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Theorem List for Metamath Proof Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembitsfzolem 12901* Lemma for bitsfzo 12902. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits ..^              ..^

Theorembitsfzo 12902 The bits of a number are all less than iff the number is nonnegative and less than . (Contributed by Mario Carneiro, 5-Sep-2016.)
..^ bits ..^

Theorembitsmod 12903 Truncating the bit sequence after some is equivalent to reducing the argument . (Contributed by Mario Carneiro, 6-Sep-2016.)
bits bits ..^

Theorembitsfi 12904 Every number is associated to a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitscmp 12905 The bit complement of is . (Thus, by bitsfi 12904, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorem0bits 12906 The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
bits

Theoremm1bits 12907 The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsinv1lem 12908 Lemma for bitsinv1 12909. (Contributed by Mario Carneiro, 22-Sep-2016.)
bits

Theorembitsinv1 12909* There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12905), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
bits

Theorembitsinv2 12910* There is an explicit inverse to the bits function for nonnegative integers, part 2. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits

Theorembitsf1ocnv 12911* 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 12562. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits bits

Theorembitsf1o 12912 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 12562. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits

Theorembitsf1 12913 The bits function is an injection from to . It is obviously not a bijection (by Cantor's theorem canth2 7219), and in fact its range is the set of finite and cofinite subsets of . (Contributed by Mario Carneiro, 22-Sep-2016.)
bits

Theorem2ebits 12914 The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsinv 12915* The inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits

Theorembitsinvp1 12916 Recursive definition of the inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits        ..^ ..^

Theoremsadadd2lem2 12917 The core of the proof of sadadd2 12927. 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 where is the number of true arguments, which is equivalently obtained by adding together one for each true argument, on the right side. (Contributed by Mario Carneiro, 8-Sep-2016.)

Theoremsadcf 12920* The carry sequence is a sequence of elements of encoding a "sequence of wffs". (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsadc0 12921* The initial element of the carry sequence is . (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsadcp1 12922* The carry sequence (which is a sequence of wffs, encoded as 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.)

Theoremsadval 12923* The full adder sequence is the half adder function applied to the inputs and the carry sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)

cadd               bits        ..^ ..^       ..^ ..^

Theoremsadcadd 12925* Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)

Theoremsadcl 12929 The sum of two sequences is a sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsadcom 12930 The adder sequence function is commutative. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsaddisj 12932 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.)

Theoremsadadd 12934 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.)

Theoremsadid1 12935 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.)

Theoremsadid2 12936 The adder sequence function has a right identity, the empty set, which is the representation of the integer zero. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsadeq 12939 Any element of a sequence sum only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theorembitsres 12940 Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsuz 12941 The bits of a number are all at least iff the number is divisible by . (Contributed by Mario Carneiro, 21-Sep-2016.)
bits

Theorembitsshft 12942* Shifting a bit sequence to the left (toward the more significant bits) causes the number to be multiplied by a power of two. (Contributed by Mario Carneiro, 22-Sep-2016.)
bits bits

Definitiondf-smu 12943* Define the multiplication of two bit sequences, using repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmufval 12944* Define the addition of two bit sequences, using df-had 1386 and df-cad 1387 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupf 12945* The sequence of partial sums of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmup0 12946* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupp1 12947* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval 12948* Define the addition of two bit sequences, using df-had 1386 and df-cad 1387 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval2 12949* The partial sum sequence stabilizes at after the -th element of the sequence; this stable value is the value of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupvallem 12950* If only has elements less than , then all elements of the partial sum sequence past already equal the final value. (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmucl 12951 The product of two sequences is a sequence. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01lem 12952* Lemma for smu01 12953 and smu02 12954. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01 12953 Multiplication of a sequence by on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu02 12954 Multiplication of a sequence by on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
smul

Theoremsmupval 12955* Rewrite the elements of the partial sum sequence in terms of sequence multiplication. (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmup1 12956* Rewrite smupp1 12947 using only smul instead of the internal recursive function . (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmueqlem 12957* Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmueq 12958 Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)
smul ..^ ..^ smul ..^ ..^

Theoremsmumullem 12959 Lemma for smumul 12960. (Contributed by Mario Carneiro, 22-Sep-2016.)
bits ..^ smul bits bits

Theoremsmumul 12960 For sequences that correspond to valid integers, the sequence multiplication function produces the sequence for the product. This is effectively a proof of the correctness of the multiplication process, implemented in terms of logic gates for df-sad 12918, whose correctness is verified in sadadd 12934.

Outside this range, the sequences cannot be representing integers, but the smul function still "works". This extended function is best interpreted in terms of the ring structure of the 2-adic integers. (Contributed by Mario Carneiro, 22-Sep-2016.)

bits smul bits bits

6.1.6  The greatest common divisor operator

Syntaxcgcd 12961 Extend the definition of a class to include the greatest common divisor operator.

Definitiondf-gcd 12962* Define the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdval 12963* The value of the operator. is the greatest common divisor of and . If and are both , the result is defined conventionally as . (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremgcd0val 12964 The value, by convention, of the operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0val 12965* The value of the operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem1 12966* Lemma for gcdn0cl 12969, gcddvds 12970 and dvdslegcd 12971. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem2 12967* Lemma for gcdn0cl 12969, gcddvds 12970 and dvdslegcd 12971. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem3 12968* Lemma for gcdn0cl 12969, gcddvds 12970 and dvdslegcd 12971. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0cl 12969 Closure of the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcddvds 12970 The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdslegcd 12971 An integer which divides both operands of the operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcl 12972 Closure of the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcld 12973 Closure of the operator. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremgcdf 12974 Domain and codomain of the operator. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremgcdcom 12975 The operator is commutative. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdeq0 12976 The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0gt0 12977 The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcd0id 12978 The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdid0 12979 The gcd of an integer and 0 is the integer's absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0gcdid0 12980 The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdneg 12981 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremneggcd 12982 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcdaddm 12984 Adding a multiple of one operand of the operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdadd 12985 The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.)

Theoremgcdid 12986 The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcd1 12987 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremgcdabs 12988 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdabs1 12989 of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabs2 12990 of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremmodgcd 12991 The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)

Theorem1gcd 12992 The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

6.1.7  Bézout's identity

Theorembezoutlem1 12993* Lemma for bezout 12997. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem2 12994* Lemma for bezout 12997. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem3 12995* Lemma for bezout 12997. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezoutlem4 12996* Lemma for bezout 12997. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezout 12997* Bézout's identity: For any integers and , there are integers such that . (Contributed by Mario Carneiro, 22-Feb-2014.)

Theoremdvdsgcd 12998 An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremdvdsgcdb 12999 Biconditional form of dvdsgcd 12998. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdass 13000 Associative law for operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

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