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
Theorem	eulerthlema 12801*	Lemma for eulerth 12804. (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 12802*	Lemma for eulerth 12804. 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 12803*	Lemma for eulerth 12804. 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 12804	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 12805	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 12806	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 12807	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 12808	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 12809*	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 12810*	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 12811*	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 12812*	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 12813*	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 12814	- Lemma for odzcl 12815, 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 12815	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 12816	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 12817	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 12818	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 12819	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 12820	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 12821	Show an explicit expression for the modular inverse of A mod P. This is an application of prmdiv 12806. (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 12822	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 12823	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 12824	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 12825*	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 12826*	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 12827*	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 12828*	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 12829	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 12830	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 12831	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 12832	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 12833	A prime not equal to 2 is odd. (Contributed by AV, 28-Jun-2021.)
|- ( N e. ( Prime \ { 2 } ) -> -. 2 || N )
Theorem	nnoddn2prmb 12834	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 12835	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 12836	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 12837*	Lemma for pythagtrip 12855. 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 12838*	Lemma for pythagtrip 12855. 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 12839	Lemma for pythagtrip 12855. 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 12840	Lemma for pythagtrip 12855. 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 12841	Lemma for pythagtrip 12855. 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 12842	Lemma for pythagtrip 12855. 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 12843	Lemma for pythagtrip 12855. 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 12844	Lemma for pythagtrip 12855. 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 12845	Lemma for pythagtrip 12855. 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 12846	Lemma for pythagtrip 12855. 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 12847	Lemma for pythagtrip 12855. 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 12848	Lemma for pythagtrip 12855. 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 12849	Lemma for pythagtrip 12855. 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 12850	Lemma for pythagtrip 12855. 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 12851	Lemma for pythagtrip 12855. 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 12852	Lemma for pythagtrip 12855. 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 12853*	Lemma for pythagtrip 12855. 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 12854*	Lemma for pythagtrip 12855. 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 12855*	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 12856	Extend class notation with the prime count function.
class pCnt
Definition	df-pc 12857*	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 12858*	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 12859*	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 12860*	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 12861*	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 12862*	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 12863*	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 12864*	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 12865*	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 12866*	Lemma for pceu 12867. (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 12867*	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 12868*	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 12869*	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 12870	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 12871	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 12872	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 12873	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 12874	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 12875	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 12876	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 12877	Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( P e. Prime -> ( P pCnt 1 ) = 0 )
Theorem	pcqcl 12878	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 12879	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 12880	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 12881	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 ) ) )
Theorem	pcxnn0cl 12882	Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( P e. Prime /\ N e. ZZ ) -> ( P pCnt N ) e. NN0* )
Theorem	pcxcl 12883	Extended real closure of the general prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ N e. QQ ) -> ( P pCnt N ) e. RR* )
Theorem	pcxqcl 12884	The general prime count function is an integer or infinite. (Contributed by Jim Kingdon, 6-Jun-2025.)
|- ( ( P e. Prime /\ N e. QQ ) -> ( ( P pCnt N ) e. ZZ \/ ( P pCnt N ) = +oo ) )
Theorem	pcge0 12885	The prime count of an integer is greater than or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ N e. ZZ ) -> 0 <_ ( P pCnt N ) )
Theorem	pczdvds 12886	Defining property of the prime count function. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( P pCnt N ) ) || N )
Theorem	pcdvds 12887	Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> ( P ^ ( P pCnt N ) ) || N )
Theorem	pczndvds 12888	Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
Theorem	pcndvds 12889	Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
Theorem	pczndvds2 12890	The remainder after dividing out all factors of P is not divisible by P. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ ( P pCnt N ) ) ) )
Theorem	pcndvds2 12891	The remainder after dividing out all factors of P is not divisible by P. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> -. P || ( N / ( P ^ ( P pCnt N ) ) ) )
Theorem	pcdvdsb 12892	P ^ A divides N if and only if A is at most the count of P. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
Theorem	pcelnn 12893	There are a positive number of powers of a prime P in N iff P divides N. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> ( ( P pCnt N ) e. NN <-> P || N ) )
Theorem	pceq0 12894	There are zero powers of a prime P in N iff P does not divide N. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( P e. Prime /\ N e. NN ) -> ( ( P pCnt N ) = 0 <-> -. P || N ) )
Theorem	pcidlem 12895	The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)
|- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
Theorem	pcid 12896	The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt ( P ^ A ) ) = A )
Theorem	pcneg 12897	The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt -u A ) = ( P pCnt A ) )
Theorem	pcabs 12898	The prime count of an absolute value. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt ( abs ` A ) ) = ( P pCnt A ) )
Theorem	pcdvdstr 12899	The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> ( P pCnt A ) <_ ( P pCnt B ) )
Theorem	pcgcd1 12900	The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
|- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )