P. 130 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prmrp 12901	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 12902	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 12903*	A number is prime iff it satisfies Euclid's lemma euclemma 12902. (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 12904	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 12905	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 12906	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 12907	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 12908	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 12909	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 12910	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 12911	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 12912	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 12913	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 12914	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 12915	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 12916	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 12917	Lemma for sqrt2irr 12918. 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 12918	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 12936 for the proof that the square root of two is irrational. The proof's core is proven in sqrt2irrlem 12917, 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 12919	The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
|- ( sqr ` 2 ) e. RR
Theorem	sqrt2irr0 12920	The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
|- ( sqr ` 2 ) e. ( RR \ QQ )
Theorem	pw2dvdslemn 12921*	Lemma for pw2dvds 12922. 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 12922*	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 12923	Lemma for pw2dvdseu 12924. 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 12924*	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 12925*	Lemma for oddpwdc 12930. 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 12926*	Lemma for oddpwdc 12930. 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 12927*	Lemma for oddpwdc 12930. 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 12928*	Lemma for oddpwdc 12930. 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 12929*	Lemma for oddpwdc 12930. 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 12930*	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 12931*	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 12932*	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 12933	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 12934	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 12935	Lemma for sqrt2irrap 12936. 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 12936	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 12918. (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 12937	Extend class notation to include canonical numerator function.
class numer
Syntax	cdenom 12938	Extend class notation to include canonical denominator function.
class denom
Definition	df-numer 12939*	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 12940*	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 12941*	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 12942*	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 12943	Lemma for qnumcl 12944 and qdencl 12945. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> ( (numer ` A ) e. ZZ /\ (denom ` A ) e. NN ) )
Theorem	qnumcl 12944	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 12945	The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. QQ -> (denom ` A ) e. NN )
Theorem	fnum 12946	Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- numer : QQ --> ZZ
Theorem	fden 12947	Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- denom : QQ --> NN
Theorem	qnumdenbi 12948	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 12949	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 12950	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 12951	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 12952	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 12953	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 12954	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 12955	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 12956	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 12957	nn0gcdsq 12956 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 12958	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 12959	Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (numer ` ( A ^ 2 ) ) = ( (numer ` A ) ^ 2 ) )
Theorem	densq 12960	Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> (denom ` ( A ^ 2 ) ) = ( (denom ` A ) ^ 2 ) )
Theorem	qden1elz 12961	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 12962	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 12963	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 12964	Extend class notation with the order function on the class of integers modulo N.
class odZ
Syntax	cphi 12965	Extend class notation with the Euler phi function.
class phi
Definition	df-odz 12966*	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 12967*	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 12968*	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 12969*	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 12970	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 12971	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 12972*	Lemma for phibnd 12973. (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 12973	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 12974	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 12975	Value of the Euler phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( phi ` 1 ) = 1
Theorem	dfphi2 12976*	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 12977*	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 12978	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 12979	Value of the Euler phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
Theorem	crth 12980*	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 12981*	Lemma for phimul 12982. (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 12982	The Euler phi function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. Theorem 2.5(c) in [ApostolNT] p. 28. (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 12983*	Lemma for eulerth 12989. (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	eulerthlemfi 12984*	Lemma for eulerth 12989. The set S is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   =>    |- ( ph -> S e. Fin )
Theorem	eulerthlemrprm 12985*	Lemma for eulerth 12989. N and prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( N gcd prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) ) = 1 )
Theorem	eulerthlema 12986*	Lemma for eulerth 12989. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( ( ( A ^ ( phi ` N ) ) x. prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) ) mod N ) = ( prod_ x e. ( 1 ... ( phi ` N ) ) ( ( A x. ( F ` x ) ) mod N ) mod N ) )
Theorem	eulerthlemh 12987*	Lemma for eulerth 12989. A permutation of ( 1 ... ( phi ` N ) ). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   &    |- H = ( `' F o. ( y e. ( 1 ... ( phi ` N ) ) |-> ( ( A x. ( F ` y ) ) mod N ) ) )   =>    |- ( ph -> H : ( 1 ... ( phi ` N ) ) -1-1-onto-> ( 1 ... ( phi ` N ) ) )
Theorem	eulerthlemth 12988*	Lemma for eulerth 12989. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
Theorem	eulerth 12989	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. Also called Euler-Fermat theorem, see theorem 5.17 in [ApostolNT] p. 113. (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 ) )
Theorem	fermltl 12990	Fermat's little theorem. When P is prime, A ^ P == A (mod P) for any A, see theorem 5.19 in [ApostolNT] p. 114. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 19-Mar-2022.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
Theorem	prmdiv 12991	Show an explicit expression for the modular inverse of A mod P. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
Theorem	prmdiveq 12992	The modular inverse of A mod P is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) <-> S = R ) )
Theorem	prmdivdiv 12993	The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A = ( ( R ^ ( P - 2 ) ) mod P ) )
Theorem	hashgcdlem 12994*	A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)
|- A = { y e. ( 0..^ ( M / N ) ) | ( y gcd ( M / N ) ) = 1 }   &    |- B = { z e. ( 0..^ M ) | ( z gcd M ) = N }   &    |- F = ( x e. A |-> ( x x. N ) )   =>    |- ( ( M e. NN /\ N e. NN /\ N || M ) -> F : A -1-1-onto-> B )
Theorem	dvdsfi 12995*	A natural number has finitely many divisors. (Contributed by Jim Kingdon, 9-Oct-2025.)
|- ( N e. NN -> { x e. NN | x || N } e. Fin )
Theorem	hashgcdeq 12996*	Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
|- ( ( M e. NN /\ N e. NN ) -> (♯ ` { x e. ( 0..^ M ) | ( x gcd M ) = N } ) = if ( N || M , ( phi ` ( M / N ) ) , 0 ) )
Theorem	phisum 12997*	The divisor sum identity of the totient function. Theorem 2.2 in [ApostolNT] p. 26. (Contributed by Stefan O'Rear, 12-Sep-2015.)
|- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( phi ` d ) = N )
Theorem	odzval 12998*	Value of the order function. This is a function of functions; the inner argument selects the base (i.e., mod N for some N, often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod N. In order to ensure the supremum is well-defined, we only define the expression when A and N are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
Theorem	odzcllem 12999	- Lemma for odzcl 13000, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( ( odZ ` N ) ` A ) e. NN /\ N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
Theorem	odzcl 13000	The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) e. NN )