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Theorem List for Metamath Proof Explorer - 38001-38100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremetransclem25 38001*  P factorial divides the  N-th derivative of  F applied to  J. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  C : ( 0 ...
 M ) --> ( 0
 ... N ) )   &    |-  ( ph  ->  sum_ j  e.  ( 0 ... M ) ( C `  j )  =  N )   &    |-  T  =  ( ( ( ! `  N )  /  prod_ j  e.  (
 0 ... M ) ( ! `  ( C `
  j ) ) )  x.  ( if ( ( P  -  1 )  <  ( C `
  0 ) ,  0 ,  ( ( ( ! `  ( P  -  1 ) ) 
 /  ( ! `  ( ( P  -  1 )  -  ( C `  0 ) ) ) )  x.  ( J ^ ( ( P  -  1 )  -  ( C `  0 ) ) ) ) )  x.  prod_ j  e.  (
 1 ... M ) if ( P  <  ( C `  j ) ,  0 ,  ( ( ( ! `  P )  /  ( ! `  ( P  -  ( C `  j ) ) ) )  x.  (
 ( J  -  j
 ) ^ ( P  -  ( C `  j ) ) ) ) ) ) )   &    |-  ( ph  ->  J  e.  ( 1 ... M ) )   =>    |-  ( ph  ->  ( ! `  P )  ||  T )
 
Theoremetransclem26 38002* Every term in the sum of the  N-th derivative of  F applied to  J is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  J  e.  ZZ )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n ) 
 ^m  ( 0 ...
 M ) )  | 
 sum_ j  e.  (
 0 ... M ) ( c `  j )  =  n } )   &    |-  ( ph  ->  D  e.  ( C `  N ) )   =>    |-  ( ph  ->  ( (
 ( ! `  N )  /  prod_ j  e.  (
 0 ... M ) ( ! `  ( D `
  j ) ) )  x.  ( if ( ( P  -  1 )  <  ( D `
  0 ) ,  0 ,  ( ( ( ! `  ( P  -  1 ) ) 
 /  ( ! `  ( ( P  -  1 )  -  ( D `  0 ) ) ) )  x.  ( J ^ ( ( P  -  1 )  -  ( D `  0 ) ) ) ) )  x.  prod_ j  e.  (
 1 ... M ) if ( P  <  ( D `  j ) ,  0 ,  ( ( ( ! `  P )  /  ( ! `  ( P  -  ( D `  j ) ) ) )  x.  (
 ( J  -  j
 ) ^ ( P  -  ( D `  j ) ) ) ) ) ) )  e.  ZZ )
 
Theoremetransclem27 38003* 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 )   &    |-  H  =  ( j  e.  ( 0 ...
 M )  |->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) ) )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ph  ->  C : dom  C --> ( NN0  ^m  (
 0 ... M ) ) )   &    |-  G  =  ( x  e.  X  |->  sum_ l  e.  dom  C prod_ j  e.  ( 0 ...
 M ) ( ( ( S  Dn
 ( H `  j
 ) ) `  (
 ( C `  l
 ) `  j )
 ) `  x )
 )   &    |-  ( ph  ->  J  e.  X )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  ( G `  J )  e.  ZZ )
 
Theoremetransclem28 38004*  ( P  -  1 ) factorial divides the  N-th derivative of  F applied to  J. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  P  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n )  ^m  (
 0 ... M ) )  |  sum_ j  e.  (
 0 ... M ) ( c `  j )  =  n } )   &    |-  ( ph  ->  D  e.  ( C `  N ) )   &    |-  ( ph  ->  J  e.  ( 0 ... M ) )   &    |-  T  =  ( ( ( ! `  N )  /  prod_ j  e.  ( 0 ... M ) ( ! `  ( D `  j ) ) )  x.  ( if ( ( P  -  1 )  <  ( D `
  0 ) ,  0 ,  ( ( ( ! `  ( P  -  1 ) ) 
 /  ( ! `  ( ( P  -  1 )  -  ( D `  0 ) ) ) )  x.  ( J ^ ( ( P  -  1 )  -  ( D `  0 ) ) ) ) )  x.  prod_ j  e.  (
 1 ... M ) if ( P  <  ( D `  j ) ,  0 ,  ( ( ( ! `  P )  /  ( ! `  ( P  -  ( D `  j ) ) ) )  x.  (
 ( J  -  j
 ) ^ ( P  -  ( D `  j ) ) ) ) ) ) )   =>    |-  ( ph  ->  ( ! `  ( P  -  1
 ) )  ||  T )
 
Theoremetransclem29 38005* The  N-th derivative of  F. (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 } )   &    |-  E  =  ( x  e.  X  |->  prod_ j  e.  ( 0
 ... M ) ( ( H `  j
 ) `  x )
 )   =>    |-  ( ph  ->  (
 ( S  Dn F ) `  N )  =  ( x  e.  X  |->  sum_ c  e.  ( C `  N ) ( ( ( ! `  N )  /  prod_ j  e.  ( 0 ... M ) ( ! `  ( c `  j
 ) ) )  x. 
 prod_ j  e.  (
 0 ... M ) ( ( ( S  Dn ( H `  j ) ) `  ( c `  j
 ) ) `  x ) ) ) )
 
Theoremetransclem30 38006* The  N-th derivative of  F. (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  ->  ( ( S  Dn F ) `
  N )  =  ( x  e.  X  |->  sum_
 c  e.  ( C `
  N ) ( ( ( ! `  N )  /  prod_ j  e.  ( 0 ... M ) ( ! `  ( c `  j
 ) ) )  x. 
 prod_ j  e.  (
 0 ... M ) ( ( ( S  Dn ( H `  j ) ) `  ( c `  j
 ) ) `  x ) ) ) )
 
Theoremetransclem31 38007* The  N-th derivative of  H applied to  Y. (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  ->  Y  e.  X )   =>    |-  ( ph  ->  (
 ( ( S  Dn F ) `  N ) `  Y )  = 
 sum_ c  e.  ( C `  N ) ( ( ( ! `  N )  /  prod_ j  e.  ( 0 ... M ) ( ! `  ( c `  j
 ) ) )  x.  ( if ( ( P  -  1 )  <  ( c `  0 ) ,  0 ,  ( ( ( ! `  ( P  -  1 ) ) 
 /  ( ! `  ( ( P  -  1 )  -  (
 c `  0 )
 ) ) )  x.  ( Y ^ (
 ( P  -  1
 )  -  ( c `
  0 ) ) ) ) )  x. 
 prod_ j  e.  (
 1 ... M ) if ( P  <  (
 c `  j ) ,  0 ,  (
 ( ( ! `  P )  /  ( ! `  ( P  -  ( c `  j
 ) ) ) )  x.  ( ( Y  -  j ) ^
 ( P  -  (
 c `  j )
 ) ) ) ) ) ) )
 
Theoremetransclem32 38008* This is the proof for the last equation in the proof of the derivative calculated in [Juillerat] p. 12, just after equation *(6) . (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  ->  ( ( M  x.  P )  +  ( P  -  1 ) )  <  N )   &    |-  H  =  ( j  e.  (
 0 ... M )  |->  ( x  e.  X  |->  ( ( x  -  j
 ) ^ if (
 j  =  0 ,  ( P  -  1
 ) ,  P ) ) ) )   =>    |-  ( ph  ->  ( ( S  Dn F ) `  N )  =  ( x  e.  X  |->  0 ) )
 
Theoremetransclem33 38009*  F is smooth. (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  ->  (
 ( S  Dn F ) `  N ) : X --> CC )
 
Theoremetransclem34 38010* 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 )   &    |-  H  =  ( k  e.  ( 0
 ... M )  |->  ( x  e.  X  |->  ( ( x  -  k
 ) ^ if (
 k  =  0 ,  ( P  -  1
 ) ,  P ) ) ) )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n ) 
 ^m  ( 0 ...
 M ) )  | 
 sum_ k  e.  (
 0 ... M ) ( c `  k )  =  n } )   =>    |-  ( ph  ->  ( ( S  Dn F ) `
  N )  e.  ( X -cn-> CC )
 )
 
Theoremetransclem35 38011*  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 38012* 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 38013*  ( 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 38014*  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 38015*  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 38016* 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 38017*  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 38018* 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 38019*  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 38020* 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 38021*  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 38022* 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 38023*  _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
 ) )
 
Theoremetransclem48OLD 38024*  _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.) Obsolete version of etransclem48 38025 as of 28-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 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
 ) )
 
Theoremetransclem48 38025*  _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.) (Revised by AV, 28-Sep-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  = inf ( { 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 38026  _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.) (Proof shortened by AV, 28-Sep-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 38027 Extend class notation with the class of all sigma-algebras.
 class SAlg
 
Definitiondf-salg 38028* 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 38029 Extend class notation with the class of sigma-algebras on a set.
 class SalOn
 
Definitiondf-salon 38030* 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 38031 Extend class notation with the class of sigma-algebra generator.
 class SalGen
 
Definitiondf-salgen 38032* 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 38033* 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 38034 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 38035 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 38036 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 38037 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 38038 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 38039 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( S  e. SAlg  ->  (/)  e.  S )
 
Theoremsalgenval 38040* 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 38041* 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 38042 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 38043 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 38044* 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 38045 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 38046* 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 38047* The arbitrary intersection of sigma-algebra (on the same set  X) is a sigma-algebra ( on the same set  X, see intsaluni 38046). (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 38048 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 38049 Extend class notation to include the sum of nonnegative extended reals.
 class Σ^
 
Definitiondf-sumge0 38050* 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 38051* 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 38052 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 38053* 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 38054 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 38055* 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 38056* 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 38057 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 38058 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 38059* 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 38060 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 38061 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 38062* 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 38063 The sum of nonnegative extended reals is zero when applied to the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (Σ^ `  (/) )  =  0
 
Theoremfsumlesge0 38064* 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 38065* 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 38066 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 38067 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 38068 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 38069* 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 38070* 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 38071 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 38072 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 38073 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 38074* 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 38075 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 38076* 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 38078, 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 38077* 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 38078* 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 38079 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 38080* 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 38081* 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 38082* 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 38083* 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 38084 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 38085* 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 38086* 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 38087* 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 38088* 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 38081. (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 38089* 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 38090 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 38091 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 38092* 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 38093 Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 38096. (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 38094* 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 38095* 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 38096 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 38097* 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 38098* 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 38099* 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 38100* 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 ) ) ) ) )
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40127
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