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Theorem List for Intuitionistic Logic Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-dvds 11501* 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 11502* Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 12972). As proven in dvdsval3 11504, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 11502 and dvdsval2 11503 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 11503 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 11504 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 11505 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  ( X  ||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ )
 )
 
Theoremnndivdvds 11506 Strong form of dvdsval2 11503 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  ||  A 
 <->  ( A  /  B )  e.  NN )
 )
 
Theoremnndivides 11507* 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 11508 Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremmoddvds 11509 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 11510 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 11511* 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 11512* 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 11513 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 11514 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 11515 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 11516 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 11517 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 11518 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 11519 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 11520 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 11521 Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremdvdsmul1 11522 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 11523 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 11524 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 11525 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 11526 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 11527 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 11528 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 11529 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 11530 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 11531 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 11532 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 11533 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 11534 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 11535 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 11536 Natural deduction form of dvds2sub 11535. (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 11537 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 11538 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 11539 Natural deduction form of dvdsmultr1 11538. (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 11540 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 11541 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 11542 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 11543 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 11544 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 11545 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 11546 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 11547 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 11548 Lemma for dvdsle 11549. (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 11549 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 11550 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 11551 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 11552 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 11553 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 11554 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 11555* 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 11556* 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 11557* 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 11558 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( M  e.  NN0  ->  ( M  ||  1  <->  M  =  1
 ) )
 
Theoremalzdvds 11559* 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 11560* 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 11561 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 11562 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 11563 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 11564 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 10178 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 11565 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
 |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>=
 `  K ) ) 
 ->  K  ||  ( ! `  N ) )
 
Theoremdvdsexp 11566 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 11567 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 11568 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 11569 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 11570 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 )
 )
 
5.1.2  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 11574. 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 11571) 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 9159, zeo 9163, zeo2 9164, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 11593 and oddp1d2 11594. The corresponding theorems are zeneo 11575, zeo3 11572 and zeo4 11574.

 
Theoremevenelz 11571 An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11505. (Contributed by AV, 22-Jun-2021.)
 |-  ( 2  ||  N  ->  N  e.  ZZ )
 
Theoremzeo3 11572 An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  \/  -.  2  ||  N ) )
 
Theoremzeoxor 11573 An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  \/_  -.  2  ||  N ) )
 
Theoremzeo4 11574 An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  <->  -. 
 -.  2  ||  N ) )
 
Theoremzeneo 11575 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 9159 follows immediately from the fact that a contradiction implies anything, see pm2.21i 635. (Contributed by AV, 22-Jun-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2 
 ||  A  /\  -.  2  ||  B )  ->  A  =/=  B ) )
 
Theoremodd2np1lem 11576* Lemma for odd2np1 11577. (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 11577* 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 11578* 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 11579 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 11580 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 11581 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 11582 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 11583 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 11584* 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 11585* 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 11586* 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 11587* 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 11588 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 11589 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 11590 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 11591 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 11592 An integer is either even or odd, version of zeo3 11572 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 11593 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 9163 and zeo2 9164. (Contributed by AV, 22-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 2  ||  N  <->  ( N  /  2 )  e.  ZZ ) )
 
Theoremoddp1d2 11594 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 9163 and zeo2 9164. (Contributed by AV, 22-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( -.  2  ||  N 
 <->  ( ( N  +  1 )  /  2
 )  e.  ZZ )
 )
 
Theoremzob 11595 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 11596 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 11597 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 11598 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 )
 )
 
Theoremopoe 11599 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\ 
 -.  2  ||  B ) )  ->  2  ||  ( A  +  B ) )
 
Theoremomoe 11600 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\ 
 -.  2  ||  B ) )  ->  2  ||  ( A  -  B ) )
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