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Theorem List for Intuitionistic Logic Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrec2uzf1od 9801*  G (see frec2uz0d 9794) 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 9802*  G (see frec2uz0d 9794) 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 9803* 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 9804* 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 9794 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 9805* 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 9806* 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 9807* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9794 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 9808* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 9794 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 9809* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9794 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 9810* 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 9805 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 9811* 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 9812* 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 9813* 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 9814* 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 9815* 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 9816* 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 9817* 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 9818* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9794 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 9819* 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 9820 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremfrecfzennn 9821 The cardinality of a finite set of sequential integers. (See frec2uz0d 9794 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 9822 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 9823  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 9824* The mapping  G (see frec2uz0d 9794) 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 9825 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 9826 Deduction form of fzfig 9825. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( M ... N )  e.  Fin )
 
Theoremfzofig 9827 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M..^ N )  e.  Fin )
 
Theoremnn0ennn 9828 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
 |- 
 NN0  ~~  NN
 
Theoremnnenom 9829 The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |- 
 NN  ~~  om
 
Theoremnnct 9830  NN is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |- 
 NN  ~<_  om
 
Theoremfnn0nninf 9831* 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 9832* 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 ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  I :NN0* -->
 
Theorem0tonninf 9833* 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 9834* 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 9835* 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 )
 
3.6.4  Strong induction over upper sets of integers
 
Theoremuzsinds 9836* 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 9837* 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 9838* 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 )
 
3.6.5  The infinite sequence builder "seq"
 
Syntaxcseq4 9839 Extend class notation with four-argument recursive sequence builder.
 class  seq M (  .+  ,  F ,  S )
 
Syntaxcseq 9840 Extend class notation with three-argument recursive sequence builder.
 class  seq M (  .+  ,  F )
 
Definitiondf-iseq 9841* 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 iseq1 9863 and iseqp1 9870. 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 ,  QQ ) with values 1, 3/2, 7/4, 15/8,.., so that  (  seq 1
(  +  ,  F ,  QQ ) `  1
)  =  1,  (  seq 1
(  +  ,  F ,  QQ ) `  2
)  = 3/2, etc. In other words,  seq M (  +  ,  F ,  QQ ) transforms a sequence  F into an infinite series.

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 Jim Kingdon, 29-May-2020.) Use df-seq3 9842 or dfseq3-2 9843 instead. (New usage is discouraged.)

 |- 
 seq M (  .+  ,  F ,  S )  =  ran frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. ( x  +  1 ) ,  (
 y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )
 
Definitiondf-seq3 9842 Define a three-argument version of 
seq. By theorems such as iseqsst 9874, it should be capable of doing pretty much everything that the four-argument version can, and may eventually replace the four-argument version entirely. (Contributed by Jim Kingdon, 3-Oct-2022.)
 |- 
 seq M (  .+  ,  F )  =  seq M (  .+  ,  F ,  _V )
 
Theoremdfseq3-2 9843* 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 9868, seq3-1 9865 and seq3p1 9872. 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.

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.

Eventually, this will be the definition of  seq, replacing df-iseq 9841 and df-seq3 9842.

(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 ) >. )
 
Theoremiseqex 9844 Existence of the sequence builder operation.

New proofs should use seqex 9845 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 20-Aug-2021.) (New usage is discouraged.)

 |- 
 seq M (  .+  ,  F ,  S )  e.  _V
 
Theoremseqex 9845 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |- 
 seq M (  .+  ,  F )  e.  _V
 
Theoremiseqeq1 9846 Equality theorem for the sequence builder operation.

New proofs should use seqeq1 9849 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

 |-  ( M  =  N  ->  seq M (  .+  ,  F ,  S )  =  seq N ( 
 .+  ,  F ,  S ) )
 
Theoremiseqeq2 9847 Equality theorem for the sequence builder operation.

New proofs should use seqeq2 9850 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

 |-  (  .+  =  Q  ->  seq M (  .+  ,  F ,  S )  =  seq M ( Q ,  F ,  S ) )
 
Theoremiseqeq3 9848 Equality theorem for the sequence builder operation.

New proofs should use seqeq3 9851 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

 |-  ( F  =  G  ->  seq M (  .+  ,  F ,  S )  =  seq M ( 
 .+  ,  G ,  S ) )
 
Theoremseqeq1 9849 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( M  =  N  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
 
Theoremseqeq2 9850 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  (  .+  =  Q  ->  seq M (  .+  ,  F )  =  seq M ( Q ,  F ) )
 
Theoremseqeq3 9851 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( F  =  G  ->  seq M (  .+  ,  F )  =  seq M (  .+  ,  G ) )
 
Theoremseqeq1d 9852 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 9853 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 9854 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 9855 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 ) )
 
Theoremnfiseq 9856 Hypothesis builder for the sequence builder operation.

New proofs should use nfseq 9857 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

 |-  F/_ x M   &    |-  F/_ x  .+   &    |-  F/_ x F   &    |-  F/_ x S   =>    |-  F/_ x  seq M ( 
 .+  ,  F ,  S )
 
Theoremnfseq 9857 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 9858* 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 )
 
Theoremiseqval 9859* Value of the sequence builder function.

There should be no need for new usages of this theorem because once we have proved theorems seqf 9868, seq3-1 9865 and seq3p1 9872 future development can be done in terms of those.

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

 |-  R  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. ( 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 ,  S )  =  ran  R )
 
Theoremiseqvalcbv 9860* 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 ) >. )
 
Theoremiseqvalt 9861* Value of the sequence builder function.

There should be no need for new usages of this theorem because once we have proved theorems seqf 9868, seq3-1 9865 and seq3p1 9872 future development can be done in terms of those.

(Contributed by Jim Kingdon, 27-Apr-2022.) (New usage is discouraged.)

 |-  ( ph  ->  M  e.  ZZ )   &    |-  R  = 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 ) >. )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ph  ->  S  C_  T )   =>    |-  ( ph  ->  seq M (  .+  ,  F ,  T )  =  ran  R )
 
Theoremseq3val 9862* Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 9868, seq3-1 9865 and seq3p1 9872, 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 )
 
Theoremiseq1 9863* Value of the sequence builder function at its initial value.

New proofs should use seq3-1 9865 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 31-May-2020.) (New usage is discouraged.)

 |-  ( 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 ,  S ) `
  M )  =  ( F `  M ) )
 
Theoremiseq1t 9864* Value of the sequence builder function at its initial value.

New proofs should use seq3-1 9865 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 31-May-2020.) (New usage is discouraged.)

 |-  ( 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  ->  S  C_  T )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ,  T ) `  M )  =  ( F `  M ) )
 
Theoremseq3-1 9865* 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 ) )
 
Theoremiseqfcl 9866* Range of the recursive sequence builder. New proofs should use seqf 9868 instead (together with iseqsst 9874 or iseqseq3 9890 if need be). (Contributed by Jim Kingdon, 11-Apr-2022.) (New usage is discouraged.)
 |-  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 ,  S ) : Z --> S )
 
Theoremiseqfclt 9867* Range of the recursive sequence builder.

New proofs should use seqf 9868 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 26-Apr-2022.) (New usage is discouraged.)

 |-  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  ->  S 
 C_  T )   =>    |-  ( ph  ->  seq
 M (  .+  ,  F ,  T ) : Z --> S )
 
Theoremseqf 9868* 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 )
 
Theoremiseqcl 9869* Closure property of the recursive sequence builder.

New proofs should use seqf 9868 or seq3clss 9875 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 1-Jun-2020.) (New usage is discouraged.)

 |-  ( 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 ,  S ) `  N )  e.  S )
 
Theoremiseqp1 9870* Value of the sequence builder function at a successor.

New proofs should use seq3p1 9872 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 31-May-2020.) (New usage is discouraged.)

 |-  ( 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 ,  S ) `  ( N  +  1 ) )  =  ( (  seq M ( 
 .+  ,  F ,  S ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremiseqp1t 9871* Value of the sequence builder function at a successor.

New proofs should use seq3p1 9872 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 30-Apr-2022.) (New usage is discouraged.)

 |-  ( 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  ->  S  C_  T )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( (  seq M ( 
 .+  ,  F ,  T ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremseq3p1 9872* 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
 ) ) ) )
 
Theoremiseqoveq 9873* Equality theorem for the sequence builder operation. This is similar to iseqeq2 9847 but  .+ and  Q only have to agree on elements of  S.

We intend to replace it with seqfeq3 from the Metamath Proof Explorer, which is why new usages are discouraged.

(Contributed by Jim Kingdon, 21-Apr-2022.) (New usage is discouraged.)

 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  =  ( x Q y ) )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x Q y )  e.  S )   =>    |-  ( ph  ->  seq
 M (  .+  ,  F ,  S )  =  seq M ( Q ,  F ,  S ) )
 
Theoremiseqsst 9874* Specifying a larger universe for 
seq. As long as  F and  .+ are closed over  S, then any class which contains  S can be used as the last argument to 
seq.

Together with df-seq3 9842 it can be used to convert between the df-iseq 9841 syntax and the df-seq3 9842 syntax (in many cases iseqseq3 9890 is an even more convenient way to do this).

(Contributed by Jim Kingdon, 28-Apr-2022.)

 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  S 
 C_  T )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M (  .+  ,  F ,  T ) )
 
Theoremseq3clss 9875* 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 )
 
Theoremiseqm1 9876* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) Use seq3m1 9877 instead. (New usage is discouraged.)
 |-  ( 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 ,  S ) `
  N )  =  ( (  seq M (  .+  ,  F ,  S ) `  ( N  -  1 ) ) 
 .+  ( F `  N ) ) )
 
Theoremseq3m1 9877* 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 ) ) )
 
Theoremiseqfveq2 9878* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) Use seq3fveq2 9880 instead. (New usage is discouraged.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  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 ,  S ) `  N )  =  ( 
 seq K (  .+  ,  G ,  S ) `
  N ) )
 
Theoremiseqfeq2 9879* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) Use seq3feq2 9881 instead. (New usage is discouraged.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  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 ,  S )  |`  ( ZZ>= `  K )
 )  =  seq K (  .+  ,  G ,  S ) )
 
Theoremseq3fveq2 9880* 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 9881* 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 ) )
 
Theoremiseqfveq 9882* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.) Use seq3fveq 9883 instead. (New usage is discouraged.)
 |-  ( 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 ,  S ) `  N )  =  ( 
 seq M (  .+  ,  G ,  S ) `
  N ) )
 
Theoremseq3fveq 9883* 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 ) )
 
Theoremiseqfeq 9884* Equality of sequences.

New proofs should use seq3feq 9885 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 15-Aug-2021.) (New usage is discouraged.)

 |-  ( 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 ,  S )  =  seq M (  .+  ,  G ,  S ) )
 
Theoremseq3feq 9885* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
 |-  ( 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 )
 )
 
Theoremiseqshft2 9886* Shifting the index set of a sequence.

New proofs should use seq3shft2 9887 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 15-Aug-2021.) (New usage is discouraged.)

 |-  ( 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 ,  S ) `
  N )  =  (  seq ( M  +  K ) ( 
 .+  ,  G ,  S ) `  ( N  +  K )
 ) )
 
Theoremseq3shft2 9887* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 17-Oct-2022.)
 |-  ( 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 9888* 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 9889* 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 )
 
Theoremiseqseq3 9890* Equality of  seq M (  +  ,  F ,  CC ) and  seq M (  +  ,  F ). (Contributed by Jim Kingdon, 4-Oct-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  CC )   =>    |-  ( ph  ->  seq
 M (  +  ,  F ,  CC )  =  seq M (  +  ,  F ) )
 
Theoremiserf 9891* An infinite series of complex terms is a function from  NN to  CC. (Contributed by Jim Kingdon, 15-Aug-2021.) Use serf 9888 instead. (New usage is discouraged.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  seq M (  +  ,  F ,  CC ) : Z --> CC )
 
Theoremmonoord 9892* 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 9893* 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 ) )
 
Theoremisermono 9894* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( 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 ,  RR ) `  K )  <_  (  seq M (  +  ,  F ,  RR ) `  N ) )
 
Theoremseq3split 9895* 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 ) ) )
 
Theoremiseqsplit 9896* Split a sequence into two sequences. New proofs should use seq3split 9895 instead. (Contributed by Jim Kingdon, 16-Aug-2021.) (New usage is discouraged.)
 |-  ( ( 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 ,  S ) `
  N )  =  ( (  seq K (  .+  ,  F ,  S ) `  M )  .+  (  seq ( M  +  1 )
 (  .+  ,  F ,  S ) `  N ) ) )
 
Theoremseq3-1p 9897* 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 ) ) )
 
Theoremiseqcaopr3 9898* Lemma for iseqcaopr2 . (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 )
 )  ->  ( 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 ,  S ) `  n ) Q (  seq M (  .+  ,  G ,  S ) `  n ) )  .+  ( ( F `  ( n  +  1 ) ) Q ( G `  ( n  +  1
 ) ) ) )  =  ( ( ( 
 seq M (  .+  ,  F ,  S ) `
  n )  .+  ( F `  ( n  +  1 ) ) ) Q ( ( 
 seq M (  .+  ,  G ,  S ) `
  n )  .+  ( G `  ( n  +  1 ) ) ) ) )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ,  S ) `  N )  =  ( (  seq M ( 
 .+  ,  F ,  S ) `  N ) Q (  seq M (  .+  ,  G ,  S ) `  N ) ) )
 
Theoremiseqcaopr2 9899* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
 |-  ( ( 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  ->  S  e.  V )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  H ,  S ) `
  N )  =  ( (  seq M (  .+  ,  F ,  S ) `  N ) Q (  seq M (  .+  ,  G ,  S ) `  N ) ) )
 
Theoremiseqcaopr 9900* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Jim Kingdon, 17-Aug-2021.)
 |-  ( ( 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  ->  S  e.  V )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ,  S ) `  N )  =  ( (  seq M ( 
 .+  ,  F ,  S ) `  N )  .+  (  seq M (  .+  ,  G ,  S ) `  N ) ) )
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