P. 123 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12201-12300   *Has distinct variable group(s)

Type	Label	Description
Statement
5.1.9  The least common multiple 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 12202). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention.
Syntax	clcm 12201	Extend the definition of a class to include the least common multiple operator.
class lcm
Definition	df-lcm 12202*	Define the lcm operator. For example, ( 6 lcm 9 ) = 1 8. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
Theorem	lcmmndc 12203	Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
Theorem	lcmval 12204*	Value of the lcm operator. ( M lcm N ) is the least common multiple of M and N. If either M or N is 0, the result is defined conventionally as 0. Contrast with df-gcd 12083 and gcdval 12099. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
Theorem	lcmcom 12205	The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
Theorem	lcm0val 12206	The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 12205 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( M e. ZZ -> ( M lcm 0 ) = 0 )
Theorem	lcmn0val 12207*	The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
Theorem	lcmcllem 12208*	Lemma for lcmn0cl 12209 and dvdslcm 12210. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } )
Theorem	lcmn0cl 12209	Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
Theorem	dvdslcm 12210	The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
Theorem	lcmledvds 12211	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.)
|- ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) <_ K ) )
Theorem	lcmeq0 12212	The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = 0 <-> ( M = 0 \/ N = 0 ) ) )
Theorem	lcmcl 12213	Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
Theorem	gcddvdslcm 12214	The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || ( M lcm N ) )
Theorem	lcmneg 12215	Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )
Theorem	neglcm 12216	Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( M lcm N ) )
Theorem	lcmabs 12217	The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
Theorem	lcmgcdlem 12218	Lemma for lcmgcd 12219 and lcmdvds 12220. Prove them for positive M, N, and K. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( M e. NN /\ N e. NN ) -> ( ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) /\ ( ( K e. NN /\ ( M || K /\ N || K ) ) -> ( M lcm N ) || K ) ) )
Theorem	lcmgcd 12219	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 ( M gcd N ) = 1. 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 12151; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12151 and https://math.stackexchange.com/a/470827 12151. This proof uses the latter to first confirm it for positive integers M and N (the "Second Proof" in the above Stack Exchange page), then shows that implies it for all nonzero integer inputs, then finally uses lcm0val 12206 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
Theorem	lcmdvds 12220	The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
Theorem	lcmid 12221	The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) )
Theorem	lcm1 12222	The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.)
|- ( M e. ZZ -> ( M lcm 1 ) = ( abs ` M ) )
Theorem	lcmgcdnn 12223	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.)
|- ( ( M e. NN /\ N e. NN ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( M x. N ) )
Theorem	lcmgcdeq 12224	Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M gcd N ) <-> ( abs ` M ) = ( abs ` N ) ) )
Theorem	lcmdvdsb 12225	Biconditional form of lcmdvds 12220. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K ) <-> ( M lcm N ) || K ) )
Theorem	lcmass 12226	Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) = ( N lcm ( M lcm P ) ) )
Theorem	3lcm2e6woprm 12227	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.)
|- ( 3 lcm 2 ) = 6
Theorem	6lcm4e12 12228	The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
|- ( 6 lcm 4 ) = ; 1 2
5.1.10  Coprimality and Euclid's lemma 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 A e. ZZ and B e. ZZ are coprime (or relatively prime) if ( A gcd B ) = 1. The equivalence of the definitions is shown by coprmgcdb 12229. The negation, i.e. two integers are not coprime, can be expressed either by ( A gcd B ) =/= 1, see ncoprmgcdne1b 12230, or equivalently by 1 < ( A gcd B ), see ncoprmgcdgt1b 12231. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 12233 (as opposed to Euclid's lemma for primes).
Theorem	coprmgcdb 12229*	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.)
|- ( ( A e. NN /\ B e. NN ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) <-> ( A gcd B ) = 1 ) )
Theorem	ncoprmgcdne1b 12230*	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.)
|- ( ( A e. NN /\ B e. NN ) -> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) <-> ( A gcd B ) =/= 1 ) )
Theorem	ncoprmgcdgt1b 12231*	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.)
|- ( ( A e. NN /\ B e. NN ) -> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) <-> 1 < ( A gcd B ) ) )
Theorem	coprmdvds1 12232	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.)
|- ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) -> ( ( I e. NN /\ I || F /\ I || G ) -> I = 1 ) )
Theorem	coprmdvds 12233	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.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M x. N ) /\ ( K gcd M ) = 1 ) -> K || N ) )
Theorem	coprmdvds2 12234	If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( M || K /\ N || K ) -> ( M x. N ) || K ) )
Theorem	mulgcddvds 12235	One half of rpmulgcd2 12236, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) )
Theorem	rpmulgcd2 12236	If M is relatively prime to N, then the GCD of K with M x. N is the product of the GCDs with M and N respectively. (Contributed by Mario Carneiro, 2-Jul-2015.)
|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd ( M x. N ) ) = ( ( K gcd M ) x. ( K gcd N ) ) )
Theorem	qredeq 12237	Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)
|- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) /\ ( M / N ) = ( P / Q ) ) -> ( M = P /\ N = Q ) )
Theorem	qredeu 12238*	Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)
|- ( A e. QQ -> E! x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) )
Theorem	rpmul 12239	If K is relatively prime to M and to N, it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K gcd M ) = 1 /\ ( K gcd N ) = 1 ) -> ( K gcd ( M x. N ) ) = 1 ) )
Theorem	rpdvds 12240	If K is relatively prime to N then it is also relatively prime to any divisor M of N. (Contributed by Mario Carneiro, 19-Jun-2015.)
|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) = 1 )
5.1.11  Cancellability of congruences
Theorem	congr 12241*	Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer A is congruent to an integer B modulo M if their difference is a multiple of M. See also the definition in [ApostolNT] p. 104: "... a is congruent to b modulo m, and we write a == b (mod m) if m divides the difference a - b", 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.)
|- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> E. n e. ZZ ( n x. M ) = ( A - B ) ) )
Theorem	divgcdcoprm0 12242	Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
Theorem	divgcdcoprmex 12243*	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.)
|- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> E. a e. ZZ E. b e. ZZ ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) )
Theorem	cncongr1 12244	One direction of the bicondition in cncongr 12246. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) ) -> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) -> ( A mod M ) = ( B mod M ) ) )
Theorem	cncongr2 12245	The other direction of the bicondition in cncongr 12246. (Contributed by AV, 11-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) ) -> ( ( A mod M ) = ( B mod M ) -> ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) ) )
Theorem	cncongr 12246	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.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) ) -> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) <-> ( A mod M ) = ( B mod M ) ) )
Theorem	cncongrcoprm 12247	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.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ ( C gcd N ) = 1 ) ) -> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) <-> ( A mod N ) = ( B mod N ) ) )
5.2  Elementary prime number theory
5.2.1  Elementary properties Remark: to represent odd prime numbers, i.e., all prime numbers except 2, the idiom P e. ( Prime \ { 2 } ) is used. It is a little bit shorter than ( P e. Prime /\ P =/= 2 ). Both representations can be converted into each other by eldifsn 3746.
Syntax	cprime 12248	Extend the definition of a class to include the set of prime numbers.
class Prime
Definition	df-prm 12249*	Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
|- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
Theorem	isprm 12250*	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.)
|- ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) )
Theorem	prmnn 12251	A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( P e. Prime -> P e. NN )
Theorem	prmz 12252	A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.)
|- ( P e. Prime -> P e. ZZ )
Theorem	prmssnn 12253	The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
|- Prime C_ NN
Theorem	prmex 12254	The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
|- Prime e. _V
Theorem	1nprm 12255	1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
|- -. 1 e. Prime
Theorem	1idssfct 12256*	The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. NN -> { 1 , N } C_ { n e. NN | n || N } )
Theorem	isprm2lem 12257*	Lemma for isprm2 12258. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( P e. NN /\ P =/= 1 ) -> ( { n e. NN | n || P } ~~ 2o <-> { n e. NN | n || P } = { 1 , P } ) )
Theorem	isprm2 12258*	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.)
|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
Theorem	isprm3 12259*	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.)
|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. ( 2 ... ( P - 1 ) ) -. z || P ) )
Theorem	isprm4 12260*	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.)
|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) ) )
Theorem	prmind2 12261*	A variation on prmind 12262 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = z -> ( ph <-> th ) )   &    |- ( x = ( y x. z ) -> ( ph <-> ta ) )   &    |- ( x = A -> ( ph <-> et ) )   &    |- ps   &    |- ( ( x e. Prime /\ A. y e. ( 1 ... ( x - 1 ) ) ch ) -> ph )   &    |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( ch /\ th ) -> ta ) )   =>    |- ( A e. NN -> et )
Theorem	prmind 12262*	Perform induction over the multiplicative structure of NN. If a property ph ( x ) holds for the primes and 1 and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = z -> ( ph <-> th ) )   &    |- ( x = ( y x. z ) -> ( ph <-> ta ) )   &    |- ( x = A -> ( ph <-> et ) )   &    |- ps   &    |- ( x e. Prime -> ph )   &    |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( ch /\ th ) -> ta ) )   =>    |- ( A e. NN -> et )
Theorem	dvdsprime 12263	If M divides a prime, then M is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)
|- ( ( P e. Prime /\ M e. NN ) -> ( M || P <-> ( M = P \/ M = 1 ) ) )
Theorem	nprm 12264	A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> -. ( A x. B ) e. Prime )
Theorem	nprmi 12265	An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
|- A e. NN   &    |- B e. NN   &    |- 1 < A   &    |- 1 < B   &    |- ( A x. B ) = N   =>    |- -. N e. Prime
Theorem	dvdsnprmd 12266	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.)
|- ( ph -> 1 < A )   &    |- ( ph -> A < N )   &    |- ( ph -> A || N )   =>    |- ( ph -> -. N e. Prime )
Theorem	prm2orodd 12267	A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
|- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
Theorem	2prm 12268	2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
|- 2 e. Prime
Theorem	3prm 12269	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|- 3 e. Prime
Theorem	4nprm 12270	4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
|- -. 4 e. Prime
Theorem	prmdc 12271	Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( N e. NN -> DECID N e. Prime )
Theorem	prmuz2 12272	A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
Theorem	prmgt1 12273	A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( P e. Prime -> 1 < P )
Theorem	prmm2nn0 12274	Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
|- ( P e. Prime -> ( P - 2 ) e. NN0 )
Theorem	oddprmgt2 12275	An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
|- ( P e. ( Prime \ { 2 } ) -> 2 < P )
Theorem	oddprmge3 12276	An odd prime is greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.)
|- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) )
Theorem	sqnprm 12277	A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )
Theorem	dvdsprm 12278	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.)
|- ( ( N e. ( ZZ>= ` 2 ) /\ P e. Prime ) -> ( N || P <-> N = P ) )
Theorem	exprmfct 12279*	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.)
|- ( N e. ( ZZ>= ` 2 ) -> E. p e. Prime p || N )
Theorem	prmdvdsfz 12280*	Each integer greater than 1 and less then or equal to a fixed number is divisible by a prime less then or equal to this fixed number. (Contributed by AV, 15-Aug-2020.)
|- ( ( N e. NN /\ I e. ( 2 ... N ) ) -> E. p e. Prime ( p <_ N /\ p || I ) )
Theorem	nprmdvds1 12281	No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
|- ( P e. Prime -> -. P || 1 )
Theorem	isprm5lem 12282*	Lemma for isprm5 12283. The interesting direction (showing that one only needs to check prime divisors up to the square root of P). (Contributed by Jim Kingdon, 20-Oct-2024.)
|- ( ph -> P e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A. z e. Prime ( ( z ^ 2 ) <_ P -> -. z || P ) )   &    |- ( ph -> X e. ( 2 ... ( P - 1 ) ) )   =>    |- ( ph -> -. X || P )
Theorem	isprm5 12283*	One need only check prime divisors of P up to sqr P in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. Prime ( ( z ^ 2 ) <_ P -> -. z || P ) ) )
Theorem	divgcdodd 12284	Either A / ( A gcd B ) is odd or B / ( A gcd B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( -. 2 || ( A / ( A gcd B ) ) \/ -. 2 || ( B / ( A gcd B ) ) ) )
5.2.2  Coprimality and Euclid's lemma (cont.) This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 12287.
Theorem	coprm 12285	A prime number either divides an integer or is coprime to it, but not both. Theorem 1.8 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
Theorem	prmrp 12286	Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ Q e. Prime ) -> ( ( P gcd Q ) = 1 <-> P =/= Q ) )
Theorem	euclemma 12287	Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. Theorem 1.9 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( ( P e. Prime /\ M e. ZZ /\ N e. ZZ ) -> ( P || ( M x. N ) <-> ( P || M \/ P || N ) ) )
Theorem	isprm6 12288*	A number is prime iff it satisfies Euclid's lemma euclemma 12287. (Contributed by Mario Carneiro, 6-Sep-2015.)
|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. x e. ZZ A. y e. ZZ ( P || ( x x. y ) -> ( P || x \/ P || y ) ) ) )
Theorem	prmdvdsexp 12289	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.)
|- ( ( P e. Prime /\ A e. ZZ /\ N e. NN ) -> ( P || ( A ^ N ) <-> P || A ) )
Theorem	prmdvdsexpb 12290	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.)
|- ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P = Q ) )
Theorem	prmdvdsexpr 12291	If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)
|- ( ( P e. Prime /\ Q e. Prime /\ N e. NN0 ) -> ( P || ( Q ^ N ) -> P = Q ) )
Theorem	prmexpb 12292	Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)
|- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( Q ^ N ) <-> ( P = Q /\ M = N ) ) )
Theorem	prmfac1 12293	The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)
|- ( ( N e. NN0 /\ P e. Prime /\ P || ( ! ` N ) ) -> P <_ N )
Theorem	rpexp 12294	If two numbers A and B are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) )
Theorem	rpexp1i 12295	Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN0 ) -> ( ( A gcd B ) = 1 -> ( ( A ^ M ) gcd B ) = 1 ) )
Theorem	rpexp12i 12296	Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ ( M e. NN0 /\ N e. NN0 ) ) -> ( ( A gcd B ) = 1 -> ( ( A ^ M ) gcd ( B ^ N ) ) = 1 ) )
Theorem	prmndvdsfaclt 12297	A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.)
|- ( ( P e. Prime /\ N e. NN0 ) -> ( N < P -> -. P || ( ! ` N ) ) )
Theorem	cncongrprm 12298	Corollary 2 of Cancellability of Congruences: Two products with a common factor are congruent modulo a prime number not dividing the common factor iff the other factors are congruent modulo the prime number. (Contributed by AV, 13-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( P e. Prime /\ -. P || C ) ) -> ( ( ( A x. C ) mod P ) = ( ( B x. C ) mod P ) <-> ( A mod P ) = ( B mod P ) ) )
Theorem	isevengcd2 12299	The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
|- ( Z e. ZZ -> ( 2 || Z <-> ( 2 gcd Z ) = 2 ) )
Theorem	isoddgcd1 12300	The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
|- ( Z e. ZZ -> ( -. 2 || Z <-> ( 2 gcd Z ) = 1 ) )