P. 111 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11001-11100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prmex 11001	The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
|- Prime e. _V
Theorem	1nprm 11002	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 11003*	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 11004*	Lemma for isprm2 11005. (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 11005*	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 11006*	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 11007*	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 11008*	A variation on prmind 11009 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 11009*	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 11010	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 11011	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 11012	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 11013	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 11014	A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
|- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
Theorem	2prm 11015	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 11016	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|- 3 e. Prime
Theorem	4nprm 11017	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	prmuz2 11018	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 11019	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 11020	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 11021	An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
|- ( P e. ( Prime \ { 2 } ) -> 2 < P )
Theorem	oddprmge3 11022	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 11023	A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )
Theorem	dvdsprm 11024	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 11025*	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 11026*	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 11027	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	divgcdodd 11028	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 ) ) ) )
4.2.2  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 11031.
Theorem	coprm 11029	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 11030	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 11031	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 11032*	A number is prime iff it satisfies Euclid's lemma euclemma 11031. (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 11033	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 11034	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 11035	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 11036	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 11037	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 11038	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 11039	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 11040	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 11041	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 11042	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 11043	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 11044	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 11045	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
4.2.3  Non-rationality of square root of 2
Theorem	sqrt2irrlem 11046	Lemma for sqrt2irr 11047. 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 11047	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 11064 for the proof that the square root of two is irrational. The proof's core is proven in sqrt2irrlem 11046, 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 11048	The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
|- ( sqr ` 2 ) e. RR
Theorem	pw2dvdslemn 11049*	Lemma for pw2dvds 11050. 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 11050*	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 11051	Lemma for pw2dvdseu 11052. 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 11052*	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 11053*	Lemma for oddpwdc 11058. 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 11054*	Lemma for oddpwdc 11058. 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 11055*	Lemma for oddpwdc 11058. 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 11056*	Lemma for oddpwdc 11058. 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 11057*	Lemma for oddpwdc 11058. 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 11058*	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 11059*	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 11060*	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 11061	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 11062	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 11063	Lemma for sqrt2irrap 11064. 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 11064	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 11047. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- ( Q e. QQ -> ( sqr ` 2 ) # Q )
4.2.4  Properties of the canonical representation of a rational
Syntax	cnumer 11065	Extend class notation to include canonical numerator function.
class numer
Syntax	cdenom 11066	Extend class notation to include canonical denominator function.
class denom
Definition	df-numer 11067*	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 11068*	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 11069*	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 11070*	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 11071	Lemma for qnumcl 11072 and qdencl 11073. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> ( (numer ` A ) e. ZZ /\ (denom ` A ) e. NN ) )
Theorem	qnumcl 11072	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 11073	The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> (denom ` A ) e. NN )
Theorem	fnum 11074	Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- numer : QQ --> ZZ
Theorem	fden 11075	Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- denom : QQ --> NN
Theorem	qnumdenbi 11076	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 11077	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 11078	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 11079	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 11080	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 11081	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 11082	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 11083	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 11084	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 11085	nn0gcdsq 11084 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 11086	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 11087	Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (numer ` ( A ^ 2 ) ) = ( (numer ` A ) ^ 2 ) )
Theorem	densq 11088	Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (denom ` ( A ^ 2 ) ) = ( (denom ` A ) ^ 2 ) )
Theorem	qden1elz 11089	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 11090	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 11091	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 )
4.2.5  Euler's theorem
Syntax	cphi 11092	Extend class notation with the Euler phi function.
class phi
Definition	df-phi 11093*	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 11094*	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 11095*	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 11096	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 11097	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 11098*	Lemma for phibnd 11099. (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 11099	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 11100	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 )