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Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvds2add 12801 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 12802 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 12803 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 12804 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 12805 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 12806 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 12807 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 12808 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 12809 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 12810 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 12811 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 12812 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 12813* 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 12814 Lemma for dvdsle 12815. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  M  e.  ZZ   &    |-  N  e.  NN   &    |-  K  e.  ZZ   =>    |-  ( N  <  M  ->  ( K  x.  M )  =/= 
 N )
 
Theoremdvdsle 12815 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 12816 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 12817 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 12818 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( M  e.  NN0  ->  ( M  ||  1  <->  M  =  1
 ) )
 
Theoremalzdvds 12819* 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 12820* 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 12821 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 12822 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 12823 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 12824 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
 |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>=
 `  K ) ) 
 ->  K  ||  ( ! `  N ) )
 
Theoremdvdsexp 12825 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 12826 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 12827* Lemma for odd2np1 12828. (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 12828* 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 12829 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 12830 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 12831 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 12832* 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 12833 Lemma for divalg 12843. (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 12834 Lemma for divalg 12843. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  N  e.  ZZ   &    |-  D  e.  ZZ   &    |-  D  =/=  0   =>    |-  0  <_  ( N  +  ( abs `  ( N  x.  D ) ) )
 
Theoremdivalglem2 12835* Lemma for divalg 12843. (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 12836* Lemma for divalg 12843. (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 12837* Lemma for divalg 12843. (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 12838 Lemma for divalg 12843. (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 12839 Lemma for divalg 12843. (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 12840* Lemma for divalg 12843. (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 12841* Lemma for divalg 12843. (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 12842* Lemma for divalg 12843. (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 12843* 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 12844* Express the division algorithm as stated in divalg 12843 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 12845* 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 12846* The result of the  mod operator satisfies the requirements for the remainder  r in the division algorithm for a positive divisor (compare divalg2 12845 and divalgb 12844). 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 12847 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 12848 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 12849 Special case of ndvdsadd 12848. 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 12850 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 12851 Define the binary bits of an integer.
 class bits
 
Syntaxcsad 12852 Define the sequence addition on bit sequences.
 class sadd
 
Syntaxcsmu 12853 Define the sequence multiplication on bit sequences.
 class smul
 
Definitiondf-bits 12854* 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 12855* 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 12856 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 12857 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 12858 The set of bits of an integer is a subset of  NN0. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  N )  C_ 
 NN0
 
Theorembitsf 12859 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 12860 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( 0  e.  (bits `  N )  <->  -.  2  ||  N ) )
 
Theorembits0e 12861 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 12862 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 12863 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 12864 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 12865 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 12866* Lemma for bitsfzo 12867. (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 12867 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 12868 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 12869 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 12870 The bit complement of  N is  -u N  - 
1. (Thus, by bitsfi 12869, 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 12871 The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  (bits `  0 )  =  (/)
 
Theoremm1bits 12872 The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  -u 1 )  =  NN0
 
Theorembitsinv1lem 12873 Lemma for bitsinv1 12874. (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 12874* There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12870), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ n  e.  (bits `  N ) ( 2 ^ n )  =  N )
 
Theorembitsinv2 12875* 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 12876* 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 12527. (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 12877 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 12527. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )
 
Theorembitsf1 12878 The bits function is an injection from  ZZ to  ~P NN0. It is obviously not a bijection (by Cantor's theorem canth2 7189), 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
 
Theorem2ebits 12879 The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  ( 2 ^ N ) )  =  { N } )
 
Theorembitsinv 12880* The inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  K  =  `' (bits  |` 
 NN0 )   =>    |-  ( A  e.  ( ~P NN0  i^i  Fin )  ->  ( K `  A )  =  sum_ k  e.  A  ( 2 ^ k
 ) )
 
Theorembitsinvp1 12881 Recursive definition of the inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  K  =  `' (bits  |` 
 NN0 )   =>    |-  ( ( A  C_  NN0  /\  N  e.  NN0 )  ->  ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  if ( N  e.  A ,  (
 2 ^ N ) ,  0 ) ) )
 
Theoremsadadd2lem2 12882 The core of the proof of sadadd2 12892. The intuitive justification for this is that cadd is true if at least two arguments are true, and hadd is true if an odd number of arguments are true, so altogether the result is  n  x.  A where  n is the number of true arguments, which is equivalently obtained by adding together one  A for each true argument, on the right side. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( A  e.  CC  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A , 
 0 )  +  if (cadd ( ph ,  ps ,  ch ) ,  (
 2  x.  A ) ,  0 ) )  =  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  if ( ch ,  A ,  0 ) ) )
 
Definitiondf-sad 12883* Define the addition of two bit sequences, using df-had 1386 and df-cad 1387 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |- sadd  =  ( x  e.  ~P NN0
 ,  y  e.  ~P NN0  |->  { k  e.  NN0  | hadd ( k  e.  x ,  k  e.  y ,  (/)  e.  (  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  x ,  m  e.  y ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) ) `  k ) ) } )
 
Theoremsadfval 12884* Define the addition of two bit sequences, using df-had 1386 and df-cad 1387 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  ( A sadd  B )  =  {
 k  e.  NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
  k ) ) } )
 
Theoremsadcf 12885* The carry sequence is a sequence of elements of  2o encoding a "sequence of wffs". (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  C : NN0 --> 2o )
 
Theoremsadc0 12886* The initial element of the carry sequence is  F.. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  -.  (/)  e.  ( C `  0 ) )
 
Theoremsadcp1 12887* The carry sequence (which is a sequence of wffs, encoded as  1o and  (/)) is defined recursively as the carry operation applied to the previous carry and the two current inputs. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  ( N  +  1
 ) )  <-> cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
  N ) ) ) )
 
Theoremsadval 12888* The full adder sequence is the half adder function applied to the inputs and the carry sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A sadd  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
  N ) ) ) )
 
Theoremsadcaddlem 12889* Lemma for sadcadd 12890. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  ( N  +  1
 ) )  <->  ( 2 ^
 ( N  +  1 ) )  <_  (
 ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( N  +  1 ) ) ) ) ) ) )
 
Theoremsadcadd 12890* Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) )
 
Theoremsadadd2lem 12891* Lemma for sadadd2 12892. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  ( ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
  N ) ,  ( 2 ^ N ) ,  0 )
 )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  if ( (/)  e.  ( C `
  ( N  +  1 ) ) ,  ( 2 ^ ( N  +  1 )
 ) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( N  +  1 ) ) ) ) ) )
 
Theoremsadadd2 12892* Sum of initial segments of the sadd sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
  N ) ,  ( 2 ^ N ) ,  0 )
 )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) )
 
Theoremsadadd3 12893* Sum of initial segments of the sadd sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  mod  (
 2 ^ N ) )  =  ( ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) 
 mod  ( 2 ^ N ) ) )
 
Theoremsadcl 12894 The sum of two sequences is a sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A sadd  B )  C_  NN0 )
 
Theoremsadcom 12895 The adder sequence function is commutative. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A sadd  B )  =  ( B sadd  A ) )
 
Theoremsaddisjlem 12896* Lemma for sadadd 12899. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A sadd  B )  <->  N  e.  ( A  u.  B ) ) )
 
Theoremsaddisj 12897 The sum of disjoint sequences is the union of the sequences. (In this case, there are no carried bits.) (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( A sadd  B )  =  ( A  u.  B ) )
 
Theoremsadaddlem 12898* Lemma for sadadd 12899. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  C  =  seq  0
 ( ( c  e. 
 2o ,  m  e. 
 NN0  |->  if (cadd ( m  e.  (bits `  A ) ,  m  e.  (bits `  B ) ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e. 
 NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( (bits `  A ) sadd  (bits `  B )
 )  i^i  ( 0..^ N ) )  =  (bits `  ( ( A  +  B )  mod  ( 2 ^ N ) ) ) )
 
Theoremsadadd 12899 For sequences that correspond to valid integers, the adder sequence function produces the sequence for the sum. This is effectively a proof of the correctness of the ripple carry adder, implemented with logic gates corresponding to df-had 1386 and df-cad 1387.

It is interesting to consider in what sense the sadd function can be said to be "adding" things outside the range of the bits function, that is, when adding sequences that are not eventually constant and so do not denote any integer. The correct interpretation is that the sequences are representations of 2-adic integers, which have a natural ring structure. (Contributed by Mario Carneiro, 9-Sep-2016.)

 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( (bits `  A ) sadd  (bits `  B ) )  =  (bits `  ( A  +  B ) ) )
 
Theoremsadid1 12900 The adder sequence function has a left identity, the empty set, which is the representation of the integer zero. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( A sadd  (/) )  =  A )
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