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Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclimcj 10601* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( * `
  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( * `  A ) )
 
Theoremclimre 10602* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Re
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Re `  A ) )
 
Theoremclimim 10603* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Im
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Im `  A ) )
 
Theoremclimrecl 10604* The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremclimge0 10605* A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremclimadd 10606* Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  +  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  +  B ) )
 
Theoremclimmul 10607* Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
 
Theoremclimsub 10608* Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  -  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  -  B ) )
 
Theoremclimaddc1 10609* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  +  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  +  C ) )
 
Theoremclimaddc2 10610* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  +  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  +  A ) )
 
Theoremclimmulc2 10611* Limit of a sequence multiplied by a constant  C. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  x.  A ) )
 
Theoremclimsubc1 10612* Limit of a constant  C subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  -  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  -  C ) )
 
Theoremclimsubc2 10613* Limit of a constant  C minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  -  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  -  A ) )
 
Theoremclimle 10614* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremclimsqz 10615* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  <_  A )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimsqz2 10616* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  <_  ( G `  k ) )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclim2iser 10617* The limit of an infinite series with an initial segment removed. (Contributed by Jim Kingdon, 20-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq
 M (  +  ,  F ,  CC )  ~~>  A )   =>    |-  ( ph  ->  seq ( N  +  1 )
 (  +  ,  F ,  CC )  ~~>  ( A  -  (  seq M (  +  ,  F ,  CC ) `  N ) ) )
 
Theoremclim2iser2 10618* The limit of an infinite series with an initial segment added. (Contributed by Jim Kingdon, 21-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq ( N  +  1 ) (  +  ,  F ,  CC )  ~~>  A )   =>    |-  ( ph  ->  seq M (  +  ,  F ,  CC )  ~~>  ( A  +  (  seq M (  +  ,  F ,  CC ) `  N ) ) )
 
Theoremiiserex 10619* An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ,  CC )  e.  dom  ~~> 
 <-> 
 seq N (  +  ,  F ,  CC )  e.  dom  ~~>  ) )
 
Theoremiisermulc2 10620* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  seq M (  +  ,  F ,  CC )  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  seq M (  +  ,  G ,  CC )  ~~>  ( C  x.  A ) )
 
Theoremclimlec2 10621* Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  F  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremiserile 10622* Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  seq M (  +  ,  F ,  CC )  ~~>  A )   &    |-  ( ph  ->  seq
 M (  +  ,  G ,  CC )  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A 
 <_  B )
 
Theoremiserige0 10623* The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  seq M (  +  ,  F ,  CC )  ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremclimub 10624* The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   =>    |-  ( ph  ->  ( F `  N ) 
 <_  A )
 
Theoremclimserile 10625* The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by Jim Kingdon, 22-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  seq M (  +  ,  F ,  CC )  ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  <_  A )
 
Theoremclimcau 10626* A converging sequence of complex numbers is a Cauchy sequence. The converse would require excluded middle or a different definition of Cauchy sequence (for example, fixing a rate of convergence as in climcvg1n 10629). Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  ) 
 ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x )
 
Theoremclimrecvg1n 10627* A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within  C  /  n of the nth term, where  C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcvg1nlem 10628* Lemma for climcvg1n 10629. We construct sequences of the real and imaginary parts of each term of  F, show those converge, and use that to show that  F converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
 |-  ( ph  ->  F : NN --> CC )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   &    |-  G  =  ( x  e.  NN  |->  ( Re `  ( F `  x ) ) )   &    |-  H  =  ( x  e.  NN  |->  ( Im `  ( F `
  x ) ) )   &    |-  J  =  ( x  e.  NN  |->  ( _i  x.  ( H `
  x ) ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcvg1n 10629* A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within  C  /  n of the nth term, where  C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
 |-  ( ph  ->  F : NN --> CC )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( abs `  ( ( F `  k )  -  ( F `  n ) ) )  <  ( C  /  n ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremclimcaucn 10630* A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 10626 but adds the part that  ( F `  k ) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  ) 
 ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x ) )
 
Theoremserif0 10631* If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  seq M (  +  ,  F ,  CC )  e.  dom  ~~>  )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  F  ~~>  0 )
 
3.8.2  Finite and infinite sums
 
Syntaxcsu 10632 Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
 class  sum_ k  e.  A  B
 
Definitiondf-isum 10633* Define the sum of a series with an index set of integers  A.  k is normally a free variable in  B, i.e.  B can be thought of as  B ( k ). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. In both cases we have an  if expression so that we only need  B to be defined where  k  e.  A. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e. finite sets of integers). Examples: 
sum_ k  e.  {
1 ,  2 ,  4 }  k means  1  +  2  +  4  =  7, and  sum_ k  e.  NN  (
1  /  ( 2 ^ k ) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1. (Contributed by Jim Kingdon, 15-Feb-2022.)
 |- 
 sum_ k  e.  A  B  =  ( iota x ( E. m  e. 
 ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>=
 `  m )DECID  j  e.  A  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
 ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1
 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC ) `  m ) ) ) )
 
Theoremsumeq1 10634 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  ( A  =  B  -> 
 sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremnfsum1 10635 Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  F/_ k A   =>    |-  F/_ k sum_ k  e.  A  B
 
Theoremnfsum 10636 Bound-variable hypothesis builder for sum: if  x is (effectively) not free in  A and  B, it is not free in  sum_ k  e.  A B. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x sum_ k  e.  A  B
 
Theoremsumdc 10637* Decidability of a subset of upper integers. (Contributed by Jim Kingdon, 1-Jan-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A. x  e.  ( ZZ>= `  M )DECID  x  e.  A )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  -> DECID  N  e.  A )
 
Theoremsumeq2 10638* Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( A. k  e.  A  B  =  C  -> 
 sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremcbvsum 10639 Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
 |-  ( j  =  k 
 ->  B  =  C )   &    |-  F/_ k A   &    |-  F/_ j A   &    |-  F/_ k B   &    |-  F/_ j C   =>    |- 
 sum_ j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremcbvsumv 10640* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( j  =  k 
 ->  B  =  C )   =>    |-  sum_
 j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremcbvsumi 10641* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
 |-  F/_ k B   &    |-  F/_ j C   &    |-  (
 j  =  k  ->  B  =  C )   =>    |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremsumeq1i 10642* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
 |-  A  =  B   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  C
 
Theoremsumeq2i 10643* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
 |-  ( k  e.  A  ->  B  =  C )   =>    |-  sum_
 k  e.  A  B  =  sum_ k  e.  A  C
 
Theoremsumeq12i 10644* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
 |-  A  =  B   &    |-  (
 k  e.  A  ->  C  =  D )   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  D
 
Theoremsumeq1d 10645* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremsumeq2d 10646* Equality deduction for sum. Note that unlike sumeq2dv 10647, 
k may occur in  ph. (Contributed by NM, 1-Nov-2005.)
 |-  ( ph  ->  A. k  e.  A  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2dv 10647* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2ad 10648* Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2sdv 10649* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theorem2sumeq2dv 10650* Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( ph  /\  j  e.  A  /\  k  e.  B )  ->  C  =  D )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  C  =  sum_ j  e.  A  sum_ k  e.  B  D )
 
Theoremsumeq12dv 10651* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  =  D )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  D )
 
Theoremsumeq12rdv 10652* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  B ) 
 ->  C  =  D )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  D )
 
Theoremsumfct 10653* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 18-Sep-2022.)
 |-  ( A. k  e.  A  B  e.  CC  -> 
 sum_ j  e.  A  ( ( k  e.  A  |->  B ) `  j )  =  sum_ k  e.  A  B )
 
Theoremfz1f1o 10654* A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)
 |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
 ( `  A )  e. 
 NN  /\  E. f  f : ( 1 ... ( `  A )
 )
 -1-1-onto-> A ) ) )
 
Theoremisumrblem 10655* Lemma for isumrb 10657. (Contributed by Mario Carneiro, 12-Aug-2013.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ( ph  /\  A  C_  ( ZZ>= `  N )
 )  ->  (  seq M (  +  ,  F ,  CC )  |`  ( ZZ>= `  N ) )  = 
 seq N (  +  ,  F ,  CC )
 )
 
Theoremfisumcvg 10656* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A 
 C_  ( M ... N ) )   =>    |-  ( ph  ->  seq M (  +  ,  F ,  CC )  ~~>  (  seq M (  +  ,  F ,  CC ) `  N ) )
 
Theoremisumrb 10657* Rebase the starting point of a sum. (Contributed by Jim Kingdon, 5-Mar-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  N )
 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  -> DECID  k  e.  A )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  N )
 )  -> DECID  k  e.  A )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ,  CC )  ~~>  C 
 <-> 
 seq N (  +  ,  F ,  CC )  ~~>  C ) )
 
Theoremisummolemnm 10658* Lemma for isummo 10662. (Contributed by Jim Kingdon, 15-Aug-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN )
 )   &    |-  ( ph  ->  f : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ph  ->  K : ( 1 ...
 N ) -1-1-onto-> A )   =>    |-  ( ph  ->  N  =  M )
 
Theoremisummolem3 10659* Lemma for isummo 10662. (Contributed by Mario Carneiro, 29-Mar-2014.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN )
 )   &    |-  ( ph  ->  f : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ph  ->  K : ( 1 ...
 N ) -1-1-onto-> A )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  M ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  H  =  ( n  e.  NN  |->  if ( n  <_  N ,  [_ ( K `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ph  ->  (  seq 1 (  +  ,  G ,  CC ) `  M )  =  ( 
 seq 1 (  +  ,  H ,  CC ) `  N ) )
 
Theoremisummolem2a 10660* Lemma for isummo 10662. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 3-Sep-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> DECID  k  e.  A )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   &    |-  H  =  ( n  e.  NN  |->  if ( n  <_  N ,  [_ ( K `
  n )  /  k ]_ B ,  0 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  f : ( 1 ...
 N ) -1-1-onto-> A )   &    |-  ( ph  ->  K 
 Isom  <  ,  <  (
 ( 1 ... ( `  A ) ) ,  A ) )   =>    |-  ( ph  ->  seq
 M (  +  ,  F ,  CC )  ~~>  (  seq 1 (  +  ,  G ,  CC ) `  N ) )
 
Theoremisummolem2 10661* Lemma for isummo 10662. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 8-Sep-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ( ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  ,  F ,  CC )  ~~>  x )
 )  ->  ( E. m  e.  NN  E. f
 ( f : ( 1 ... m ) -1-1-onto-> A 
 /\  y  =  ( 
 seq 1 (  +  ,  G ,  CC ) `  m ) )  ->  x  =  y )
 )
 
Theoremisummo 10662* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 10-Sep-2022.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  G  =  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )   =>    |-  ( ph  ->  E* x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  ,  F ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  x  =  ( 
 seq 1 (  +  ,  G ,  CC ) `  m ) ) ) )
 
Theoremzisum 10663* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
 0 ) )   &    |-  ( ph  ->  A. x  e.  Z DECID  x  e.  A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  (  ~~>  `  seq M (  +  ,  F ,  CC ) ) )
 
Theoremiisum 10664* Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  Z  B  =  (  ~~>  `  seq M (  +  ,  F ,  CC ) ) )
 
Theoremfisum 10665* The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 14-Sep-2022.)
 |-  ( k  =  ( F `  n ) 
 ->  B  =  C )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> A )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... M ) )  ->  ( G `
  n )  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  ( 
 seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  M ,  ( G `  n ) ,  0 )
 ) ,  CC ) `  M ) )
 
Theoremsum0 10666 Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
 |- 
 sum_ k  e.  (/)  A  =  0
 
Theoremisumz 10667* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 16-Sep-2022.)
 |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>=
 `  M )DECID  j  e.  A )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  =  0 )
 
Theoremfsumf1o 10668* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
 |-  ( k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ( ph  /\  n  e.  C ) 
 ->  ( F `  n )  =  G )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ n  e.  C  D )
 
Theoremisumss 10669* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  B 
 C_  ( ZZ>= `  M ) )   &    |-  ( ph  ->  A. j  e.  ( ZZ>= `  M )DECID  j  e.  B )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremfisumss 10670* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( B  \  A ) )  ->  C  =  0 )   &    |-  ( ph  ->  A. j  e.  B DECID  j  e.  A )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  sum_
 k  e.  A  C  =  sum_ k  e.  B  C )
 
PART 4  ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

 
4.1  Elementary properties of divisibility
 
4.1.1  The divides relation
 
Syntaxcdvds 10671 Extend the definition of a class to include the divides relation. See df-dvds 10672.
 class  ||
 
Definitiondf-dvds 10672* Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ||  =  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) }
 
Theoremdivides 10673* Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 11095). As proven in dvdsval3 10675, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 10673 and dvdsval2 10674 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <-> 
 E. n  e.  ZZ  ( n  x.  M )  =  N )
 )
 
Theoremdvdsval2 10674 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
 
Theoremdvdsval3 10675 One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  ( N  mod  M )  =  0 )
 )
 
Theoremdvdszrcl 10676 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  ( X  ||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ )
 )
 
Theoremnndivdvds 10677 Strong form of dvdsval2 10674 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  ||  A 
 <->  ( A  /  B )  e.  NN )
 )
 
Theoremnndivides 10678* Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  ||  N 
 <-> 
 E. n  e.  NN  ( n  x.  M )  =  N )
 )
 
Theoremdvdsdc 10679 Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremmoddvds 10680 Two ways to say  A  ==  B (mod  N), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  mod  N )  =  ( B 
 mod  N )  <->  N  ||  ( A  -  B ) ) )
 
Theoremdvds0lem 10681 A lemma to assist theorems of 
|| with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  x.  M )  =  N )  ->  M  ||  N )
 
Theoremdvds1lem 10682* A lemma to assist theorems of 
|| with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ph  ->  ( J  e.  ZZ  /\  K  e.  ZZ ) )   &    |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )   &    |-  ( ( ph  /\  x  e.  ZZ )  ->  Z  e.  ZZ )   &    |-  (
 ( ph  /\  x  e. 
 ZZ )  ->  (
 ( x  x.  J )  =  K  ->  ( Z  x.  M )  =  N ) )   =>    |-  ( ph  ->  ( J  ||  K  ->  M  ||  N ) )
 
Theoremdvds2lem 10683* A lemma to assist theorems of 
|| with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ph  ->  ( I  e.  ZZ  /\  J  e.  ZZ ) )   &    |-  ( ph  ->  ( K  e.  ZZ  /\  L  e.  ZZ ) )   &    |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ )
 )   &    |-  ( ( ph  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  Z  e.  ZZ )   &    |-  (
 ( ph  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( ( ( x  x.  I )  =  J  /\  ( y  x.  K )  =  L )  ->  ( Z  x.  M )  =  N ) )   =>    |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L )  ->  M  ||  N )
 )
 
Theoremiddvds 10684 An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  N  ||  N )
 
Theorem1dvds 10685 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  1  ||  N )
 
Theoremdvds0 10686 Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  N  ||  0 )
 
Theoremnegdvdsb 10687 An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  -u M  ||  N ) )
 
Theoremdvdsnegb 10688 An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  M  ||  -u N ) )
 
Theoremabsdvdsb 10689 An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  ( abs `  M )  ||  N ) )
 
Theoremdvdsabsb 10690 An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  M  ||  ( abs `  N ) ) )
 
Theorem0dvds 10691 Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( 0  ||  N  <->  N  =  0 ) )
 
Theoremzdvdsdc 10692 Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremdvdsmul1 10693 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M  x.  N ) )
 
Theoremdvdsmul2 10694 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M  x.  N ) )
 
Theoremiddvdsexp 10695 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  ||  ( M ^ N ) )
 
Theoremmuldvds1 10696 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  ||  N  ->  K 
 ||  N ) )
 
Theoremmuldvds2 10697 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  ||  N  ->  M 
 ||  N ) )
 
Theoremdvdscmul 10698 Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N ) ) )
 
Theoremdvdsmulc 10699 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K ) ) )
 
Theoremdvdscmulr 10700 Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( ( K  x.  M )  ||  ( K  x.  N ) 
 <->  M  ||  N )
 )
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