P. 128 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12701-12800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nprmi 12701	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 12702	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 12703	A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
|- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
Theorem	2prm 12704	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 12705	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|- 3 e. Prime
Theorem	4nprm 12706	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 12707	Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( N e. NN -> DECID N e. Prime )
Theorem	prmuz2 12708	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 12709	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 12710	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 12711	An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
|- ( P e. ( Prime \ { 2 } ) -> 2 < P )
Theorem	oddprmge3 12712	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 12713	A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )
Theorem	dvdsprm 12714	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 12715*	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 12716*	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 12717	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 12718*	Lemma for isprm5 12719. 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 12719*	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 12720	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 12723.
Theorem	coprm 12721	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 12722	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 12723	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 12724*	A number is prime iff it satisfies Euclid's lemma euclemma 12723. (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 12725	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 12726	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 12727	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 12728	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 12729	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 12730	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 12731	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 12732	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 12733	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 12734	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 12735	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 12736	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 ) )
Theorem	3lcm2e6 12737	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
5.2.3  Non-rationality of square root of 2
Theorem	sqrt2irrlem 12738	Lemma for sqrt2irr 12739. This is the core of the proof: - if A / B = sqr ( 2 ), then A and B are even, so A / 2 and B / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. NN )   &    |- ( ph -> ( sqr ` 2 ) = ( A / B ) )   =>    |- ( ph -> ( ( A / 2 ) e. ZZ /\ ( B / 2 ) e. NN ) )
Theorem	sqrt2irr 12739	The square root of 2 is not rational. That is, for any rational number, ( sqr ` 2 ) does not equal it. However, if we were to say "the square root of 2 is irrational" that would mean something stronger: "for any rational number, ( sqr ` 2 ) is apart from it" (the two statements are equivalent given excluded middle). See sqrt2irrap 12757 for the proof that the square root of two is irrational. The proof's core is proven in sqrt2irrlem 12738, which shows that if A / B = sqr ( 2 ), then A and B are even, so A / 2 and B / 2 are smaller representatives, which is absurd. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
|- ( sqr ` 2 ) e/ QQ
Theorem	sqrt2re 12740	The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
|- ( sqr ` 2 ) e. RR
Theorem	sqrt2irr0 12741	The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
|- ( sqr ` 2 ) e. ( RR \ QQ )
Theorem	pw2dvdslemn 12742*	Lemma for pw2dvds 12743. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)
|- ( ( N e. NN /\ A e. NN /\ -. ( 2 ^ A ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
Theorem	pw2dvds 12743*	A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.)
|- ( N e. NN -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
Theorem	pw2dvdseulemle 12744	Lemma for pw2dvdseu 12745. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
|- ( ph -> N e. NN )   &    |- ( ph -> A e. NN0 )   &    |- ( ph -> B e. NN0 )   &    |- ( ph -> ( 2 ^ A ) || N )   &    |- ( ph -> -. ( 2 ^ ( B + 1 ) ) || N )   =>    |- ( ph -> A <_ B )
Theorem	pw2dvdseu 12745*	A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.)
|- ( N e. NN -> E! m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
Theorem	oddpwdclemxy 12746*	Lemma for oddpwdc 12751. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.)
|- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) -> ( X = ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) /\ Y = ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) )
Theorem	oddpwdclemdvds 12747*	Lemma for oddpwdc 12751. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
|- ( A e. NN -> ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) || A )
Theorem	oddpwdclemndvds 12748*	Lemma for oddpwdc 12751. A natural number is not divisible by one more than the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
|- ( A e. NN -> -. ( 2 ^ ( ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) + 1 ) ) || A )
Theorem	oddpwdclemodd 12749*	Lemma for oddpwdc 12751. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.)
|- ( A e. NN -> -. 2 || ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) )
Theorem	oddpwdclemdc 12750*	Lemma for oddpwdc 12751. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.)
|- ( ( ( ( X e. NN /\ -. 2 || X ) /\ Y e. NN0 ) /\ A = ( ( 2 ^ Y ) x. X ) ) <-> ( A e. NN /\ ( X = ( A / ( 2 ^ ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) /\ Y = ( iota_ z e. NN0 ( ( 2 ^ z ) || A /\ -. ( 2 ^ ( z + 1 ) ) || A ) ) ) ) )
Theorem	oddpwdc 12751*	The function F that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.)
|- J = { z e. NN | -. 2 || z }   &    |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )   =>    |- F : ( J X. NN0 ) -1-1-onto-> NN
Theorem	sqpweven 12752*	The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
|- J = { z e. NN | -. 2 || z }   &    |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )   =>    |- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) )
Theorem	2sqpwodd 12753*	The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
|- J = { z e. NN | -. 2 || z }   &    |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )   =>    |- ( A e. NN -> -. 2 || ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) )
Theorem	sqne2sq 12754	The square of a natural number can never be equal to two times the square of a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
|- ( ( A e. NN /\ B e. NN ) -> ( A ^ 2 ) =/= ( 2 x. ( B ^ 2 ) ) )
Theorem	znege1 12755	The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.)
|- ( ( A e. ZZ /\ B e. ZZ /\ A =/= B ) -> 1 <_ ( abs ` ( A - B ) ) )
Theorem	sqrt2irraplemnn 12756	Lemma for sqrt2irrap 12757. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- ( ( A e. NN /\ B e. NN ) -> ( sqr ` 2 ) # ( A / B ) )
Theorem	sqrt2irrap 12757	The square root of 2 is irrational. That is, for any rational number, ( sqr ` 2 ) is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 12739. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- ( Q e. QQ -> ( sqr ` 2 ) # Q )
5.2.4  Properties of the canonical representation of a rational
Syntax	cnumer 12758	Extend class notation to include canonical numerator function.
class numer
Syntax	cdenom 12759	Extend class notation to include canonical denominator function.
class denom
Definition	df-numer 12760*	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 12761*	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 12762*	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 12763*	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 12764	Lemma for qnumcl 12765 and qdencl 12766. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> ( (numer ` A ) e. ZZ /\ (denom ` A ) e. NN ) )
Theorem	qnumcl 12765	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 12766	The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> (denom ` A ) e. NN )
Theorem	fnum 12767	Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- numer : QQ --> ZZ
Theorem	fden 12768	Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- denom : QQ --> NN
Theorem	qnumdenbi 12769	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 12770	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 12771	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 12772	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 12773	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 12774	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 12775	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 12776	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 12777	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 12778	nn0gcdsq 12777 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 12779	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 12780	Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (numer ` ( A ^ 2 ) ) = ( (numer ` A ) ^ 2 ) )
Theorem	densq 12781	Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (denom ` ( A ^ 2 ) ) = ( (denom ` A ) ^ 2 ) )
Theorem	qden1elz 12782	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	nn0sqrtelqelz 12783	If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.)
|- ( ( A e. NN0 /\ ( sqr ` A ) e. QQ ) -> ( sqr ` A ) e. ZZ )
Theorem	nonsq 12784	Any integer strictly between two adjacent squares has a non-rational 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 )
5.2.5  Euler's theorem
Syntax	codz 12785	Extend class notation with the order function on the class of integers modulo N.
class odZ
Syntax	cphi 12786	Extend class notation with the Euler phi function.
class phi
Definition	df-odz 12787*	Define the order function on the class of integers modulo N. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
|- odZ = ( n e. NN |-> ( x e. { x e. ZZ | ( x gcd n ) = 1 } |-> inf ( { m e. NN | n || ( ( x ^ m ) - 1 ) } , RR , < ) ) )
Definition	df-phi 12788*	Define the Euler phi function (also called "Euler totient function"), which counts the number of integers less than n and coprime to it, see definition in [ApostolNT] p. 25. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- phi = ( n e. NN |-> (♯ ` { x e. ( 1 ... n ) | ( x gcd n ) = 1 } ) )
Theorem	phivalfi 12789*	Finiteness of an expression used to define the Euler phi function. (Contributed by Jim Kingon, 28-May-2022.)
|- ( N e. NN -> { x e. ( 1 ... N ) | ( x gcd N ) = 1 } e. Fin )
Theorem	phival 12790*	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 12791	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 12792	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 12793*	Lemma for phibnd 12794. (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 12794	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 12795	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 12796	Value of the Euler phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( phi ` 1 ) = 1
Theorem	dfphi2 12797*	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 12798*	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 12799	Value of the Euler phi function at a prime power. Theorem 2.5(a) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.)
|- ( ( P e. Prime /\ K e. NN ) -> ( phi ` ( P ^ K ) ) = ( ( P ^ ( K - 1 ) ) x. ( P - 1 ) ) )
Theorem	phiprm 12800	Value of the Euler phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )