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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | gcddiv 11101 | Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | gcdmultiple 11102 | 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.) |
Theorem | gcdmultiplez 11103 | Extend gcdmultiple 11102 so can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | gcdzeq 11104 | A positive integer is equal to its gcd with an integer if and only if divides . Generalization of gcdeq 11105. (Contributed by AV, 1-Jul-2020.) |
Theorem | gcdeq 11105 | is equal to its gcd with if and only if divides . (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by AV, 8-Aug-2021.) |
Theorem | dvdssqim 11106 | Unidirectional form of dvdssq 11113. (Contributed by Scott Fenton, 19-Apr-2014.) |
Theorem | dvdsmulgcd 11107 | Relationship between the order of an element and that of a multiple. (a divisibility equivalent). (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Theorem | rpmulgcd 11108 | 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.) |
Theorem | rplpwr 11109 | If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | rppwr 11110 | If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | sqgcd 11111 | Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | dvdssqlem 11112 | Lemma for dvdssq 11113. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | dvdssq 11113 | Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | bezoutr 11114 | Partial converse to bezout 11093. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | bezoutr1 11115 | Converse of bezout 11093 for when the greater common divisor is one (sufficient condition for relative primality). (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | nn0seqcvgd 11116* | A strictly-decreasing nonnegative integer sequence with initial term reaches zero by the th term. Deduction version. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | ialgrlem1st 11117 | Lemma for ialgr0 11119. Expressing algrflemg 5977 in a form suitable for theorems such as iseq1 9840 or iseqfcl 9843. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Theorem | ialgrlemconst 11118 | Lemma for ialgr0 11119. Closure of a constant function, in a form suitable for theorems such as iseq1 9840 or iseqfcl 9843. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Theorem | ialgr0 11119 | 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.) |
Theorem | ialgrf 11120 |
An algorithm is a step 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.) |
Theorem | ialgrp1 11121 | The value of the algorithm iterator at . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Theorem | ialginv 11122* | If is an invariant of , its value is unchanged after any number of iterations of . (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | ialgcvg 11123* |
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.) |
Theorem | algcvgblem 11124 | Lemma for algcvgb 11125. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | algcvgb 11125 | 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 nonzero, 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.) |
Theorem | ialgcvga 11126* | The countdown function remains after steps. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ialgfx 11127* | If reaches a fixed point when the countdown function reaches , remains fixed after steps. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | eucalgval2 11128* | 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.) |
Theorem | eucalgval 11129* |
Euclid's Algorithm eucialg 11134 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.) |
Theorem | eucalgf 11130* | 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.) |
Theorem | eucalginv 11131* | 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.) |
Theorem | eucalglt 11132* | 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.) |
Theorem | eucialgcvga 11133* | Once Euclid's Algorithm halts after steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.) |
Theorem | eucialg 11134* |
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. Theorem
1.15 in [ApostolNT] p. 20.
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 Jim Kingdon, 11-Jan-2022.) |
According to Wikipedia ("Least common multiple", 27-Aug-2020, https://en.wikipedia.org/wiki/Least_common_multiple): "In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility." In this section, an operation calculating the least common multiple of two integers (df-lcm 11136). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention. | ||
Syntax | clcm 11135 | Extend the definition of a class to include the least common multiple operator. |
lcm | ||
Definition | df-lcm 11136* | Define the lcm operator. For example, lcm . (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmmndc 11137 | Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.) |
DECID | ||
Theorem | lcmval 11138* | Value of the lcm operator. lcm is the least common multiple of and . If either or is , the result is defined conventionally as . Contrast with df-gcd 11032 and gcdval 11044. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmcom 11139 | The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm | ||
Theorem | lcm0val 11140 | The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 11139 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmn0val 11141* | The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmcllem 11142* | Lemma for lcmn0cl 11143 and dvdslcm 11144. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmn0cl 11143 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | dvdslcm 11144 | The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmledvds 11145 | A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmeq0 11146 | The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmcl 11147 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | gcddvdslcm 11148 | The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmneg 11149 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | neglcm 11150 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmabs 11151 | The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmgcdlem 11152 | Lemma for lcmgcd 11153 and lcmdvds 11154. Prove them for positive , , and . (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm | ||
Theorem | lcmgcd 11153 |
The product of two numbers' least common multiple and greatest common
divisor is the absolute value of the product of the two numbers. In
particular, that absolute value is the least common multiple of two
coprime numbers, for which .
Multiple methods exist for proving this, and it is often proven either as a consequence of the fundamental theorem of arithmetic or of Bézout's identity bezout 11093; see e.g. https://proofwiki.org/wiki/Product_of_GCD_and_LCM and https://math.stackexchange.com/a/470827. This proof uses the latter to first confirm it for positive integers and (the "Second Proof" in the above Stack Exchange page), then shows that implies it for all nonzero integer inputs, then finally uses lcm0val 11140 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmdvds 11154 | The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmid 11155 | The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcm1 11156 | The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.) |
lcm | ||
Theorem | lcmgcdnn 11157 | The product of two positive integers' least common multiple and greatest common divisor is the product of the two integers. (Contributed by AV, 27-Aug-2020.) |
lcm | ||
Theorem | lcmgcdeq 11158 | Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmdvdsb 11159 | Biconditional form of lcmdvds 11154. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmass 11160 | Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm lcm lcm | ||
Theorem | 3lcm2e6woprm 11161 | The least common multiple of three and two is six. This proof does not use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.) |
lcm | ||
Theorem | 6lcm4e12 11162 | The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.) |
lcm ; | ||
According to Wikipedia "Coprime integers", see https://en.wikipedia.org/wiki/Coprime_integers (16-Aug-2020) "[...] two integers a and b are said to be relatively prime, mutually prime, or coprime [...] if the only positive integer (factor) that divides both of them is 1. Consequently, any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.". In the following, we use this equivalent characterization to say that and are coprime (or relatively prime) if . The equivalence of the definitions is shown by coprmgcdb 11163. The negation, i.e. two integers are not coprime, can be expressed either by , see ncoprmgcdne1b 11164, or equivalently by , see ncoprmgcdgt1b 11165. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 11167 (as opposed to Euclid's lemma for primes). | ||
Theorem | coprmgcdb 11163* | Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | ncoprmgcdne1b 11164* | Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | ncoprmgcdgt1b 11165* | Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | coprmdvds1 11166 | If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.) |
Theorem | coprmdvds 11167 | Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.) |
Theorem | coprmdvds2 11168 | If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | mulgcddvds 11169 | One half of rpmulgcd2 11170, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | rpmulgcd2 11170 | If is relatively prime to , then the GCD of with is the product of the GCDs with and respectively. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | qredeq 11171 | Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Theorem | qredeu 11172* | Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Theorem | rpmul 11173 | If is relatively prime to and to , it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | rpdvds 11174 | If is relatively prime to then it is also relatively prime to any divisor of . (Contributed by Mario Carneiro, 19-Jun-2015.) |
Theorem | congr 11175* | Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer is congruent to an integer modulo if their difference is a multiple of . See also the definition in [ApostolNT] p. 104: "... is congruent to modulo , and we write (mod ) if divides the difference ", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.) |
Theorem | divgcdcoprm0 11176 | Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.) |
Theorem | divgcdcoprmex 11177* | Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.) |
Theorem | cncongr1 11178 | One direction of the bicondition in cncongr 11180. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
Theorem | cncongr2 11179 | The other direction of the bicondition in cncongr 11180. (Contributed by AV, 11-Jul-2021.) |
Theorem | cncongr 11180 | Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
Theorem | cncongrcoprm 11181 | Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.) |
Remark: to represent odd prime numbers, i.e., all prime numbers except , the idiom is used. It is a little bit shorter than . Both representations can be converted into each other by eldifsn 3562. | ||
Syntax | cprime 11182 | Extend the definition of a class to include the set of prime numbers. |
Definition | df-prm 11183* | Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm 11184* | The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | prmnn 11185 | A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | prmz 11186 | A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.) |
Theorem | prmssnn 11187 | The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.) |
Theorem | prmex 11188 | The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) |
Theorem | 1nprm 11189 | 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
Theorem | 1idssfct 11190* | The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2lem 11191* | Lemma for isprm2 11192. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2 11192* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. Definition in [ApostolNT] p. 16. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | isprm3 11193* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | isprm4 11194* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | prmind2 11195* | A variation on prmind 11196 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | prmind 11196* | Perform induction over the multiplicative structure of . If a property holds for the primes and and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprime 11197 | If divides a prime, then is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.) |
Theorem | nprm 11198 | A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | nprmi 11199 | An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsnprmd 11200 | If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.) |
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