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Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulgcd 13001 Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremabsmulgcd 13002 Distribute absolute value of multiplication over gcd. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmulgcdr 13003 Reverse distribution law for the operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcddiv 13004 Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdmultiple 13005 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 13006 Extend gcdmultiple 13005 so can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdeq 13007 is equal to its gcd with if and only if divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdssqim 13008 Unidirectional form of dvdssq 13015. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremdvdsmulgcd 13009 A divisibility equivalent for odmulg 15147. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremrpmulgcd 13010 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 13011 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremrppwr 13012 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsqgcd 13013 Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssqlem 13014 Lemma for dvdssq 13015. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssq 13015 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 13016* 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 13017 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 13018 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 13019 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 13020 The value of the algorithm iterator at . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremalginv 13021* If is an invariant of , its value is unchanged after any number of iterations of . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvg 13022* 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 13023 Lemma for algcvgb 13024. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvgb 13024 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 13025* The countdown function remains after steps. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremalgfx 13026* 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 13027* 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 13028* Euclid's Algorithm eucalg 13033 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 13029* 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 13030* 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 13031* 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 13032* 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 13033* 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 . (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 13034 Extend the definition of a class to include the set of prime numbers.

Definitiondf-prm 13035* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremisprm 13036* 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.)

Theoremprmnn 13037 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremprmz 13038 A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.)

Theorem1nprm 13039 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)

Theorem1idssfct 13040* The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremisprm2lem 13041* Lemma for isprm2 13042. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremisprm2 13042* 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. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremisprm3 13043* 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.)

Theoremisprm4 13044* 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.)

Theoremprmind2 13045* A variation on prmind 13046 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremprmind 13046* 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 natural number. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvdsprime 13047 If divides a prime, then is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)

Theoremnprm 13048 A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremnprmi 13049 An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theorem2prm 13050 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)

Theorem3prm 13051 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremprmuz2 13052 A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremsqnprm 13053 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvdsprm 13054 An integer greater than or equal to 2 divides a prime number iff it is equal to it. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremcoprm 13055 A prime number either divides an integer or is coprime to it, but not both. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremprmrp 13056 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremcoprmdvds 13057 If an integer divides the product of two integers and is coprime to one of them, then it divides the other. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremcoprmdvds2 13058 If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremmulgcddvds 13059 One half of rpmulgcd2 13060, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremrpmulgcd2 13060 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.)

Theoremqredeq 13061 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremqredeu 13062* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremeuclemma 13063 Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremisprm6 13064* A number is prime iff it satisfies Euclid's lemma euclemma 13063. (Contributed by Mario Carneiro, 6-Sep-2015.)

Theoremexprmfct 13065* Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.)

Theoremnprmdvds1 13066 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)

Theoremisprm5 13067* One need only check prime divisors of up to in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremmaxprmfct 13068* The set of prime factors of an integer greater than or equal to 2 satisfies the conditions to have a supremum, and that supremum is a member of the set. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremprmdvdsexp 13069 A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.)

Theoremprmdvdsexpb 13070 A prime divides a positive power of another iff they are equal. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 24-Feb-2014.)

Theoremprmdvdsexpr 13071 If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theoremprmexpb 13072 Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremprmfac1 13073 The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)

Theoremdivgcdodd 13074 Either is odd or is odd. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremrpexp 13075 If two numbers and are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremrpexp1i 13076 Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)

Theoremrpexp12i 13077 Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)

Theoremrpmul 13078 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.)

Theoremrpdvds 13079 If is relatively prime to then it is also relatively prime to any divisor of . (Contributed by Mario Carneiro, 19-Jun-2015.)

6.2.2  Properties of the canonical representation of a rational

Syntaxcnumer 13080 Extend class notation to include canonical numerator function.
numer

Syntaxcdenom 13081 Extend class notation to include canonical denominator function.
denom

Definitiondf-numer 13082* The canonical numerator of a rational is the numerator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Definitiondf-denom 13083* The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumval 13084* Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Theoremqdenval 13085* Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumdencl 13086 Lemma for qnumcl 13087 and qdencl 13088. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqnumcl 13087 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Theoremqdencl 13088 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremfnum 13089 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer

Theoremfden 13090 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumdenbi 13091 Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqnumdencoprm 13092 The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqeqnumdivden 13093 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremqmuldeneqnum 13094 Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom numer

Theoremdivnumden 13095 Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.)
numer denom

Theoremdivdenle 13096 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremqnumgt0 13097 A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
numer

Theoremqgt0numnn 13098 A rational is positive iff its canonical numerator is a natural number. (Contributed by Stefan O'Rear, 15-Sep-2014.)
numer

Theoremnn0gcdsq 13099 Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremzgcdsq 13100 nn0gcdsq 13099 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.)

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