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Theorem List for Intuitionistic Logic Explorer - 12201-12300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqnumcl 12201 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  e.  ZZ )
 
Theoremqdencl 12202 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 12203 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 12204 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 12205 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 ) ) )
 
Theoremqnumdencoprm 12206 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
 )
 
Theoremqeqnumdivden 12207 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  A  =  ( (numer `  A )  /  (denom `  A ) ) )
 
Theoremqmuldeneqnum 12208 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 )
 )
 
Theoremdivnumden 12209 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 ) ) ) )
 
Theoremdivdenle 12210 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 )
 
Theoremqnumgt0 12211 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 ) ) )
 
Theoremqgt0numnn 12212 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 )
 
Theoremnn0gcdsq 12213 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 ) ) )
 
Theoremzgcdsq 12214 nn0gcdsq 12213 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 ) ) )
 
Theoremnumdensq 12215 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
 ) ) )
 
Theoremnumsq 12216 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 ) )
 
Theoremdensq 12217 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) )
 
Theoremqden1elz 12218 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 ) )
 
Theoremnn0sqrtelqelz 12219 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 )
 
Theoremnonsq 12220 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
 
Syntaxcodz 12221 Extend class notation with the order function on the class of integers modulo N.
 class  odZ
 
Syntaxcphi 12222 Extend class notation with the Euler phi function.
 class  phi
 
Definitiondf-odz 12223* 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 ,  <  ) ) )
 
Definitiondf-phi 12224* 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 } ) )
 
Theoremphivalfi 12225* 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 )
 
Theoremphival 12226* 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 } ) )
 
Theoremphicl2 12227 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 ) )
 
Theoremphicl 12228 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
 
Theoremphibndlem 12229* Lemma for phibnd 12230. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  C_  ( 1 ... ( N  -  1 ) ) )
 
Theoremphibnd 12230 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
 ) )
 
Theoremphicld 12231 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 )
 
Theoremphi1 12232 Value of the Euler  phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( phi `  1
 )  =  1
 
Theoremdfphi2 12233* 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 } ) )
 
Theoremhashdvds 12234* 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 ) ) ) )
 
Theoremphiprmpw 12235 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
 ) ) )
 
Theoremphiprm 12236 Value of the Euler  phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1
 ) )
 
Theoremcrth 12237* 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 )
 
Theoremphimullem 12238* Lemma for phimul 12239. (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 ) ) )
 
Theoremphimul 12239 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 ) ) )
 
Theoremeulerthlem1 12240* Lemma for eulerth 12246. (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 )
 
Theoremeulerthlemfi 12241* Lemma for eulerth 12246. 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 )
 
Theoremeulerthlemrprm 12242* Lemma for eulerth 12246. 
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 )
 
Theoremeulerthlema 12243* Lemma for eulerth 12246. (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 ) )
 
Theoremeulerthlemh 12244* Lemma for eulerth 12246. 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 ) ) )
 
Theoremeulerthlemth 12245* Lemma for eulerth 12246. 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 ) )
 
Theoremeulerth 12246 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 ) )
 
Theoremfermltl 12247 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 ) )
 
Theoremprmdiv 12248 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 ) ) )
 
Theoremprmdiveq 12249 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 ) )
 
Theoremprmdivdiv 12250 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 ) )
 
Theoremhashgcdlem 12251* 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 )
 
Theoremhashgcdeq 12252* 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 ) )
 
Theoremphisum 12253* 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 )
 
Theoremodzval 12254* 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 ,  <  ) )
 
Theoremodzcllem 12255 - Lemma for odzcl 12256, 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 ) ) )
 
Theoremodzcl 12256 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 )
 
Theoremodzid 12257 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 ) )
 
Theoremodzdvds 12258 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 ) )
 
Theoremodzphi 12259 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
 
Theoremmodprm1div 12260 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 ) ) )
 
Theoremm1dvdsndvds 12261 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 ) )
 
Theoremmodprminv 12262 Show an explicit expression for the modular inverse of  A  mod  P. This is an application of prmdiv 12248. (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 ) )
 
Theoremmodprminveq 12263 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 ) )
 
Theoremvfermltl 12264 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 )
 
Theorempowm2modprm 12265 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 ) )
 
Theoremreumodprminv 12266* 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 )
 
Theoremmodprm0 12267* 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 )
 
Theoremnnnn0modprm0 12268* 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 )
 
Theoremmodprmn0modprm0 12269* 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
 
Theoremcoprimeprodsq 12270 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 ) ) )
 
Theoremcoprimeprodsq2 12271 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 ) ) )
 
Theoremoddprm 12272 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 )
 
Theoremnnoddn2prm 12273 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 ) )
 
Theoremoddn2prm 12274 A prime not equal to  2 is odd. (Contributed by AV, 28-Jun-2021.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  -.  2  ||  N )
 
Theoremnnoddn2prmb 12275 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 ) )
 
Theoremprm23lt5 12276 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 ) )
 
Theoremprm23ge5 12277 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 ) ) )
 
Theorempythagtriplem1 12278* Lemma for pythagtrip 12296. 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 ) )
 
Theorempythagtriplem2 12279* Lemma for pythagtrip 12296. 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 ) ) )
 
Theorempythagtriplem3 12280 Lemma for pythagtrip 12296. 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 )
 
Theorempythagtriplem4 12281 Lemma for pythagtrip 12296. 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 )
 
Theorempythagtriplem10 12282 Lemma for pythagtrip 12296. 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 ) )
 
Theorempythagtriplem6 12283 Lemma for pythagtrip 12296. 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 ) )
 
Theorempythagtriplem7 12284 Lemma for pythagtrip 12296. 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 ) )
 
Theorempythagtriplem8 12285 Lemma for pythagtrip 12296. 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 )
 
Theorempythagtriplem9 12286 Lemma for pythagtrip 12296. 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 )
 
Theorempythagtriplem11 12287 Lemma for pythagtrip 12296. 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 )
 
Theorempythagtriplem12 12288 Lemma for pythagtrip 12296. 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 ) )
 
Theorempythagtriplem13 12289 Lemma for pythagtrip 12296. 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 )
 
Theorempythagtriplem14 12290 Lemma for pythagtrip 12296. 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 ) )
 
Theorempythagtriplem15 12291 Lemma for pythagtrip 12296. 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 ) ) )
 
Theorempythagtriplem16 12292 Lemma for pythagtrip 12296. 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 ) ) )
 
Theorempythagtriplem17 12293 Lemma for pythagtrip 12296. 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 ) ) )
 
Theorempythagtriplem18 12294* Lemma for pythagtrip 12296. 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 ) ) ) )
 
Theorempythagtriplem19 12295* Lemma for pythagtrip 12296. 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 ) ) ) ) )
 
Theorempythagtrip 12296* 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
 
Syntaxcpc 12297 Extend class notation with the prime count function.
 class  pCnt
 
Definitiondf-pc 12298* 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 ,  <  ) ) ) ) ) )
 
Theorempclem0 12299* 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 )
 
Theorempclemub 12300* 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 )
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