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Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmoddvds 12501 Two ways to say  A  ==  B
(  mod  N ). (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 12502 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 12503* 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 12504* 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 12505 An integer divides itself. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  N  ||  N )
 
Theorem1dvds 12506 1 divides any integer. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  1  ||  N )
 
Theoremdvds0 12507 Any integer divides 0. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  N  ||  0 )
 
Theoremnegdvdsb 12508 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 12509 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 12510 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 12511 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 12512 Only 0 is divisible by 0 . (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( 0  ||  N  <->  N  =  0 ) )
 
Theoremdvdsmul1 12513 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 12514 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 12515 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 12516 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 12517 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 12518 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  ->  ( K  x.  M )  ||  ( K  x.  N ) ) )
 
Theoremdvdsmulc 12519 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 12520 Cancellation law for the divides relation. (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 12521 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 )
 )
 
Theoremdvds2ln 12522 If an integer divides each of two other integers, it divides any linear combination of them. (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 12523 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 12524 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 ) ) )
 
Theoremdvdstr 12525 The divides relation is transitive. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  /\  M  ||  N )  ->  K  ||  N ) )
 
Theoremdvdsmultr1 12526 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 ) ) )
 
Theoremdvdsmultr2 12527 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 12528 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 12529 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 12530 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 12531 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 12532 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 12533 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 12534 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 ) ) )
 
Theoremfsumdvds 12535* If every term in a sum is divisible by  N, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  N  ||  B )   =>    |-  ( ph  ->  N  ||  sum_ k  e.  A  B )
 
Theoremdvdslelem 12536 Lemma for dvdsle 12537. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  M  e.  ZZ   &    |-  N  e.  NN   &    |-  K  e.  ZZ   =>    |-  ( N  <  M  ->  ( K  x.  M )  =/= 
 N )
 
Theoremdvdsle 12537 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 12538 The divisors of a nonzero integer are bounded by its absolute value. (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 ) ) )
 
Theoremdvdseq 12539 If two integers divide each other, they must be equal, up to a difference in sign. (Contributed by Mario Carneiro, 30-May-2014.)
 |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M 
 ||  N  /\  N  ||  M ) )  ->  M  =  N )
 
Theoremdvds1 12540 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( M  e.  NN0  ->  ( M  ||  1  <->  M  =  1
 ) )
 
Theoremalzdvds 12541* 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 12542* 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 12543 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 12544 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 12545 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 ) )
 
Theoremdvdsfac 12546 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
 |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>=
 `  K ) ) 
 ->  K  ||  ( ! `  N ) )
 
Theoremdvdsexp 12547 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 12548 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 ) )
 
Theoremodd2np1lem 12549* Lemma for odd2np1 12550. (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 12550* 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 ) )
 
Theoremoddm1even 12551 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 12552 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 12553 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 ) )
 
Theorem3dvds 12554* A rule for divisibility by 3 of a number written in base 10. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.)
 |-  ( ( N  e.  NN0  /\  F : ( 0
 ... N ) --> ZZ )  ->  ( 3  ||  sum_ k  e.  ( 0 ... N ) ( ( F `
  k )  x.  ( 10 ^ k
 ) )  <->  3  ||  sum_ k  e.  ( 0 ... N ) ( F `  k ) ) )
 
6.1.4  The division algorithm
 
Theoremdivalglem0 12555 Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   =>    |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) ) ) )
 
Theoremdivalglem1 12556 Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   =>    |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
 
Theoremdivalglem2 12557* Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  sup ( S ,  RR ,  `'  <  )  e.  S
 
Theoremdivalglem4 12558* Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  S  =  {
 r  e.  NN0  |  E. q  e.  ZZ  N  =  ( (
 q  x.  D )  +  r ) }
 
Theoremdivalglem5 12559* Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   &    |-  R  =  sup ( S ,  RR ,  `'  <  )   =>    |-  (
 0  <_  R  /\  R  <  ( abs `  D ) )
 
Theoremdivalglem6 12560 Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  A  e.  NN   &    |-  X  e.  ( 0 ... ( A  -  1 ) )   &    |-  K  e.  ZZ   =>    |-  ( K  =/=  0  ->  -.  ( X  +  ( K  x.  A ) )  e.  (
 0 ... ( A  -  1 ) ) )
 
Theoremdivalglem7 12561 Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  D  e.  ZZ   &    |-  D  =/=  0   =>    |-  ( ( X  e.  ( 0 ... (
 ( abs `  D )  -  1 ) ) 
 /\  K  e.  ZZ )  ->  ( K  =/=  0  ->  -.  ( X  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0
 ... ( ( abs `  D )  -  1
 ) ) ) )
 
Theoremdivalglem8 12562* Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  ( ( ( X  e.  S  /\  Y  e.  S )  /\  ( X  <  ( abs `  D )  /\  Y  <  ( abs `  D ) ) )  ->  ( K  e.  ZZ  ->  ( ( K  x.  ( abs `  D )
 )  =  ( Y  -  X )  ->  X  =  Y )
 ) )
 
Theoremdivalglem9 12563* Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   &    |-  R  =  sup ( S ,  RR ,  `'  <  )   =>    |-  E! x  e.  S  x  <  ( abs `  D )
 
Theoremdivalglem10 12564* Lemma for divalg 12565. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   &    |-  S  =  { r  e.  NN0  |  D  ||  ( N  -  r ) }   =>    |-  E! r  e. 
 ZZ  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r
 ) )
 
Theoremdivalg 12565* The division algorithm (theorem). Dividing an integer  N by a nonzero integer  D produces a (unique) quotient  q and a unique remainder  0  <_  r  <  ( abs `  D
). The proof does not use  /,  |_ or  mod. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 ) 
 ->  E! r  e.  ZZ  E. q  e.  ZZ  (
 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r )
 ) )
 
Theoremdivalgb 12566* Express the division algorithm as stated in divalg 12565 in terms of  ||. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 ) 
 ->  ( E! r  e. 
 ZZ  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r
 ) )  <->  E! r  e.  NN0  ( r  <  ( abs `  D )  /\  D  ||  ( N  -  r
 ) ) ) )
 
Theoremdivalg2 12567* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e. 
 NN0  ( r  <  D  /\  D  ||  ( N  -  r ) ) )
 
Theoremdivalgmod 12568* The result of the  mod operator satisfies the requirements for the remainder  r in the division algorithm for a positive divisor (compare divalg2 12567 and divalgb 12566). This demonstration theorem justifies the use of  mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( r  =  ( N  mod  D ) 
 <->  ( r  e.  NN0  /\  ( r  <  D  /\  D  ||  ( N  -  r ) ) ) ) )
 
Theoremndvdssub 12569 Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  -  1,  N  -  2...  N  -  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  ||  N  ->  -.  D  ||  ( N  -  K ) ) )
 
Theoremndvdsadd 12570 Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  +  1,  N  +  2...  N  +  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  ||  N  ->  -.  D  ||  ( N  +  K ) ) )
 
Theoremndvdsp1 12571 Special case of ndvdsadd 12570. If an integer  D greater than  1 divides  N, it does not divide  N  +  1. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  1  <  D ) 
 ->  ( D  ||  N  ->  -.  D  ||  ( N  +  1 )
 ) )
 
Theoremndvdsi 12572 A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  Q  e.  NN0   &    |-  R  e.  NN   &    |-  (
 ( A  x.  Q )  +  R )  =  B   &    |-  R  <  A   =>    |-  -.  A  ||  B
 
6.1.5  Bit sequences
 
Syntaxcbits 12573 Define the binary bits of an integer.
 class bits
 
Syntaxcsad 12574 Define the sequence addition on bit sequences.
 class sadd
 
Syntaxcsmu 12575 Define the sequence multiplication on bit sequences.
 class smul
 
Definitiondf-bits 12576* Define the binary bits of an integer. The expression  M  e.  (bits `  N ) means that the  M-th bit of  N is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.)
 |- bits  =  ( n  e.  ZZ  |->  { m  e.  NN0  |  -.  2  ||  ( |_ `  ( n  /  (
 2 ^ m ) ) ) } )
 
Theorembitsfval 12577* Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  (bits `  N )  =  { m  e.  NN0  |  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ m ) ) ) }
 )
 
Theorembitsval 12578 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( M  e.  (bits `  N )  <->  ( N  e.  ZZ  /\  M  e.  NN0  /\ 
 -.  2  ||  ( |_ `  ( N  /  ( 2 ^ M ) ) ) ) )
 
Theorembitsval2 12579 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( M  e.  (bits `  N )  <->  -.  2  ||  ( |_ `  ( N  /  ( 2 ^ M ) ) ) ) )
 
Theorembitsss 12580 The set of bits of an integer is a subset of  NN0. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  N )  C_ 
 NN0
 
Theorembitsf 12581 The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |- bits : ZZ --> ~P NN0
 
Theorembits0 12582 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( 0  e.  (bits `  N )  <->  -.  2  ||  N ) )
 
Theorembits0e 12583 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  -.  0  e.  (bits `  ( 2  x.  N ) ) )
 
Theorembits0o 12584 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  0  e.  (bits `  ( ( 2  x.  N )  +  1 ) ) )
 
Theorembitsp1 12585 The  M  +  1-th bit of  N is the  M-th bit of  |_ ( N  / 
2 ). (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  N )  <->  M  e.  (bits `  ( |_ `  ( N  / 
 2 ) ) ) ) )
 
Theorembitsp1e 12586 The  M  +  1-th bit of  2 N is the  M-th bit of  N. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  ( 2  x.  N ) )  <->  M  e.  (bits `  N ) ) )
 
Theorembitsp1o 12587 The  M  +  1-th bit of  2 N  +  1 is the  M-th bit of  N. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  ( (
 2  x.  N )  +  1 ) )  <->  M  e.  (bits `  N ) ) )
 
Theorembitsfzolem 12588* Lemma for bitsfzo 12589. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  (bits `  N )  C_  (
 0..^ M ) )   &    |-  S  =  sup ( { n  e.  NN0  |  N  <  ( 2 ^ n ) } ,  RR ,  `'  <  )   =>    |-  ( ph  ->  N  e.  ( 0..^ ( 2 ^ M ) ) )
 
Theorembitsfzo 12589 The bits of a number are all less than  M iff the number is nonnegative and less than  2 ^ M. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( N  e.  ( 0..^ ( 2 ^ M ) )  <->  (bits `  N )  C_  ( 0..^ M ) ) )
 
Theorembitsmod 12590 Truncating the bit sequence after some  M is equivalent to reducing the argument  mod  2 ^ M. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  (bits `  ( N  mod  ( 2 ^ M ) ) )  =  ( (bits `  N )  i^i  ( 0..^ M ) ) )
 
Theorembitsfi 12591 Every number is associated to a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  N )  e. 
 Fin )
 
Theorembitscmp 12592 The bit complement of  N is  -u N  - 
1. (Thus, by bitsfi 12591, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( NN0  \  (bits `  N ) )  =  (bits `  ( -u N  -  1 ) ) )
 
Theorem0bits 12593 The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  (bits `  0 )  =  (/)
 
Theoremm1bits 12594 The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  -u 1 )  =  NN0
 
Theorembitsinv1lem 12595 Lemma for bitsinv1 12596. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( N  mod  ( 2 ^ ( M  +  1 )
 ) )  =  ( ( N  mod  (
 2 ^ M ) )  +  if ( M  e.  (bits `  N ) ,  ( 2 ^ M ) ,  0 ) ) )
 
Theorembitsinv1 12596* There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12592), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ n  e.  (bits `  N ) ( 2 ^ n )  =  N )
 
Theorembitsinv2 12597* There is an explicit inverse to the bits function for nonnegative integers, part 2. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( A  e.  ( ~P NN0  i^i  Fin )  ->  (bits `  sum_ n  e.  A  ( 2 ^ n ) )  =  A )
 
Theorembitsf1ocnv 12598* The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 12252. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( (bits  |`  NN0 ) : NN0
 -1-1-onto-> ( ~P NN0  i^i  Fin )  /\  `' (bits  |`  NN0 )  =  ( x  e.  ( ~P
 NN0  i^i  Fin )  |->  sum_ n  e.  x  ( 2 ^ n ) ) )
 
Theorembitsf1o 12599 The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 12252. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )
 
Theorembitsf1 12600 The bits function is an injection from  ZZ to  ~P NN0. It is obviously not a bijection (by Cantor's theorem canth2 6982), and in fact its range is the set of finite and cofinite subsets of  NN0. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |- bits : ZZ -1-1-> ~P NN0
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