P. 129 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12801-12900   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-phi 12801*	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 12802*	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 12803*	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 12804	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 12805	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 12806*	Lemma for phibnd 12807. (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 12807	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 12808	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 12809	Value of the Euler phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( phi ` 1 ) = 1
Theorem	dfphi2 12810*	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 12811*	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 12812	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 12813	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 12814*	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 12815*	Lemma for phimul 12816. (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 12816	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 12817*	Lemma for eulerth 12823. (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 12818*	Lemma for eulerth 12823. 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 12819*	Lemma for eulerth 12823. 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 12820*	Lemma for eulerth 12823. (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 12821*	Lemma for eulerth 12823. 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 12822*	Lemma for eulerth 12823. 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 12823	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 12824	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 12825	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 12826	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 12827	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 12828*	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 12829*	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 12830*	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 12831*	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 12832*	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 12833	- Lemma for odzcl 12834, 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 12834	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 )
Theorem	odzid 12835	Any element raised to the power of its order is 1. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) )
Theorem	odzdvds 12836	The only powers of A that are congruent to 1 are the multiples of the order of A. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)
|- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ K e. NN0 ) -> ( N || ( ( A ^ K ) - 1 ) <-> ( ( odZ ` N ) ` A ) || K ) )
Theorem	odzphi 12837	The order of any group element is a divisor of the Euler phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) || ( phi ` N ) )
5.2.6  Arithmetic modulo a prime number
Theorem	modprm1div 12838	A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.) (Proof shortened by AV, 30-May-2023.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A mod P ) = 1 <-> P || ( A - 1 ) ) )
Theorem	m1dvdsndvds 12839	If an integer minus 1 is divisible by a prime number, the integer itself is not divisible by this prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> -. P || A ) )
Theorem	modprminv 12840	Show an explicit expression for the modular inverse of A mod P. This is an application of prmdiv 12825. (Contributed by Alexander van der Vekens, 15-May-2018.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ ( ( A x. R ) mod P ) = 1 ) )
Theorem	modprminveq 12841	The modular inverse of A mod P is unique. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- R = ( ( A ^ ( P - 2 ) ) mod P )   =>    |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ ( ( A x. S ) mod P ) = 1 ) <-> S = R ) )
Theorem	vfermltl 12842	Variant of Fermat's little theorem if A is not a multiple of P, see theorem 5.18 in [ApostolNT] p. 113. (Contributed by AV, 21-Aug-2020.) (Proof shortened by AV, 5-Sep-2020.)
|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 1 ) ) mod P ) = 1 )
Theorem	powm2modprm 12843	If an integer minus 1 is divisible by a prime number, then the integer to the power of the prime number minus 2 is 1 modulo the prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> ( ( A ^ ( P - 2 ) ) mod P ) = 1 ) )
Theorem	reumodprminv 12844*	For any prime number and for any positive integer less than this prime number, there is a unique modular inverse of this positive integer. (Contributed by Alexander van der Vekens, 12-May-2018.)
|- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E! i e. ( 1 ... ( P - 1 ) ) ( ( N x. i ) mod P ) = 1 )
Theorem	modprm0 12845*	For two positive integers less than a given prime number there is always a nonnegative integer (less than the given prime number) so that the sum of one of the two positive integers and the other of the positive integers multiplied by the nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( ( P e. Prime /\ N e. ( 1..^ P ) /\ I e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
Theorem	nnnn0modprm0 12846*	For a positive integer and a nonnegative integer both less than a given prime number there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the positive integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 8-Nov-2018.)
|- ( ( P e. Prime /\ N e. ( 1..^ P ) /\ I e. ( 0..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
Theorem	modprmn0modprm0 12847*	For an integer not being 0 modulo a given prime number and a nonnegative integer less than the prime number, there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 10-Nov-2018.)
|- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( I e. ( 0..^ P ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
5.2.7  Pythagorean Triples
Theorem	coprimeprodsq 12848	If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN0 /\ B e. ZZ /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> A = ( ( A gcd C ) ^ 2 ) ) )
Theorem	coprimeprodsq2 12849	If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ B e. NN0 /\ C e. NN0 ) /\ ( ( A gcd B ) gcd C ) = 1 ) -> ( ( C ^ 2 ) = ( A x. B ) -> B = ( ( B gcd C ) ^ 2 ) ) )
Theorem	oddprm 12850	A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 10-Jul-2022.)
|- ( N e. ( Prime \ { 2 } ) -> ( ( N - 1 ) / 2 ) e. NN )
Theorem	nnoddn2prm 12851	A prime not equal to 2 is an odd positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( N e. ( Prime \ { 2 } ) -> ( N e. NN /\ -. 2 || N ) )
Theorem	oddn2prm 12852	A prime not equal to 2 is odd. (Contributed by AV, 28-Jun-2021.)
|- ( N e. ( Prime \ { 2 } ) -> -. 2 || N )
Theorem	nnoddn2prmb 12853	A number is a prime number not equal to 2 iff it is an odd prime number. Conversion theorem for two representations of odd primes. (Contributed by AV, 14-Jul-2021.)
|- ( N e. ( Prime \ { 2 } ) <-> ( N e. Prime /\ -. 2 || N ) )
Theorem	prm23lt5 12854	A prime less than 5 is either 2 or 3. (Contributed by AV, 5-Jul-2021.)
|- ( ( P e. Prime /\ P < 5 ) -> ( P = 2 \/ P = 3 ) )
Theorem	prm23ge5 12855	A prime is either 2 or 3 or greater than or equal to 5. (Contributed by AV, 5-Jul-2021.)
|- ( P e. Prime -> ( P = 2 \/ P = 3 \/ P e. ( ZZ>= ` 5 ) ) )
Theorem	pythagtriplem1 12856*	Lemma for pythagtrip 12874. Prove a weaker version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( E. n e. NN E. m e. NN E. k e. NN ( A = ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) /\ B = ( k x. ( 2 x. ( m x. n ) ) ) /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) )
Theorem	pythagtriplem2 12857*	Lemma for pythagtrip 12874. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( E. n e. NN E. m e. NN E. k e. NN ( { A , B } = { ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) , ( k x. ( 2 x. ( m x. n ) ) ) } /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) )
Theorem	pythagtriplem3 12858	Lemma for pythagtrip 12874. Show that C and B are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( B gcd C ) = 1 )
Theorem	pythagtriplem4 12859	Lemma for pythagtrip 12874. Show that C - B and C + B are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( ( C - B ) gcd ( C + B ) ) = 1 )
Theorem	pythagtriplem10 12860	Lemma for pythagtrip 12874. Show that C - B is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) ) -> 0 < ( C - B ) )
Theorem	pythagtriplem6 12861	Lemma for pythagtrip 12874. Calculate ( sqr ` ( C - B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) = ( ( C - B ) gcd A ) )
Theorem	pythagtriplem7 12862	Lemma for pythagtrip 12874. Calculate ( sqr ` ( C + B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C + B ) ) = ( ( C + B ) gcd A ) )
Theorem	pythagtriplem8 12863	Lemma for pythagtrip 12874. Show that ( sqr ` ( C - B ) ) is a positive integer. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C - B ) ) e. NN )
Theorem	pythagtriplem9 12864	Lemma for pythagtrip 12874. Show that ( sqr ` ( C + B ) ) is a positive integer. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( sqr ` ( C + B ) ) e. NN )
Theorem	pythagtriplem11 12865	Lemma for pythagtrip 12874. Show that M (which will eventually be closely related to the m in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> M e. NN )
Theorem	pythagtriplem12 12866	Lemma for pythagtrip 12874. Calculate the square of M. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( M ^ 2 ) = ( ( C + A ) / 2 ) )
Theorem	pythagtriplem13 12867	Lemma for pythagtrip 12874. Show that N (which will eventually be closely related to the n in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> N e. NN )
Theorem	pythagtriplem14 12868	Lemma for pythagtrip 12874. Calculate the square of N. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> ( N ^ 2 ) = ( ( C - A ) / 2 ) )
Theorem	pythagtriplem15 12869	Lemma for pythagtrip 12874. Show the relationship between M, N, and A. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   &    |- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> A = ( ( M ^ 2 ) - ( N ^ 2 ) ) )
Theorem	pythagtriplem16 12870	Lemma for pythagtrip 12874. Show the relationship between M, N, and B. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   &    |- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> B = ( 2 x. ( M x. N ) ) )
Theorem	pythagtriplem17 12871	Lemma for pythagtrip 12874. Show the relationship between M, N, and C. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- M = ( ( ( sqr ` ( C + B ) ) + ( sqr ` ( C - B ) ) ) / 2 )   &    |- N = ( ( ( sqr ` ( C + B ) ) - ( sqr ` ( C - B ) ) ) / 2 )   =>    |- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> C = ( ( M ^ 2 ) + ( N ^ 2 ) ) )
Theorem	pythagtriplem18 12872*	Lemma for pythagtrip 12874. Wrap the previous M and N up in quantifiers. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ ( ( A gcd B ) = 1 /\ -. 2 || A ) ) -> E. n e. NN E. m e. NN ( A = ( ( m ^ 2 ) - ( n ^ 2 ) ) /\ B = ( 2 x. ( m x. n ) ) /\ C = ( ( m ^ 2 ) + ( n ^ 2 ) ) ) )
Theorem	pythagtriplem19 12873*	Lemma for pythagtrip 12874. Introduce k and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. NN ) /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) /\ -. 2 || ( A / ( A gcd B ) ) ) -> E. n e. NN E. m e. NN E. k e. NN ( A = ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) /\ B = ( k x. ( 2 x. ( m x. n ) ) ) /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) )
Theorem	pythagtrip 12874*	Parameterize the Pythagorean triples. If A, B, and C are naturals, then they obey the Pythagorean triple formula iff they are parameterized by three naturals. This proof follows the Isabelle proof at http://afp.sourceforge.net/entries/Fermat3_4.shtml. This is Metamath 100 proof #23. (Contributed by Scott Fenton, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN /\ C e. NN ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( C ^ 2 ) <-> E. n e. NN E. m e. NN E. k e. NN ( { A , B } = { ( k x. ( ( m ^ 2 ) - ( n ^ 2 ) ) ) , ( k x. ( 2 x. ( m x. n ) ) ) } /\ C = ( k x. ( ( m ^ 2 ) + ( n ^ 2 ) ) ) ) ) )
5.2.8  The prime count function
Syntax	cpc 12875	Extend class notation with the prime count function.
class pCnt
Definition	df-pc 12876*	Define the prime count function, which returns the largest exponent of a given prime (or other positive integer) that divides the number. For rational numbers, it returns negative values according to the power of a prime in the denominator. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- pCnt = ( p e. Prime , r e. QQ |-> if ( r = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( r = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) ) ) ) )
Theorem	pclem0 12877*	Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
|- A = { n e. NN0 | ( P ^ n ) || N }   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> 0 e. A )
Theorem	pclemub 12878*	Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
|- A = { n e. NN0 | ( P ^ n ) || N }   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ A. y e. A y <_ x )
Theorem	pclemdc 12879*	Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.)
|- A = { n e. NN0 | ( P ^ n ) || N }   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )
Theorem	pcprecl 12880*	Closure of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- A = { n e. NN0 | ( P ^ n ) || N }   &    |- S = sup ( A , RR , < )   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
Theorem	pcprendvds 12881*	Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- A = { n e. NN0 | ( P ^ n ) || N }   &    |- S = sup ( A , RR , < )   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )
Theorem	pcprendvds2 12882*	Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- A = { n e. NN0 | ( P ^ n ) || N }   &    |- S = sup ( A , RR , < )   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ S ) ) )
Theorem	pcpre1 12883*	Value of the prime power pre-function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 26-Apr-2016.)
|- A = { n e. NN0 | ( P ^ n ) || N }   &    |- S = sup ( A , RR , < )   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ N = 1 ) -> S = 0 )
Theorem	pcpremul 12884*	Multiplicative property of the prime count pre-function. Note that the primality of P is essential for this property; ( 4 pCnt 2 ) = 0 but ( 4 pCnt ( 2 x. 2 ) ) = 1 =/= 2 x. ( 4 pCnt 2 ) = 0. Since this is needed to show uniqueness for the real prime count function (over QQ), we don't bother to define it off the primes. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- S = sup ( { n e. NN0 | ( P ^ n ) || M } , RR , < )   &    |- T = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < )   &    |- U = sup ( { n e. NN0 | ( P ^ n ) || ( M x. N ) } , RR , < )   =>    |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) = U )
Theorem	pceulem 12885*	Lemma for pceu 12886. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )   &    |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )   &    |- U = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )   &    |- V = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> N =/= 0 )   &    |- ( ph -> ( x e. ZZ /\ y e. NN ) )   &    |- ( ph -> N = ( x / y ) )   &    |- ( ph -> ( s e. ZZ /\ t e. NN ) )   &    |- ( ph -> N = ( s / t ) )   =>    |- ( ph -> ( S - T ) = ( U - V ) )
Theorem	pceu 12886*	Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )   &    |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )   =>    |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> E! z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) )
Theorem	pcval 12887*	The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)
|- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )   &    |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )   =>    |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( P pCnt N ) = ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) )
Theorem	pczpre 12888*	Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- S = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < )   =>    |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) = S )
Theorem	pczcl 12889	Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) e. NN0 )
Theorem	pccl 12890	Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> ( P pCnt N ) e. NN0 )
Theorem	pccld 12891	Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> P e. Prime )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( P pCnt N ) e. NN0 )
Theorem	pcmul 12892	Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = ( ( P pCnt A ) + ( P pCnt B ) ) )
Theorem	pcdiv 12893	Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)
|- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ B e. NN ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )
Theorem	pcqmul 12894	Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = ( ( P pCnt A ) + ( P pCnt B ) ) )
Theorem	pc0 12895	The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( P e. Prime -> ( P pCnt 0 ) = +oo )
Theorem	pc1 12896	Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( P e. Prime -> ( P pCnt 1 ) = 0 )
Theorem	pcqcl 12897	Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( P pCnt N ) e. ZZ )
Theorem	pcqdiv 12898	Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )
Theorem	pcrec 12899	Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt ( 1 / A ) ) = -u ( P pCnt A ) )
Theorem	pcexp 12900	Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)
|- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ N e. ZZ ) -> ( P pCnt ( A ^ N ) ) = ( N x. ( P pCnt A ) ) )