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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsadid2 14101 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 14104 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 14105 Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsuz 14106 The bits of a number are all at least iff the number is divisible by . (Contributed by Mario Carneiro, 21-Sep-2016.)
bits

Theorembitsshft 14107* 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 14108* Define the multiplication of two bit sequences, using repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmufval 14109* Define the addition of two bit sequences, using df-had 1435 and df-cad 1436 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupf 14110* The sequence of partial sums of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmup0 14111* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmupp1 14112* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval 14113* Define the addition of two bit sequences, using df-had 1435 and df-cad 1436 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremsmuval2 14114* 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 14115* 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 14116 The product of two sequences is a sequence. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01lem 14117* Lemma for smu01 14118 and smu02 14119. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu01 14118 Multiplication of a sequence by on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
smul

Theoremsmu02 14119 Multiplication of a sequence by on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
smul

Theoremsmupval 14120* Rewrite the elements of the partial sum sequence in terms of sequence multiplication. (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmup1 14121* Rewrite smupp1 14112 using only smul instead of the internal recursive function . (Contributed by Mario Carneiro, 20-Sep-2016.)

Theoremsmueqlem 14122* 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 14123 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 14124 Lemma for smumul 14125. (Contributed by Mario Carneiro, 22-Sep-2016.)
bits ..^ smul bits bits

Theoremsmumul 14125 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 14083, whose correctness is verified in sadadd 14099.

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 14126 Extend the definition of a class to include the greatest common divisor operator.

Definitiondf-gcd 14127* Define the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdval 14128* 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 14129 The value, by convention, of the operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0val 14130* The value of the operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem1 14131* Lemma for gcdn0cl 14134, gcddvds 14135 and dvdslegcd 14136. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem2 14132* Lemma for gcdn0cl 14134, gcddvds 14135 and dvdslegcd 14136. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcllem3 14133* Lemma for gcdn0cl 14134, gcddvds 14135 and dvdslegcd 14136. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0cl 14134 Closure of the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcddvds 14135 The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdslegcd 14136 An integer which divides both operands of the operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcl 14137 Closure of the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdcld 14138 Closure of the operator. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremgcdf 14139 Domain and codomain of the operator. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremgcdcom 14140 The operator is commutative. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdeq0 14141 The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0gt0 14142 The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcd0id 14143 The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdid0 14144 The gcd of an integer and 0 is the integer's absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0gcdid0 14145 The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdneg 14146 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremneggcd 14147 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcdaddm 14149 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 14150 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 14151 The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcd1 14152 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremgcdabs 14153 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdabs1 14154 of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabs2 14155 of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremmodgcd 14156 The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)

Theorem1gcd 14157 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 14158* Lemma for bezout 14162. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem2 14159* Lemma for bezout 14162. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem3 14160* Lemma for bezout 14162. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezoutlem4 14161* Lemma for bezout 14162. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezout 14162* Bézout's identity: For any integers and , there are integers such that . This is Metamath 100 proof #60. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theoremdvdsgcd 14163 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 14164 Biconditional form of dvdsgcd 14163. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdass 14165 Associative law for operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremmulgcd 14166 Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremabsmulgcd 14167 Distribute absolute value of multiplication over gcd. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmulgcdr 14168 Reverse distribution law for the operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcddiv 14169 Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdmultiple 14170 The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdmultiplez 14171 Extend gcdmultiple 14170 so can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdeq 14172 is equal to its gcd with if and only if divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdssqim 14173 Unidirectional form of dvdssq 14180. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremdvdsmulgcd 14174 A divisibility equivalent for odmulg 16557. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremrpmulgcd 14175 If and are relatively prime, then the GCD of and is the GCD of and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremrplpwr 14176 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremrppwr 14177 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsqgcd 14178 Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssqlem 14179 Lemma for dvdssq 14180. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssq 14180 Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

6.1.8  Algorithms

Theoremnn0seqcvgd 14181* A strictly-decreasing nonnegative integer sequence with initial term reaches zero by the th term. Deduction version. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremseq1st 14182 A sequence whose iteration function ignores the second argument is only affected by the first point of the initial value function. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremalgr0 14183 The value of the algorithm iterator at is the initial state . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremalgrf 14184 An algorithm is step a function on a state space . An algorithm acts on an initial state by iteratively applying to give , , and so on. An algorithm is said to halt if a fixed point of is reached after a finite number of iterations.

The algorithm iterator "runs" the algorithm so that is the state after iterations of on the initial state .

Domain and codomain of the algorithm iterator . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremalgrp1 14185 The value of the algorithm iterator at . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremalginv 14186* If is an invariant of , its value is unchanged after any number of iterations of . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvg 14187* One way to prove that an algorithm halts is to construct a countdown function whose value is guaranteed to decrease for each iteration of until it reaches . That is, if is not a fixed point of , then .

If is a countdown function for algorithm , the sequence reaches after at most steps, where is the value of for the initial state . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremalgcvgblem 14188 Lemma for algcvgb 14189. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvgb 14189 Two ways of expressing that is a countdown function for algorithm . The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently non-zero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvga 14190* The countdown function remains after steps. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremalgfx 14191* If reaches a fixed point when the countdown function reaches , remains fixed after steps. (Contributed by Paul Chapman, 22-Jun-2011.)

6.1.9  Euclid's Algorithm

Theoremeucalgval2 14192* The value of the step function for Euclid's Algorithm on an ordered pair. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremeucalgval 14193* Euclid's Algorithm eucalg 14198 computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0.

The value of the step function for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremeucalgf 14194* Domain and codomain of the step function for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremeucalginv 14195* The invariant of the step function for Euclid's Algorithm is the operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)

Theoremeucalglt 14196* The second member of the state decreases with each iteration of the step function for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)

Theoremeucalgcvga 14197* Once Euclid's Algorithm halts after steps, the second element of the state remains 0 . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 29-May-2014.)

Theoremeucalg 14198* Euclid's Algorithm computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0.

Upon halting, the 1st member of the final state is equal to the gcd of the values comprising the input state . This is Metamath 100 proof #69 (greatest common divisor algorithm). (Contributed by Paul Chapman, 31-Mar-2011.) (Proof shortened by Mario Carneiro, 29-May-2014.)

6.2  Elementary prime number theory

6.2.1  Elementary properties

Syntaxcprime 14199 Extend the definition of a class to include the set of prime numbers.

Definitiondf-prm 14200* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)

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