P. 123 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12201-12300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fermltl 12201	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 12202	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 12203	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 12204	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 12205*	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	hashgcdeq 12206*	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 12207*	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 12208*	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 12209	- Lemma for odzcl 12210, 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 12210	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 12211	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 12212	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 12213	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 12214	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 12215	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 12216	Show an explicit expression for the modular inverse of A mod P. This is an application of prmdiv 12202. (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 12217	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 12218	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 12219	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 12220*	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 12221*	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 12222*	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 12223*	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 12224	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 12225	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 12226	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 12227	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 12228	A prime not equal to 2 is odd. (Contributed by AV, 28-Jun-2021.)
|- ( N e. ( Prime \ { 2 } ) -> -. 2 || N )
Theorem	nnoddn2prmb 12229	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 12230	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 12231	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 12232*	Lemma for pythagtrip 12250. 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 12233*	Lemma for pythagtrip 12250. 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 12234	Lemma for pythagtrip 12250. 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 12235	Lemma for pythagtrip 12250. 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 12236	Lemma for pythagtrip 12250. 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 12237	Lemma for pythagtrip 12250. 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 12238	Lemma for pythagtrip 12250. 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 12239	Lemma for pythagtrip 12250. 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 12240	Lemma for pythagtrip 12250. 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 12241	Lemma for pythagtrip 12250. 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 12242	Lemma for pythagtrip 12250. 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 12243	Lemma for pythagtrip 12250. 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 12244	Lemma for pythagtrip 12250. 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 12245	Lemma for pythagtrip 12250. 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 12246	Lemma for pythagtrip 12250. 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 12247	Lemma for pythagtrip 12250. 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 12248*	Lemma for pythagtrip 12250. 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 12249*	Lemma for pythagtrip 12250. 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 12250*	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 12251	Extend class notation with the prime count function.
class pCnt
Definition	df-pc 12252*	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 12253*	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 12254*	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 12255*	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 12256*	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 12257*	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 12258*	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 12259*	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 12260*	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 12261*	Lemma for pceu 12262. (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 12262*	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 12263*	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 12264*	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 12265	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 12266	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 12267	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 12268	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 12269	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 12270	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 12271	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 12272	Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( P e. Prime -> ( P pCnt 1 ) = 0 )
Theorem	pcqcl 12273	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 12274	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 12275	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 12276	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 12277	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 12278	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	pcge0 12279	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 12280	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 12281	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 12282	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 12283	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 12284	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 12285	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 12286	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 12287	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 12288	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 12289	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 12290	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 12291	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 12292	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 12293	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 12294	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 ) )
Theorem	pcgcd 12295	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 gcd B ) ) = if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) )
Theorem	pc2dvds 12296*	A characterization of divisibility in terms of prime count. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || B <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) ) )
Theorem	pc11 12297*	The prime count function, viewed as a function from NN to ( NN ^m Prime ), is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) )
Theorem	pcz 12298*	The prime count function can be used as an indicator that a given rational number is an integer. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( A e. QQ -> ( A e. ZZ <-> A. p e. Prime 0 <_ ( p pCnt A ) ) )
Theorem	pcprmpw2 12299*	Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
|- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )
Theorem	pcprmpw 12300*	Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
|- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A = ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )