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Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgcdn0gt0 12701 The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcd0id 12702 The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdid0 12703 The gcd of an integer and 0 is the integer's absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0gcdid0 12704 The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdneg 12705 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremneggcd 12706 Negating one operand of the operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremgcdaddm 12708 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 12709 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 12710 The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcd1 12711 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremgcdabs 12712 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremgcdabs1 12713 of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabs2 12714 of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremmodgcd 12715 The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)

Theorem1gcd 12716 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 12717* Lemma for bezout 12721. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem2 12718* Lemma for bezout 12721. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembezoutlem3 12719* Lemma for bezout 12721. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezoutlem4 12720* Lemma for bezout 12721. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorembezout 12721* Bézout's identity: For any integers and , there are integers such that . (Contributed by Mario Carneiro, 22-Feb-2014.)

Theoremdvdsgcd 12722 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 12723 Biconditional form of dvdsgcd 12722. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdass 12724 Associative law for operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremmulgcd 12725 Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremabsmulgcd 12726 Distribute absolute value of multiplication over gcd. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmulgcdr 12727 Reverse distribution law for the operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcddiv 12728 Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdmultiple 12729 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 12730 Extend gcdmultiple 12729 so can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdeq 12731 is equal to its gcd with if and only if divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdssqim 12732 Unidirectional form of dvdssq 12739. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremdvdsmulgcd 12733 A divisibility equivalent for odmulg 14869. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremrpmulgcd 12734 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 12735 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremrppwr 12736 If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsqgcd 12737 Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssqlem 12738 Lemma for dvdssq 12739. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdssq 12739 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 12740* 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 12741 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 12742 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 12743 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 12744 The value of the algorithm iterator at . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremalginv 12745* If is an invariant of , its value is unchanged after any number of iterations of . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvg 12746* 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 12747 Lemma for algcvgb 12748. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremalgcvgb 12748 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 12749* The countdown function remains after steps. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremalgfx 12750* 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 12751* 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 12752* Euclid's Algorithm eucalg 12757 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 12753* 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 12754* 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 12755* 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 12756* 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 12757* 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 12758 Extend the definition of a class to include the set of prime numbers.

Definitiondf-prm 12759* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremisprm 12760* 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 12761 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremprmz 12762 A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.)

Theorem1nprm 12763 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)

Theorem1idssfct 12764* The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremisprm2lem 12765* Lemma for isprm2 12766. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremisprm2 12766* 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 12767* 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 12768* 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 12769* A variation on prmind 12770 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremprmind 12770* 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 12771 If divides a prime, then is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)

Theoremnprm 12772 A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremnprmi 12773 An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theorem2prm 12774 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)

Theorem3prm 12775 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremprmuz2 12776 A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremsqnprm 12777 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvdsprm 12778 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 12779 A prime number either divides an integer or is coprime to it, but not both. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremprmrp 12780 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremcoprmdvds 12781 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 12782 If an integer is divisible by two coprime integers, than it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremmulgcddvds 12783 One half of rpmulgcd2 12784, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremrpmulgcd2 12784 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 12785 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremqredeu 12786* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremeuclemma 12787 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 12788* A number is prime iff it satisfies Euclid's lemma euclemma 12787. (Contributed by Mario Carneiro, 6-Sep-2015.)

Theoremexprmfct 12789* 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 12790 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)

Theoremisprm5 12791* One need only check prime divisors of up to in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremmaxprmfct 12792* 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 12793 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 12794 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 12795 If a prime divides a nonnegative power of another then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theoremprmexpb 12796 Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremprmfac1 12797 The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)

Theoremdivgcdodd 12798 Either is odd or is odd. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremrpexp 12799 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 12800 Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)

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