  G ) ) )
 
Theoremsge0ssrempt 37781* If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  e.  RR )   &    |-  ( ph  ->  C  C_  A )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  C  |->  B ) )  e.  RR )
 
Theoremsge0resplit 37782 Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 37785. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  U  =  ( A  u.  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  F : U --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  (Σ^ `  F )  e.  RR )   =>    |-  ( ph  ->  (Σ^ `  F )  =  ( (Σ^ `  ( F  |`  A ) )  +  (Σ^ `  ( F  |`  B ) ) ) )
 
Theoremsge0le 37783* If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  G : X --> ( 0 [,] +oo ) )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( F `  x )  <_  ( G `  x ) )   =>    |-  ( ph  ->  (Σ^ `  F )  <_  (Σ^ `  G ) )
 
Theoremsge0ltfirpmpt 37784* If the extended sum of nonnegative reals is not +oo, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ph  ->  Y  e.  RR+ )   &    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  e.  RR )   =>    |-  ( ph  ->  E. y  e.  ( ~P A  i^i  Fin )
 (Σ^ `  ( x  e.  A  |->  B ) )  < 
 ( (Σ^ `  ( x  e.  y  |->  B ) )  +  Y ) )
 
Theoremsge0split 37785 Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  U  =  ( A  u.  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  F : U --> ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  F )  =  ( (Σ^ `  ( F  |`  A ) ) +e (Σ^ `  ( F  |`  B ) ) ) )
 
Theoremsge0lempt 37786* If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  ( 0 [,] +oo ) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  <_  C )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  B ) )  <_  (Σ^ `  ( x  e.  A  |->  C ) ) )
 
Theoremsge0splitmpt 37787* Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  ( 0 [,] +oo ) )   &    |-  (
 ( ph  /\  x  e.  B )  ->  C  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  ( x  e.  ( A  u.  B )  |->  C ) )  =  ( (Σ^ `  ( x  e.  A  |->  C ) ) +e (Σ^ `  ( x  e.  B  |->  C ) ) ) )
 
Theoremsge0ss 37788* Change the index set to a subset in a sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  C  e.  ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  k  e.  ( B 
 \  A ) ) 
 ->  C  =  0 )   =>    |-  ( ph  ->  (Σ^ `  ( k  e.  A  |->  C ) )  =  (Σ^ `  ( k  e.  B  |->  C ) ) )
 
Theoremsge0iunmptlemfi 37789* Sum of nonnegative extended reals over a disjoint indexed union (in this lemma, for a finite index set) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  (
 ( ph  /\  x  e.  A  /\  k  e.  B )  ->  C  e.  ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  (Σ^ `  ( k  e.  B  |->  C ) )  e. 
 RR )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  U_ x  e.  A  B  |->  C ) )  =  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( k  e.  B  |->  C ) ) ) ) )
 
Theoremsge0p1 37790* The addition of the next term in a finite sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M ... ( N  +  1 )
 ) )  ->  A  e.  ( 0 [,] +oo ) )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  B )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  ( M
 ... ( N  +  1 ) )  |->  A ) )  =  ( (Σ^ `  ( k  e.  ( M ... N )  |->  A ) ) +e B ) )
 
Theoremsge0iunmptlemre 37791* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  (
 ( ph  /\  x  e.  A  /\  k  e.  B )  ->  C  e.  ( 0 [,] +oo ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  (Σ^ `  ( k  e.  B  |->  C ) )  e. 
 RR )   &    |-  ( ph  ->  (Σ^ `  (
 k  e.  U_ x  e.  A  B  |->  C ) )  e.  RR* )   &    |-  ( ph  ->  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( k  e.  B  |->  C ) ) ) )  e.  RR* )   &    |-  ( ph  ->  ( k  e.  U_ x  e.  A  B  |->  C ) :
 U_ x  e.  A  B
 --> ( 0 [,] +oo ) )   &    |-  ( ph  ->  U_ x  e.  A  B  e.  _V )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  U_ x  e.  A  B  |->  C ) )  =  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( k  e.  B  |->  C ) ) ) ) )
 
Theoremsge0fodjrnlem 37792* Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned  0 in the sum (this is true, for example, both for measures and outer measures) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  F/ n ph   &    |-  (
 k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  F : C -onto-> A )   &    |-  ( ph  -> Disj  n  e.  C  ( F `  n ) )   &    |-  (
 ( ph  /\  n  e.  C )  ->  ( F `  n )  =  G )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ( 0 [,] +oo ) )   &    |-  (
 ( ph  /\  k  =  (/) )  ->  B  =  0 )   &    |-  Z  =  ( `' F " { (/) } )   =>    |-  ( ph  ->  (Σ^ `  ( k  e.  A  |->  B ) )  =  (Σ^ `  ( n  e.  C  |->  D ) ) )
 
Theoremsge0fodjrn 37793* Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned  0 in the sum (this is true, for example, both for measures and outer measures) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  F/ n ph   &    |-  (
 k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  F : C -onto-> A )   &    |-  ( ph  -> Disj  n  e.  C  ( F `  n ) )   &    |-  (
 ( ph  /\  n  e.  C )  ->  ( F `  n )  =  G )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ( 0 [,] +oo ) )   &    |-  (
 ( ph  /\  k  =  (/) )  ->  B  =  0 )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  A  |->  B ) )  =  (Σ^ `  ( n  e.  C  |->  D ) ) )
 
Theoremsge0iunmpt 37794* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  (
 ( ph  /\  x  e.  A  /\  k  e.  B )  ->  C  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  (Σ^ `  (
 k  e.  U_ x  e.  A  B  |->  C ) )  =  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( k  e.  B  |->  C ) ) ) ) )
 
Theoremsge0iun 37795* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  X  =  U_ x  e.  A  B   &    |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )   &    |-  ( ph  -> Disj  x  e.  A  B )   =>    |-  ( ph  ->  (Σ^ `  F )  =  (Σ^ `  ( x  e.  A  |->  (Σ^ `  ( F  |`  B ) ) ) ) )
 
21.30.18.3  Measures

Proofs for most of the theorems in section 112 of [Fremlin1]

 
Syntaxcmea 37796 Extend class notation with the class of measures.
 class Meas
 
Definitiondf-mea 37797* Define the class of measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- Meas  =  { x  |  ( (
 ( x : dom  x --> ( 0 [,] +oo )  /\  dom  x  e. SAlg ) 
 /\  ( x `  (/) )  =  0 ) 
 /\  A. y  e.  ~P  dom 
 x ( ( y  ~<_ 
 om  /\ Disj  w  e.  y  w )  ->  ( x `
  U. y )  =  (Σ^ `  ( x  |`  y ) ) ) ) }
 
Theoremismea 37798* Express the predicate " M is a measure." Definition 112A of [Fremlin1] p. 14 (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( M  e. Meas  <->  ( ( ( M : dom  M --> ( 0 [,] +oo )  /\  dom  M  e. SAlg ) 
 /\  ( M `  (/) )  =  0 ) 
 /\  A. x  e.  ~P  dom 
 M ( ( x  ~<_ 
 om  /\ Disj  y  e.  x  y )  ->  ( M `
  U. x )  =  (Σ^ `  ( M  |`  x ) ) ) ) )
 
Theoremdmmeasal 37799 The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   =>    |-  ( ph  ->  S  e. SAlg )
 
Theoremmeaf 37800 A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e. Meas )   &    |-  S  =  dom  M   =>    |-  ( ph  ->  M : S --> ( 0 [,] +oo ) )
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