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Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfsump1 10701* The addition of the next term in a finite sum of  A ( k ) is the current term plus  B i.e.  A ( N  +  1 ). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... ( N  +  1 )
 ) )  ->  A  e.  CC )   &    |-  ( k  =  ( N  +  1 )  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  ( M ... ( N  +  1 )
 ) A  =  (
 sum_ k  e.  ( M ... N ) A  +  B ) )
 
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 10702 Extend the definition of a class to include the divides relation. See df-dvds 10703.
 class  ||
 
Definitiondf-dvds 10703* 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 10704* Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 11126). As proven in dvdsval3 10706, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 10704 and dvdsval2 10705 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 10705 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 10706 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 10707 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  ( X  ||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ )
 )
 
Theoremnndivdvds 10708 Strong form of dvdsval2 10705 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  ||  A 
 <->  ( A  /  B )  e.  NN )
 )
 
Theoremnndivides 10709* 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 10710 Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremmoddvds 10711 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 10712 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 10713* 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 10714* 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 10715 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 10716 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 10717 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 10718 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 10719 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 10720 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 10721 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 10722 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 10723 Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremdvdsmul1 10724 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 10725 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 10726 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 10727 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 10728 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 10729 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 10730 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 10731 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 )
 )
 
Theoremdvdsmulcr 10732 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( ( M  x.  K )  ||  ( N  x.  K ) 
 <->  M  ||  N )
 )
 
Theoremsummodnegmod 10733 The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A  +  B )  mod  N )  =  0  <->  ( A  mod  N )  =  ( -u B  mod  N ) ) )
 
Theoremmodmulconst 10734 Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( ( A  mod  M )  =  ( B  mod  M )  <->  ( ( C  x.  A )  mod  ( C  x.  M ) )  =  (
 ( C  x.  B )  mod  ( C  x.  M ) ) ) )
 
Theoremdvds2ln 10735 If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( I  e.  ZZ  /\  J  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )  ->  (
 ( K  ||  M  /\  K  ||  N )  ->  K  ||  ( ( I  x.  M )  +  ( J  x.  N ) ) ) )
 
Theoremdvds2add 10736 If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  +  N )
 ) )
 
Theoremdvds2sub 10737 If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  -  N ) ) )
 
Theoremdvds2subd 10738 Natural deduction form of dvds2sub 10737. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  K 
 ||  M )   &    |-  ( ph  ->  K  ||  N )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  K 
 ||  ( M  -  N ) )
 
Theoremdvdstr 10739 The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  /\  M  ||  N )  ->  K  ||  N ) )
 
Theoremdvdsmultr1 10740 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  M  ->  K  ||  ( M  x.  N ) ) )
 
Theoremdvdsmultr1d 10741 Natural deduction form of dvdsmultr1 10740. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  K  ||  M )   =>    |-  ( ph  ->  K  ||  ( M  x.  N ) )
 
Theoremdvdsmultr2 10742 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  ->  K  ||  ( M  x.  N ) ) )
 
Theoremordvdsmul 10743 If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  \/  K  ||  N )  ->  K  ||  ( M  x.  N ) ) )
 
Theoremdvdssub2 10744 If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K 
 ||  M  <->  K  ||  N ) )
 
Theoremdvdsadd 10745 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  M  ||  ( M  +  N ) ) )
 
Theoremdvdsaddr 10746 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  M  ||  ( N  +  M ) ) )
 
Theoremdvdssub 10747 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  M  ||  ( M  -  N ) ) )
 
Theoremdvdssubr 10748 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  M  ||  ( N  -  M ) ) )
 
Theoremdvdsadd2b 10749 Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C )
 )  ->  ( A  ||  B  <->  A  ||  ( C  +  B ) ) )
 
Theoremdvdslelemd 10750 Lemma for dvdsle 10751. (Contributed by Jim Kingdon, 8-Nov-2021.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  N  <  M )   =>    |-  ( ph  ->  ( K  x.  M )  =/= 
 N )
 
Theoremdvdsle 10751 The divisors of a positive integer are bounded by it. The proof does not use  /. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  ||  N  ->  M  <_  N ) )
 
Theoremdvdsleabs 10752 The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  ( M  ||  N  ->  M  <_  ( abs `  N ) ) )
 
Theoremdvdsleabs2 10753 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 ) 
 ->  ( M  ||  N  ->  ( abs `  M )  <_  ( abs `  N ) ) )
 
Theoremdvdsabseq 10754 If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)
 |-  ( ( M  ||  N  /\  N  ||  M )  ->  ( abs `  M )  =  ( abs `  N ) )
 
Theoremdvdseq 10755 If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.)
 |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M 
 ||  N  /\  N  ||  M ) )  ->  M  =  N )
 
Theoremdivconjdvds 10756 If a nonzero integer  M divides another integer  N, the other integer  N divided by the nonzero integer  M (i.e. the divisor conjugate of  N to  M) divides the other integer  N. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
 |-  ( ( M  ||  N  /\  M  =/=  0
 )  ->  ( N  /  M )  ||  N )
 
Theoremdvdsdivcl 10757* The complement of a divisor of  N is also a divisor of  N. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)
 |-  ( ( N  e.  NN  /\  A  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  A )  e.  { x  e. 
 NN  |  x  ||  N } )
 
Theoremdvdsflip 10758* An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)
 |-  A  =  { x  e.  NN  |  x  ||  N }   &    |-  F  =  ( y  e.  A  |->  ( N  /  y ) )   =>    |-  ( N  e.  NN  ->  F : A -1-1-onto-> A )
 
Theoremdvdsssfz1 10759* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( A  e.  NN  ->  { p  e.  NN  |  p  ||  A }  C_  ( 1 ... A ) )
 
Theoremdvds1 10760 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( M  e.  NN0  ->  ( M  ||  1  <->  M  =  1
 ) )
 
Theoremalzdvds 10761* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( A. x  e. 
 ZZ  x  ||  N  <->  N  =  0 ) )
 
Theoremdvdsext 10762* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  =  B  <->  A. x  e.  NN0  ( A  ||  x  <->  B  ||  x ) ) )
 
Theoremfzm1ndvds 10763 No number between  1 and  M  - 
1 divides  M. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( ( M  e.  NN  /\  N  e.  (
 1 ... ( M  -  1 ) ) ) 
 ->  -.  M  ||  N )
 
Theoremfzo0dvdseq 10764 Zero is the only one of the first 
A nonnegative integers that is divisible by  A. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( B  e.  (
 0..^ A )  ->  ( A  ||  B  <->  B  =  0
 ) )
 
Theoremfzocongeq 10765 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  ( C..^ D )  /\  B  e.  ( C..^ D ) )  ->  ( ( D  -  C )  ||  ( A  -  B )  <->  A  =  B ) )
 
TheoremaddmodlteqALT 10766 Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 9736 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N )  /\  S  e.  ZZ )  ->  ( ( ( I  +  S )  mod  N )  =  ( ( J  +  S ) 
 mod  N )  <->  I  =  J ) )
 
Theoremdvdsfac 10767 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
 |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>=
 `  K ) ) 
 ->  K  ||  ( ! `  N ) )
 
Theoremdvdsexp 10768 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  ->  ( A ^ M ) 
 ||  ( A ^ N ) )
 
Theoremdvdsmod 10769 Any number  K whose mod base  N is divisible by a divisor  P of the base is also divisible by 
P. This means that primes will also be relatively prime to the base when reduced  mod 
N for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P 
 ||  ( K  mod  N )  <->  P  ||  K ) )
 
Theoremmulmoddvds 10770 If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( N  ||  A  ->  ( ( A  x.  B )  mod  N )  =  0 ) )
 
Theorem3dvdsdec 10771 A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if  A and  B actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers  A and  B, especially if  A is itself a decimal number, e.g.  A  = ; C D. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |-  ( 3  || ; A B  <->  3  ||  ( A  +  B )
 )
 
Theorem3dvds2dec 10772 A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if  A,  B and  C actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers  A,  B and  C. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   =>    |-  ( 3  || ;; A B C  <->  3  ||  (
 ( A  +  B )  +  C )
 )
 
4.1.2  Even and odd numbers

The set  ZZ of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 10776. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom  2 
||  N to say that " N is even" (which implies  N  e.  ZZ, see evenelz 10773) and  -.  2  ||  N to say that " N is odd" (under the assumption that  N  e.  ZZ). The previously proven theorems about even and odd numbers, like zneo 8783, zeo 8787, zeo2 8788, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 10795 and oddp1d2 10796. The corresponding theorems are zeneo 10777, zeo3 10774 and zeo4 10776.

 
Theoremevenelz 10773 An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 10707. (Contributed by AV, 22-Jun-2021.)
 |-  ( 2  ||  N  ->  N  e.  ZZ )
 
Theoremzeo3 10774 An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  \/  -.  2  ||  N ) )
 
Theoremzeoxor 10775 An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  \/_  -.  2  ||  N ) )
 
Theoremzeo4 10776 An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  <->  -. 
 -.  2  ||  N ) )
 
Theoremzeneo 10777 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 8783 follows immediately from the fact that a contradiction implies anything, see pm2.21i 608. (Contributed by AV, 22-Jun-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2 
 ||  A  /\  -.  2  ||  B )  ->  A  =/=  B ) )
 
Theoremodd2np1lem 10778* Lemma for odd2np1 10779. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( N  e.  NN0  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N  \/  E. k  e.  ZZ  (
 k  x.  2 )  =  N ) )
 
Theoremodd2np1 10779* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <-> 
 E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
 
Theoremeven2n 10780* An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)
 |-  ( 2  ||  N  <->  E. n  e.  ZZ  (
 2  x.  n )  =  N )
 
Theoremoddm1even 10781 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  2  ||  ( N  -  1 ) ) )
 
Theoremoddp1even 10782 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  2  ||  ( N  +  1 ) ) )
 
Theoremoexpneg 10783 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)
 |-  ( ( A  e.  CC  /\  N  e.  NN  /\ 
 -.  2  ||  N )  ->  ( -u A ^ N )  =  -u ( A ^ N ) )
 
Theoremmod2eq0even 10784 An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
 |-  ( N  e.  ZZ  ->  ( ( N  mod  2 )  =  0  <->  2 
 ||  N ) )
 
Theoremmod2eq1n2dvds 10785 An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.)
 |-  ( N  e.  ZZ  ->  ( ( N  mod  2 )  =  1  <->  -.  2  ||  N )
 )
 
Theoremoddnn02np1 10786* A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)
 |-  ( N  e.  NN0  ->  ( -.  2  ||  N  <->  E. n  e.  NN0  (
 ( 2  x.  n )  +  1 )  =  N ) )
 
Theoremoddge22np1 10787* An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( -.  2  ||  N  <->  E. n  e.  NN  (
 ( 2  x.  n )  +  1 )  =  N ) )
 
Theoremevennn02n 10788* A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)
 |-  ( N  e.  NN0  ->  ( 2  ||  N  <->  E. n  e.  NN0  (
 2  x.  n )  =  N ) )
 
Theoremevennn2n 10789* A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)
 |-  ( N  e.  NN  ->  ( 2  ||  N  <->  E. n  e.  NN  (
 2  x.  n )  =  N ) )
 
Theorem2tp1odd 10790 A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  =  ( ( 2  x.  A )  +  1 )
 )  ->  -.  2  ||  B )
 
Theoremmulsucdiv2z 10791 An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.)
 |-  ( N  e.  ZZ  ->  ( ( N  x.  ( N  +  1
 ) )  /  2
 )  e.  ZZ )
 
Theoremsqoddm1div8z 10792 A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)
 |-  ( ( N  e.  ZZ  /\  -.  2  ||  N )  ->  ( ( ( N ^ 2
 )  -  1 ) 
 /  8 )  e. 
 ZZ )
 
Theorem2teven 10793 A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  =  ( 2  x.  A ) )  ->  2  ||  B )
 
Theoremzeo5 10794 An integer is either even or odd, version of zeo3 10774 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  \/  2  ||  ( N  +  1 ) ) )
 
Theoremevend2 10795 An integer is even iff its quotient with 2 is an integer. This is a representation of even numbers without using the divides relation, see zeo 8787 and zeo2 8788. (Contributed by AV, 22-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  <->  ( N  /  2 )  e.  ZZ ) )
 
Theoremoddp1d2 10796 An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 8787 and zeo2 8788. (Contributed by AV, 22-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  ( ( N  +  1 )  /  2
 )  e.  ZZ )
 )
 
Theoremzob 10797 Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( ( ( N  +  1 )  / 
 2 )  e.  ZZ  <->  (
 ( N  -  1
 )  /  2 )  e.  ZZ ) )
 
Theoremoddm1d2 10798 An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  ( ( N  -  1 )  /  2
 )  e.  ZZ )
 )
 
Theoremltoddhalfle 10799 An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.)
 |-  ( ( N  e.  ZZ  /\  -.  2  ||  N  /\  M  e.  ZZ )  ->  ( M  <  ( N  /  2 )  <->  M  <_  ( ( N  -  1 )  / 
 2 ) ) )
 
Theoremhalfleoddlt 10800 An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.)
 |-  ( ( N  e.  ZZ  /\  -.  2  ||  N  /\  M  e.  ZZ )  ->  ( ( N 
 /  2 )  <_  M 
 <->  ( N  /  2
 )  <  M )
 )
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