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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremq2txmodxeq0 10401 Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( X  e.  QQ  /\  0  <  X )  ->  ( ( 2  x.  X )  mod  X )  =  0 )
 
Theoremq2submod 10402 If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B 
 <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
 
Theoremmodifeq2int 10403 If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
 
Theoremmodaddmodup 10404 The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  ( ( B  +  ( A  mod  M ) )  -  M )  =  ( ( B  +  A )  mod  M ) ) )
 
Theoremmodaddmodlo 10405 The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  ( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M ) ) )
 
Theoremmodqmulmod 10406 The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  x.  B )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodqmulmodr 10407 The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  x.  ( B  mod  M ) )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodqaddmulmod 10408 The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod  M )  =  ( ( A  +  ( B  x.  C ) )  mod  M ) )
 
Theoremmodqdi 10409 Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  mod  C ) )  =  ( ( A  x.  B )  mod  ( A  x.  C ) ) )
 
Theoremmodqsubdir 10410 Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( B  mod  C )  <_  ( A  mod  C )  <->  ( ( A  -  B )  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) ) )
 
Theoremmodqeqmodmin 10411 A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( A  mod  M )  =  ( ( A  -  M )  mod  M ) )
 
Theoremmodfzo0difsn 10412* For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( J  e.  ( 0..^ N )  /\  K  e.  ( (
 0..^ N )  \  { J } ) ) 
 ->  E. i  e.  (
 1..^ N ) K  =  ( ( i  +  J )  mod  N ) )
 
Theoremmodsumfzodifsn 10413 The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( J  e.  ( 0..^ N )  /\  K  e.  ( 1..^ N ) )  ->  ( ( K  +  J )  mod  N )  e.  ( ( 0..^ N )  \  { J } ) )
 
Theoremmodlteq 10414 Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N ) ) 
 ->  ( ( I  mod  N )  =  ( J 
 mod  N )  <->  I  =  J ) )
 
Theoremaddmodlteq 10415 Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (Contributed by AV, 20-Mar-2021.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N )  /\  S  e.  ZZ )  ->  ( ( ( I  +  S )  mod  N )  =  ( ( J  +  S ) 
 mod  N )  <->  I  =  J ) )
 
4.6.3  Miscellaneous theorems about integers
 
Theoremfrec2uz0d 10416* The mapping  G is a one-to-one mapping from  om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number  C (normally 0 for the upper integers  NN0 or 1 for the upper integers  NN), 1 maps to  C + 1, etc. This theorem shows the value of  G at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  ( G `  (/) )  =  C )
 
Theoremfrec2uzzd 10417* The value of  G (see frec2uz0d 10416) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  A )  e. 
 ZZ )
 
Theoremfrec2uzsucd 10418* The value of  G (see frec2uz0d 10416) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  suc  A )  =  ( ( G `
  A )  +  1 ) )
 
Theoremfrec2uzuzd 10419* The value  G (see frec2uz0d 10416) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  A )  e.  ( ZZ>= `  C )
 )
 
Theoremfrec2uzltd 10420* Less-than relation for  G (see frec2uz0d 10416). (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  e.  B  ->  ( G `  A )  <  ( G `  B ) ) )
 
Theoremfrec2uzlt2d 10421* The mapping  G (see frec2uz0d 10416) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  e.  B  <->  ( G `  A )  <  ( G `
  B ) ) )
 
Theoremfrec2uzrand 10422* Range of  G (see frec2uz0d 10416). (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  ran  G  =  ( ZZ>= `  C )
 )
 
Theoremfrec2uzf1od 10423*  G (see frec2uz0d 10416) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  G : om
 -1-1-onto-> ( ZZ>= `  C )
 )
 
Theoremfrec2uzisod 10424*  G (see frec2uz0d 10416) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  G  Isom  _E  ,  <  ( om ,  ( ZZ>= `  C ) ) )
 
Theoremfrecuzrdgrrn 10425* The function  R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 
S. (Contributed by Jim Kingdon, 28-Mar-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ( ph  /\  D  e.  om )  ->  ( R `  D )  e.  ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrec2uzrdg 10426* A helper lemma for the value of a recursive definition generator on upper integers (typically either  NN or  NN0) with characteristic function 
F ( x ,  y ) and initial value  A. This lemma shows that evaluating  R at an element of  om gives an ordered pair whose first element is the index (translated from  om to  ( ZZ>= `  C )). See comment in frec2uz0d 10416 which describes  G and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B ) ) >. )
 
Theoremfrecuzrdgrcl 10427* The function  R (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 1-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  R : om --> ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrecuzrdglem 10428* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  B  e.  ( ZZ>= `  C ) )   =>    |-  ( ph  ->  <. B ,  ( 2nd `  ( R `  ( `' G `  B ) ) )
 >.  e.  ran  R )
 
Theoremfrecuzrdgtcl 10429* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10416 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 26-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ph  ->  T :
 ( ZZ>= `  C ) --> S )
 
Theoremfrecuzrdg0 10430* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10416 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 27-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ph  ->  ( T `  C )  =  A )
 
Theoremfrecuzrdgsuc 10431* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10416 for the description of  G as the mapping from 
om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 28-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  C )
 )  ->  ( T `  ( B  +  1 ) )  =  ( B F ( T `
  B ) ) )
 
Theoremfrecuzrdgrclt 10432* The function  R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of  S. Similar to frecuzrdgrcl 10427 except that  S and  T need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  R : om --> ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrecuzrdgg 10433* Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating  R at a natural number gives an ordered pair whose first element is the mapping of that natural number via  G. (Contributed by Jim Kingdon, 23-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  N  e.  om )   &    |-  G  = frec ( ( x  e. 
 ZZ  |->  ( x  +  1 ) ) ,  C )   =>    |-  ( ph  ->  ( 1st `  ( R `  N ) )  =  ( G `  N ) )
 
Theoremfrecuzrdgdomlem 10434* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C ) )
 
Theoremfrecuzrdgdom 10435* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C ) )
 
Theoremfrecuzrdgfunlem 10436* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  Fun  ran  R )
 
Theoremfrecuzrdgfun 10437* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  Fun  ran  R )
 
Theoremfrecuzrdgtclt 10438* The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ph  ->  P :
 ( ZZ>= `  C ) --> S )
 
Theoremfrecuzrdg0t 10439* Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ph  ->  ( P `  C )  =  A )
 
Theoremfrecuzrdgsuctlem 10440* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10416 for the description of  G as the mapping from  om to  ( ZZ>= `  C ). (Contributed by Jim Kingdon, 29-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  C ) )  ->  ( P `  ( B  +  1 ) )  =  ( B F ( P `  B ) ) )
 
Theoremfrecuzrdgsuct 10441* Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  C )
 )  ->  ( P `  ( B  +  1 ) )  =  ( B F ( P `
  B ) ) )
 
Theoremuzenom 10442 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremfrecfzennn 10443 The cardinality of a finite set of sequential integers. (See frec2uz0d 10416 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( N  e.  NN0  ->  ( 1 ... N ) 
 ~~  ( `' G `  N ) )
 
Theoremfrecfzen2 10444 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( M ... N ) 
 ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
 
Theoremfrechashgf1o 10445  G maps  om one-to-one onto  NN0. (Contributed by Jim Kingdon, 19-May-2020.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  G : om -1-1-onto-> NN0
 
Theoremfrec2uzled 10446* The mapping  G (see frec2uz0d 10416) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  C_  B  <->  ( G `  A )  <_  ( G `
  B ) ) )
 
Theoremfzfig 10447 A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  e.  Fin )
 
Theoremfzfigd 10448 Deduction form of fzfig 10447. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( M ... N )  e.  Fin )
 
Theoremfzofig 10449 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M..^ N )  e.  Fin )
 
Theoremnn0ennn 10450 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
 |- 
 NN0  ~~  NN
 
Theoremnnenom 10451 The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of natural numbers as ordinals). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |- 
 NN  ~~  om
 
Theoremnnct 10452  NN is dominated by  om. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |- 
 NN  ~<_  om
 
Theoremuzennn 10453 An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( M  e.  ZZ  ->  ( ZZ>= `  M )  ~~  NN )
 
Theoremfnn0nninf 10454* A function from  NN0 into ℕ. (Contributed by Jim Kingdon, 16-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   =>    |-  ( F  o.  `' G ) : NN0 -->
 
Theoremfxnn0nninf 10455* A function from NN0* into ℕ. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7139 instead of infnninfOLD 7140. More generally, this theorem and most theorems in this section could use an extended  G defined by  G  =  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )  u.  <. om , +oo >. ) and  F  =  ( n  e.  suc  om  |->  ( i  e.  om  |->  if ( i  e.  n ,  1o ,  (/) ) ) ) as in nnnninf2 7142.
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  I :NN0* -->
 
Theorem0tonninf 10456* The mapping of zero into ℕ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  ( I `  0
 )  =  ( x  e.  om  |->  (/) )
 
Theorem1tonninf 10457* The mapping of one into ℕ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  ( I `  1
 )  =  ( x  e.  om  |->  if ( x  =  (/) ,  1o ,  (/) ) )
 
Theoreminftonninf 10458* The mapping of +oo into ℕ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  ( I ` +oo )  =  ( x  e.  om  |->  1o )
 
4.6.4  Strong induction over upper sets of integers
 
Theoremuzsinds 10459* Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  ( ZZ>= `  M )  ->  ( A. y  e.  ( M ... ( x  -  1
 ) ) ps  ->  ph ) )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ch )
 
Theoremnnsinds 10460* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN  ->  (
 A. y  e.  (
 1 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN  ->  ch )
 
Theoremnn0sinds 10461* Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN0  ->  ( A. y  e.  (
 0 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN0 
 ->  ch )
 
4.6.5  The infinite sequence builder "seq"
 
Syntaxcseq 10462 Extend class notation with recursive sequence builder.
 class  seq M (  .+  ,  F )
 
Definitiondf-seqfrec 10463* Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as  NN or  NN0) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seqf 10478, seq3-1 10477 and seq3p1 10479. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation  +, an input sequence  F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence  seq 1 (  +  ,  F ) with values 1, 3/2, 7/4, 15/8,.., so that  (  seq 1
(  +  ,  F
) `  1 )  =  1,  (  seq 1 (  +  ,  F ) `  2
)  = 3/2, etc. In other words,  seq M (  +  ,  F ) transforms a sequence  F into an infinite series. 
seq M (  +  ,  F )  ~~>  2 means "the sum of F(n) from n = M to infinity is 2". Since limits are unique (climuni 11318), by climdm 11320 the "sum of F(n) from n = 1 to infinity" can be expressed as  (  ~~>  `  seq 1
(  +  ,  F
) ) (provided the sequence converges) and evaluates to 2 in this example.

Internally, the frec function generates as its values a set of ordered pairs starting at 
<. M ,  ( F `
 M ) >., with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain.

(Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)

 |- 
 seq M (  .+  ,  F )  =  ran frec ( ( x  e.  ( ZZ>=
 `  M ) ,  y  e.  _V  |->  <.
 ( x  +  1 ) ,  ( y 
 .+  ( F `  ( x  +  1
 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )
 
Theoremseqex 10464 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |- 
 seq M (  .+  ,  F )  e.  _V
 
Theoremseqeq1 10465 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( M  =  N  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
 
Theoremseqeq2 10466 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  (  .+  =  Q  ->  seq M (  .+  ,  F )  =  seq M ( Q ,  F ) )
 
Theoremseqeq3 10467 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( F  =  G  ->  seq M (  .+  ,  F )  =  seq M (  .+  ,  G ) )
 
Theoremseqeq1d 10468 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  seq A (  .+  ,  F )  =  seq B ( 
 .+  ,  F )
 )
 
Theoremseqeq2d 10469 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  seq M ( A ,  F )  =  seq M ( B ,  F ) )
 
Theoremseqeq3d 10470 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  seq M (  .+  ,  A )  =  seq M ( 
 .+  ,  B )
 )
 
Theoremseqeq123d 10471 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  M  =  N )   &    |-  ( ph  ->  .+  =  Q )   &    |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N ( Q ,  G ) )
 
Theoremnfseq 10472 Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x M   &    |-  F/_ x  .+   &    |-  F/_ x F   =>    |-  F/_ x  seq M (  .+  ,  F )
 
Theoremiseqovex 10473* Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
 |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  ->  ( x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y )  e.  S )
 
Theoremiseqvalcbv 10474* Changing the bound variables in an expression which appears in some  seq related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.)
 |- frec
 ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x ( z  e.  ( ZZ>=
 `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( a  e.  ( ZZ>= `  M ) ,  b  e.  T  |->  <. ( a  +  1 ) ,  (
 a ( c  e.  ( ZZ>= `  M ) ,  d  e.  S  |->  ( d  .+  ( F `
  ( c  +  1 ) ) ) ) b ) >. ) ,  <. M ,  ( F `  M ) >. )
 
Theoremseq3val 10475* Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10478, seq3-1 10477 and seq3p1 10479, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  M ) ,  y  e.  _V  |->  <.
 ( x  +  1 ) ,  ( x ( z  e.  ( ZZ>=
 `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y ) >. ) ,  <. M ,  ( F `  M ) >. )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F )  =  ran  R )
 
Theoremseqvalcd 10476* Value of the sequence builder function. Similar to seq3val 10475 but the classes  D (type of each term) and  C (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  M ) ,  y  e.  _V  |->  <.
 ( x  +  1 ) ,  ( x ( z  e.  ( ZZ>=
 `  M ) ,  w  e.  C  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y ) >. ) ,  <. M ,  ( F `  M ) >. )   &    |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D )
 )  ->  ( x  .+  y )  e.  C )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  seq M (  .+  ,  F )  =  ran  R )
 
Theoremseq3-1 10477* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  M )  =  ( F `  M ) )
 
Theoremseqf 10478* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  Z )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F ) : Z --> S )
 
Theoremseq3p1 10479* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  ( N  +  1 )
 )  =  ( ( 
 seq M (  .+  ,  F ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremseqovcd 10480* A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10481 and seq1cd 10482 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
 |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D )
 )  ->  ( x  .+  y )  e.  C )   =>    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  C ) )  ->  ( x ( z  e.  ( ZZ>= `  M ) ,  w  e.  C  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y )  e.  C )
 
Theoremseqf2 10481* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
 |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D ) )  ->  ( x 
 .+  y )  e.  C )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  seq M (  .+  ,  F ) : Z --> C )
 
Theoremseq1cd 10482* Initial value of the recursive sequence builder. A version of seq3-1 10477 which provides two classes 
D and  C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
 |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D ) )  ->  ( x 
 .+  y )  e.  C )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  M )  =  ( F `  M ) )
 
Theoremseqp1cd 10483* Value of the sequence builder function at a successor. A version of seq3p1 10479 which provides two classes  D and  C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D ) )  ->  ( x 
 .+  y )  e.  C )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1
 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  ( N  +  1 )
 )  =  ( ( 
 seq M (  .+  ,  F ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremseq3clss 10484* Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  T )   &    |-  ( ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   &    |-  ( ph  ->  S 
 C_  T )   &    |-  (
 ( ph  /\  ( x  e.  T  /\  y  e.  T ) )  ->  ( x  .+  y )  e.  T )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  N )  e.  S )
 
Theoremseq3m1 10485* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 )
 ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  N )  =  ( (  seq M (  .+  ,  F ) `  ( N  -  1 ) ) 
 .+  ( F `  N ) ) )
 
Theoremseq3fveq2 10486* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  K )  =  ( G `  K ) )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  K )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K )
 )   &    |-  ( ( ph  /\  k  e.  ( ( K  +  1 ) ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  (  seq K (  .+  ,  G ) `  N ) )
 
Theoremseq3feq2 10487* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  K )  =  ( G `  K ) )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  K )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F )  |`  ( ZZ>= `  K ) )  = 
 seq K (  .+  ,  G ) )
 
Theoremseq3fveq 10488* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  (  seq M (  .+  ,  G ) `  N ) )
 
Theoremseq3feq 10489* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  =  ( G `  k ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M ( 
 .+  ,  G )
 )
 
Theoremseq3shft2 10490* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  =  ( G `
  ( k  +  K ) ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  K ) ) ) 
 ->  ( G `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  N )  =  (  seq ( M  +  K ) (  .+  ,  G ) `  ( N  +  K ) ) )
 
Theoremserf 10491* An infinite series of complex terms is a function from  NN to  CC. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
 
Theoremserfre 10492* An infinite series of real numbers is a function from  NN to  RR. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   =>    |-  ( ph  ->  seq M (  +  ,  F ) : Z --> RR )
 
Theoremmonoord 10493* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  -  1 ) ) ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   =>    |-  ( ph  ->  ( F `  M ) 
 <_  ( F `  N ) )
 
Theoremmonoord2 10494* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  -  1 ) ) ) 
 ->  ( F `  (
 k  +  1 ) )  <_  ( F `  k ) )   =>    |-  ( ph  ->  ( F `  N ) 
 <_  ( F `  M ) )
 
Theoremser3mono 10495* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K )
 )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  RR )   &    |-  ( ( ph  /\  x  e.  ( ( K  +  1 ) ... N ) )  ->  0  <_  ( F `  x ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ) `  K )  <_  (  seq M (  +  ,  F ) `  N ) )
 
Theoremseq3split 10496* Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  K ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  K ) )  ->  ( F `  x )  e.  S )   =>    |-  ( ph  ->  ( 
 seq K (  .+  ,  F ) `  N )  =  ( (  seq K (  .+  ,  F ) `  M )  .+  (  seq ( M  +  1 )
 (  .+  ,  F ) `  N ) ) )
 
Theoremseq3-1p 10497* Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  ( ( F `  M )  .+  (  seq ( M  +  1 ) (  .+  ,  F ) `  N ) ) )
 
Theoremseq3caopr3 10498* Lemma for seq3caopr2 10499. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x Q y )  e.  S )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  ->  ( F `  k )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( H `  k )  =  ( ( F `  k
 ) Q ( G `
  k ) ) )   &    |-  ( ( ph  /\  n  e.  ( M..^ N ) )  ->  ( ( (  seq M (  .+  ,  F ) `  n ) Q (  seq M ( 
 .+  ,  G ) `  n ) )  .+  ( ( F `  ( n  +  1
 ) ) Q ( G `  ( n  +  1 ) ) ) )  =  ( ( (  seq M (  .+  ,  F ) `
  n )  .+  ( F `  ( n  +  1 ) ) ) Q ( ( 
 seq M (  .+  ,  G ) `  n )  .+  ( G `  ( n  +  1
 ) ) ) ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ) `  N )  =  ( (  seq M (  .+  ,  F ) `  N ) Q (  seq M (  .+  ,  G ) `
  N ) ) )
 
Theoremseq3caopr2 10499* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x Q y )  e.  S )   &    |-  ( ( ph  /\  ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 ) )  ->  (
 ( x Q z )  .+  ( y Q w ) )  =  ( ( x 
 .+  y ) Q ( z  .+  w ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( H `  k )  =  ( ( F `  k
 ) Q ( G `
  k ) ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ) `  N )  =  ( (  seq M (  .+  ,  F ) `  N ) Q (  seq M (  .+  ,  G ) `
  N ) ) )
 
Theoremseq3caopr 10500* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  ->  ( F `  k )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( H `  k )  =  ( ( F `  k
 )  .+  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ) `  N )  =  ( (  seq M (  .+  ,  F ) `
  N )  .+  (  seq M (  .+  ,  G ) `  N ) ) )
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