P. 147 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1idssfct 14601*	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 14602*	Lemma for isprm2 14603. (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 14603*	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.)
|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
Theorem	isprm3 14604*	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 14605*	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 14606*	A variation on prmind 14607 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 14607*	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 14608	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 14609	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 14610	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	2prm 14611	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 14612	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|- 3 e. Prime
Theorem	prmuz2 14613	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 14614	A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( P e. Prime -> 1 < P )
Theorem	prmn2uzge3 14615	A prime number which is not 2 is an integer greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
|- ( ( P e. Prime /\ P =/= 2 ) -> P e. ( ZZ>= ` 3 ) )
Theorem	prmm2nn0 14616	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	sqnprm 14617	A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )
Theorem	dvdsprm 14618	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 14619*	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 14620*	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 14621	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	isprm5 14622*	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	maxprmfct 14623*	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.)
|- S = { z e. Prime | z || N }   =>    |- ( N e. ( ZZ>= ` 2 ) -> ( ( S C_ ZZ /\ S =/= (/) /\ E. x e. ZZ A. y e. S y <_ x ) /\ sup ( S , RR , < ) e. S ) )
Theorem	divgcdodd 14624	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 ) ) ) )
6.2.2  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 14625. The negation, i.e. two integers are not coprime, can be expressed either by ( A gcd B ) =/= 1, see ncoprmgcdne1b 14626, or equivalently by 1 < ( A gcd B ), see ncoprmgcdgt1b 14627. The proof of Euclid's lemma, see euclemma 14636, is based on theorems about coprimality (e.g. on coprmdvds 14630).
Theorem	coprmgcdb 14625*	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 14626*	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 14627*	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	coprm 14628	A prime number either divides an integer or is coprime to it, but not both. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
Theorem	prmrp 14629	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	coprmdvds 14630	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.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M x. N ) /\ ( K gcd M ) = 1 ) -> K || N ) )
Theorem	coprmdvds2 14631	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 14632	One half of rpmulgcd2 14633, 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 14633	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 14634	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 14635*	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	euclemma 14636	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.)
|- ( ( P e. Prime /\ M e. ZZ /\ N e. ZZ ) -> ( P || ( M x. N ) <-> ( P || M \/ P || N ) ) )
Theorem	isprm6 14637*	A number is prime iff it satisfies Euclid's lemma euclemma 14636. (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 14638	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 14639	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 14640	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 14641	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 14642	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 14643	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 14644	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 14645	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	rpmul 14646	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 14647	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 )
Theorem	ncoprmlnprm 14648	If two positive integers are not coprime, the larger of them is not a prime number. (Contributed by AV, 9-Aug-2020.)
|- ( ( A e. NN /\ B e. NN /\ A < B ) -> ( 1 < ( A gcd B ) -> B e/ Prime ) )
Theorem	coprmprod 14649*	The product of the elements of a sequence of pairwise coprime positive integers is coprime to a positive integer which is coprime to all integers of the sequence. (Contributed by AV, 18-Aug-2020.)
|- ( ( ( M e. Fin /\ M C_ NN /\ N e. NN ) /\ F : NN --> NN /\ A. m e. M ( ( F ` m ) gcd N ) = 1 ) -> ( A. m e. M A. n e. ( M \ { m } ) ( ( F ` m ) gcd ( F ` n ) ) = 1 -> ( prod_ m e. M ( F ` m ) gcd N ) = 1 ) )
Theorem	coprmproddvdslem 14650*	Lemma for coprmproddvds 14651: Induction step. (Contributed by AV, 19-Aug-2020.)
|- ( ( y e. Fin /\ -. z e. y ) -> ( ( ( ( y C_ NN /\ ( K e. NN /\ F : NN --> NN ) ) /\ ( A. m e. y A. n e. ( y \ { m } ) ( ( F ` m ) gcd ( F ` n ) ) = 1 /\ A. m e. y ( F ` m ) || K ) ) -> prod_ m e. y ( F ` m ) || K ) -> ( ( ( ( y u. { z } ) C_ NN /\ ( K e. NN /\ F : NN --> NN ) ) /\ ( A. m e. ( y u. { z } ) A. n e. ( ( y u. { z } ) \ { m } ) ( ( F ` m ) gcd ( F ` n ) ) = 1 /\ A. m e. ( y u. { z } ) ( F ` m ) || K ) ) -> prod_ m e. ( y u. { z } ) ( F ` m ) || K ) ) )
Theorem	coprmproddvds 14651*	If a positive integer is divisible by each element of a set of pairwise coprime positive integers, then it is divisible by their product. (Contributed by AV, 19-Aug-2020.)
|- ( ( ( M C_ NN /\ M e. Fin ) /\ ( K e. NN /\ F : NN --> NN ) /\ ( A. m e. M A. n e. ( M \ { m } ) ( ( F ` m ) gcd ( F ` n ) ) = 1 /\ A. m e. M ( F ` m ) || K ) ) -> prod_ m e. M ( F ` m ) || K )
Theorem	3lcm2e6 14652	The least common multiple of three and two is six. The operands are unequal primes and thus coprime, so the result is (the absolute value of) their product. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 27-Aug-2020.)
|- ( 3 lcm 2 ) = 6
6.2.3  Properties of the canonical representation of a rational
Syntax	cnumer 14653	Extend class notation to include canonical numerator function.
class numer
Syntax	cdenom 14654	Extend class notation to include canonical denominator function.
class denom
Definition	df-numer 14655*	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 = ( y e. QQ |-> ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ y = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
Definition	df-denom 14656*	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 = ( y e. QQ |-> ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ y = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
Theorem	qnumval 14657*	Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> (numer ` A ) = ( 1st ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
Theorem	qdenval 14658*	Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> (denom ` A ) = ( 2nd ` ( iota_ x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) ) ) )
Theorem	qnumdencl 14659	Lemma for qnumcl 14660 and qdencl 14661. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> ( (numer ` A ) e. ZZ /\ (denom ` A ) e. NN ) )
Theorem	qnumcl 14660	The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> (numer ` A ) e. ZZ )
Theorem	qdencl 14661	The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> (denom ` A ) e. NN )
Theorem	fnum 14662	Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- numer : QQ --> ZZ
Theorem	fden 14663	Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- denom : QQ --> NN
Theorem	qnumdenbi 14664	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.)
|- ( ( A e. QQ /\ B e. ZZ /\ C e. NN ) -> ( ( ( B gcd C ) = 1 /\ A = ( B / C ) ) <-> ( (numer ` A ) = B /\ (denom ` A ) = C ) ) )
Theorem	qnumdencoprm 14665	The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> ( (numer ` A ) gcd (denom ` A ) ) = 1 )
Theorem	qeqnumdivden 14666	Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> A = ( (numer ` A ) / (denom ` A ) ) )
Theorem	qmuldeneqnum 14667	Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> ( A x. (denom ` A ) ) = (numer ` A ) )
Theorem	divnumden 14668	Calculate the reduced form of a quotient using gcd. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( (numer ` ( A / B ) ) = ( A / ( A gcd B ) ) /\ (denom ` ( A / B ) ) = ( B / ( A gcd B ) ) ) )
Theorem	divdenle 14669	Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( ( A e. ZZ /\ B e. NN ) -> (denom ` ( A / B ) ) <_ B )
Theorem	qnumgt0 14670	A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> ( 0 < A <-> 0 < (numer ` A ) ) )
Theorem	qgt0numnn 14671	A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( ( A e. QQ /\ 0 < A ) -> (numer ` A ) e. NN )
Theorem	nn0gcdsq 14672	Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
Theorem	zgcdsq 14673	nn0gcdsq 14672 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) ^ 2 ) = ( ( A ^ 2 ) gcd ( B ^ 2 ) ) )
Theorem	numdensq 14674	Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> ( (numer ` ( A ^ 2 ) ) = ( (numer ` A ) ^ 2 ) /\ (denom ` ( A ^ 2 ) ) = ( (denom ` A ) ^ 2 ) ) )
Theorem	numsq 14675	Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (numer ` ( A ^ 2 ) ) = ( (numer ` A ) ^ 2 ) )
Theorem	densq 14676	Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (denom ` ( A ^ 2 ) ) = ( (denom ` A ) ^ 2 ) )
Theorem	qden1elz 14677	A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> ( (denom ` A ) = 1 <-> A e. ZZ ) )
Theorem	zsqrtelqelz 14678	If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( ( A e. ZZ /\ ( sqr ` A ) e. QQ ) -> ( sqr ` A ) e. ZZ )
Theorem	nonsq 14679	Any integer strictly between two adjacent squares has an irrational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( ( B ^ 2 ) < A /\ A < ( ( B + 1 ) ^ 2 ) ) ) -> -. ( sqr ` A ) e. QQ )
6.2.4  Euler's theorem
Syntax	codz 14680	Extend class notation with the order function on the class of integers mod N.
class odZ
Syntax	cphi 14681	Extend class notation with the Euler phi function.
class phi
Definition	df-odz 14682*	Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- odZ = ( n e. NN |-> ( x e. { x e. ZZ | ( x gcd n ) = 1 } |-> sup ( { m e. NN | n || ( ( x ^ m ) - 1 ) } , RR , `' < ) ) )
Definition	df-phi 14683*	Define the Euler phi function, which counts the number of integers less than n and coprime to it. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- phi = ( n e. NN |-> ( # ` { x e. ( 1 ... n ) | ( x gcd n ) = 1 } ) )
Theorem	phival 14684*	Value of the Euler phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( N e. NN -> ( phi ` N ) = ( # ` { x e. ( 1 ... N ) | ( x gcd N ) = 1 } ) )
Theorem	phicl2 14685	Bounds and closure for the value of the Euler phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( N e. NN -> ( phi ` N ) e. ( 1 ... N ) )
Theorem	phicl 14686	Closure for the value of the Euler phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( N e. NN -> ( phi ` N ) e. NN )
Theorem	phibndlem 14687*	Lemma for phibnd 14688. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( N e. ( ZZ>= ` 2 ) -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } C_ ( 1 ... ( N - 1 ) ) )
Theorem	phibnd 14688	A slightly tighter bound on the value of the Euler phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( N e. ( ZZ>= ` 2 ) -> ( phi ` N ) <_ ( N - 1 ) )
Theorem	phicld 14689	Closure for the value of the Euler phi function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> N e. NN )   =>    |- ( ph -> ( phi ` N ) e. NN )
Theorem	phi1 14690	Value of the Euler phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( phi ` 1 ) = 1
Theorem	dfphi2 14691*	Alternate definition of the Euler phi function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.)
|- ( N e. NN -> ( phi ` N ) = ( # ` { x e. ( 0..^ N ) | ( x gcd N ) = 1 } ) )
Theorem	hashdvds 14692*	The number of numbers in a given residue class in a finite set of integers. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
|- ( ph -> N e. NN )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ( ZZ>= ` ( A - 1 ) ) )   &    |- ( ph -> C e. ZZ )   =>    |- ( ph -> ( # ` { x e. ( A ... B ) | N || ( x - C ) } ) = ( ( |_ ` ( ( B - C ) / N ) ) - ( |_ ` ( ( ( A - 1 ) - C ) / N ) ) ) )
Theorem	phiprmpw 14693	Value of the Euler phi function at a prime power. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- ( ( P e. Prime /\ K e. NN ) -> ( phi ` ( P ^ K ) ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )
Theorem	phiprm 14694	Value of the Euler phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
Theorem	crt 14695*	The Chinese Remainder Theorem: the function that maps x to its remainder classes mod M and mod N is 1-1 and onto when M and N are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.)
|- S = ( 0..^ ( M x. N ) )   &    |- T = ( ( 0..^ M ) X. ( 0..^ N ) )   &    |- F = ( x e. S |-> <. ( x mod M ) , ( x mod N ) >. )   &    |- ( ph -> ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) )   =>    |- ( ph -> F : S -1-1-onto-> T )
Theorem	phimullem 14696*	Lemma for phimul 14697. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- S = ( 0..^ ( M x. N ) )   &    |- T = ( ( 0..^ M ) X. ( 0..^ N ) )   &    |- F = ( x e. S |-> <. ( x mod M ) , ( x mod N ) >. )   &    |- ( ph -> ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) )   &    |- U = { y e. ( 0..^ M ) | ( y gcd M ) = 1 }   &    |- V = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- W = { y e. S | ( y gcd ( M x. N ) ) = 1 }   =>    |- ( ph -> ( phi ` ( M x. N ) ) = ( ( phi ` M ) x. ( phi ` N ) ) )
Theorem	phimul 14697	The Euler phi function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- ( ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( phi ` ( M x. N ) ) = ( ( phi ` M ) x. ( phi ` N ) ) )
Theorem	eulerthlem1 14698*	Lemma for eulerth 14700. (Contributed by Mario Carneiro, 8-May-2015.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- T = ( 1 ... ( phi ` N ) )   &    |- ( ph -> F : T -1-1-onto-> S )   &    |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )   =>    |- ( ph -> G : T --> S )
Theorem	eulerthlem2 14699*	Lemma for eulerth 14700. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- T = ( 1 ... ( phi ` N ) )   &    |- ( ph -> F : T -1-1-onto-> S )   &    |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )   =>    |- ( ph -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
Theorem	eulerth 14700	Euler's theorem, a generalization of Fermat's little theorem. If A and N are coprime, then A ^ phi ( N ) == 1, mod N. This is Metamath 100 proof #10. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )