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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvdsmod 12601 Any number whose mod base is divisible by a divisor of the base is also divisible by . This means that primes will also be relatively prime to the base when reduced for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremodd2np1lem 12602* Lemma for odd2np1 12603. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremodd2np1 12603* 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.)

Theoremoddm1even 12604 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoddp1even 12605 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoexpneg 12606 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)

Theorem3dvds 12607* 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.)

6.1.4  The division algorithm

Theoremdivalglem0 12608 Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem1 12609 Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem2 12610* Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem4 12611* Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem5 12612* Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem6 12613 Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem7 12614 Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem8 12615* Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem9 12616* Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem10 12617* Lemma for divalg 12618. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalg 12618* The division algorithm (theorem). Dividing an integer by a nonzero integer produces a (unique) quotient and a unique remainder . The proof does not use , or . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalgb 12619* Express the division algorithm as stated in divalg 12618 in terms of . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdivalg2 12620* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalgmod 12621* The result of the operator satisfies the requirements for the remainder in the division algorithm for a positive divisor (compare divalg2 12620 and divalgb 12619). This demonstration theorem justifies the use of to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdssub 12622 Corollary of the division algorithm. If an integer greater than divides , then it does not divide any of , ... . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsadd 12623 Corollary of the division algorithm. If an integer greater than divides , then it does not divide any of , ... . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsp1 12624 Special case of ndvdsadd 12623. If an integer greater than divides , it does not divide . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsi 12625 A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)

6.1.5  Bit sequences

Syntaxcbits 12626 Define the binary bits of an integer.
bits

Syntaxcsmu 12628 Define the sequence multiplication on bit sequences.
smul

Definitiondf-bits 12629* Define the binary bits of an integer. The expression bits means that the -th bit of is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.)
bits

Theorembitsfval 12630* Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsval 12631 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsval2 12632 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsss 12633 The set of bits of an integer is a subset of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsf 12634 The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembits0 12635 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembits0e 12636 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembits0o 12637 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsp1 12638 The -th bit of is the -th bit of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsp1e 12639 The -th bit of is the -th bit of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsp1o 12640 The -th bit of is the -th bit of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsfzolem 12641* Lemma for bitsfzo 12642. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits ..^              ..^

Theorembitsfzo 12642 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 12643 Truncating the bit sequence after some is equivalent to reducing the argument . (Contributed by Mario Carneiro, 6-Sep-2016.)
bits bits ..^

Theorembitsfi 12644 Every number is associated to a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitscmp 12645 The bit complement of is . (Thus, by bitsfi 12644, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorem0bits 12646 The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
bits

Theoremm1bits 12647 The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsinv1lem 12648 Lemma for bitsinv1 12649. (Contributed by Mario Carneiro, 22-Sep-2016.)
bits

Theorembitsinv1 12649* There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12645), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
bits

Theorembitsinv2 12650* There is an explicit inverse to the bits function for nonnegative integers, part 2. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits

Theorembitsf1ocnv 12651* 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 12302. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits bits

Theorembitsf1o 12652 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 12302. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits

Theorembitsf1 12653 The bits function is an injection from to . It is obviously not a bijection (by Cantor's theorem canth2 7030), and in fact its range is the set of finite and cofinite subsets of . (Contributed by Mario Carneiro, 22-Sep-2016.)
bits

Theorem2ebits 12654 The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsinv 12655* The inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits

Theorembitsinvp1 12656 Recursive definition of the inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
bits        ..^ ..^

Theoremsadadd2lem2 12657 The core of the proof of sadadd2 12667. 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 12660* The carry sequence is a sequence of elements of encoding a "sequence of wffs". (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsadc0 12661* The initial element of the carry sequence is . (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsadcp1 12662* 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 12663* 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 12665* Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)

Theoremsadcl 12669 The sum of two sequences is a sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsadcom 12670 The adder sequence function is commutative. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremsaddisj 12672 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 12674 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 1370 and df-cad 1371.

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 12675 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 12676 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 12679 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 12680 Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsuz 12681 The bits of a number are all at least iff the number is divisible by . (Contributed by Mario Carneiro, 21-Sep-2016.)
bits

Theorembitsshft 12682* 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 12683* Define the multiplication of two bit sequences, using repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmufval 12684* Define the addition of two bit sequences, using df-had 1370 and df-cad 1371 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupf 12685* The sequence of partial sums of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmup0 12686* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupp1 12687* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval 12688* Define the addition of two bit sequences, using df-had 1370 and df-cad 1371 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval2 12689* 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 12690* 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 12691 The product of two sequences is a sequence. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01lem 12692* Lemma for smu01 12693 and smu02 12694. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01 12693 Multiplication of a sequence by on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu02 12694 Multiplication of a sequence by on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
smul

Theoremsmupval 12695* Rewrite the elements of the partial sum sequence in terms of sequence multiplication. (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmup1 12696* Rewrite smupp1 12687 using only smul instead of the internal recursive function . (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmueqlem 12697* 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 12698 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 12699 Lemma for smumul 12700. (Contributed by Mario Carneiro, 22-Sep-2016.)
bits ..^ smul bits bits

Theoremsmumul 12700 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 12658, whose correctness is verified in sadadd 12674.

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

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