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Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdensq 12701 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) )
 
Theoremqden1elz 12702 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 ) )
 
Theoremzsqrelqelz 12703 If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  ( sqr `  A )  e.  QQ )  ->  ( sqr `  A )  e.  ZZ )
 
Theoremnonsq 12704 Any integer strictly between two adjacent squares has an irrational 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 )
 
6.2.3  Euler's theorem
 
Syntaxcodz 12705 Extend class notation with the order function on the class of integers mod N.
 class  od Z
 
Syntaxcphi 12706 Extend class notation with the Euler phi function.
 class  phi
 
Definitiondf-odz 12707* Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |- 
 od Z  =  ( n  e.  NN  |->  ( x  e.  { x  e. 
 ZZ  |  ( x 
 gcd  n )  =  1 }  |->  sup ( { m  e.  NN  |  n  ||  ( ( x ^ m )  -  1 ) } ,  RR ,  `'  <  ) ) )
 
Definitiondf-phi 12708* Define the Euler phi function, which counts the number of integers less than  n and coprime to it. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |- 
 phi  =  ( n  e.  NN  |->  ( # `  { x  e.  ( 1 ... n )  |  ( x  gcd  n )  =  1 } ) )
 
Theoremphival 12709* 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 12710 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 12711 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
 
Theoremphibndlem 12712* Lemma for phibnd 12713. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  C_  ( 1 ... ( N  -  1 ) ) )
 
Theoremphibnd 12713 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 12714 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 12715 Value of the Euler  phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( phi `  1
 )  =  1
 
Theoremdfphi2 12716* 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 12717* 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 12718 Value of the Euler  phi function at a prime power. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( phi `  ( P ^ K ) )  =  ( ( P ^ ( K  -  1 ) )  x.  ( P  -  1
 ) ) )
 
Theoremphiprm 12719 Value of the Euler  phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1
 ) )
 
Theoremcrt 12720* 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 12721* Lemma for phimul 12722. (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 12722 The Euler  phi function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. (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 12723* Lemma for eulerth 12725. (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 )
 
Theoremeulerthlem2 12724* Lemma for eulerth 12725. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( 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  ->  ( ( A ^ ( phi `  N ) ) 
 mod  N )  =  ( 1  mod  N ) )
 
Theoremeulerth 12725 Euler's theorem, a generalization of Fermat's little theorem. If  A and  N are coprime, then  A ^ phi ( N )  ==  1, mod  N. (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 12726 Fermat's little theorem. When  P is prime,  A ^ P  ==  A, mod  P for any  A. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( ( A ^ P )  mod  P )  =  ( A 
 mod  P ) )
 
Theoremprmdiv 12727 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 12728 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 12729 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 ) )
 
Theoremodzval 12730* 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.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( od Z `  N ) `  A )  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1
 ) } ,  RR ,  `'  <  ) )
 
Theoremodzcllem 12731 - Lemma for odzcl 12732, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( ( ( od
 Z `  N ) `  A )  e.  NN  /\  N  ||  ( ( A ^ ( ( od
 Z `  N ) `  A ) )  -  1 ) ) )
 
Theoremodzcl 12732 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 )  ->  ( ( od Z `  N ) `  A )  e.  NN )
 
Theoremodzid 12733 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 ^ ( ( od
 Z `  N ) `  A ) )  -  1 ) )
 
Theoremodzdvds 12734 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.)
 |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A 
 gcd  N )  =  1 )  /\  K  e.  NN0 )  ->  ( N  ||  ( ( A ^ K )  -  1
 ) 
 <->  ( ( od Z `  N ) `  A )  ||  K ) )
 
Theoremodzphi 12735 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 )  ->  ( ( od Z `  N ) `  A )  ||  ( phi `  N ) )
 
6.2.4  Pythagorean Triples
 
Theoremcoprimeprodsq 12736 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 12737 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 ) ) )
 
Theoremopoe 12738 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\ 
 -.  2  ||  B ) )  ->  2  ||  ( A  +  B ) )
 
Theoremomoe 12739 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\ 
 -.  2  ||  B ) )  ->  2  ||  ( A  -  B ) )
 
Theoremopeo 12740 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\  2  ||  B )
 )  ->  -.  2  ||  ( A  +  B ) )
 
Theoremomeo 12741 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( ( A  e.  ZZ  /\  -.  2  ||  A )  /\  ( B  e.  ZZ  /\  2  ||  B )
 )  ->  -.  2  ||  ( A  -  B ) )
 
Theoremoddprm 12742 A prime not equal to  2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  ( ( N  -  1 )  / 
 2 )  e.  NN )
 
Theorempythagtriplem1 12743* Lemma for pythagtrip 12761. 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 12744* Lemma for pythagtrip 12761. 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 12745 Lemma for pythagtrip 12761. 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 12746 Lemma for pythagtrip 12761. 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 12747 Lemma for pythagtrip 12761. 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 12748 Lemma for pythagtrip 12761. 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 12749 Lemma for pythagtrip 12761. 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 12750 Lemma for pythagtrip 12761. Show that  ( sqr `  ( C  -  B ) ) is a natural number (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 12751 Lemma for pythagtrip 12761. Show that  ( sqr `  ( C  +  B ) ) is a natural number (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 12752 Lemma for pythagtrip 12761. 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 12753 Lemma for pythagtrip 12761. 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 12754 Lemma for pythagtrip 12761. 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 12755 Lemma for pythagtrip 12761. 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 12756 Lemma for pythagtrip 12761. 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 12757 Lemma for pythagtrip 12761. 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 12758 Lemma for pythagtrip 12761. 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 12759* Lemma for pythagtrip 12761. Wrap the previous  M and  N up in quanitifers. (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 12760* Lemma for pythagtrip 12761. 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 12761* 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. (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 ) ) ) ) ) )
 
Theoremiserodd 12762* Collect the odd terms in a sequence. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( ph  /\  k  e.  NN0 )  ->  C  e.  CC )   &    |-  ( n  =  ( ( 2  x.  k )  +  1 )  ->  B  =  C )   =>    |-  ( ph  ->  (  seq  0 (  +  ,  ( k  e.  NN0  |->  C ) )  ~~>  A  <->  seq  1 (  +  ,  ( n  e.  NN  |->  if ( 2  ||  n ,  0 ,  B ) ) )  ~~>  A )
 )
 
6.2.5  The prime count function
 
Syntaxcpc 12763 Extend class notation with the prime count function.
 class  pCnt
 
Definitiondf-pc 12764* Define the prime count function, which returns the largest exponent of a given prime (or other natural number) 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 ,  <  ) ) ) ) ) )
 
Theorempclem 12765* - Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  A  =  { n  e.  NN0  |  ( P ^ n )  ||  N }   =>    |-  ( ( P  e.  ( ZZ>= `  2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( A  C_ 
 ZZ  /\  A  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  A  y 
 <_  x ) )
 
Theorempcprecl 12766* 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 ) )
 
Theorempcprendvds 12767* 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 )
 
Theorempcprendvds2 12768* 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 ) ) )
 
Theorempcpre1 12769* 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 )
 
Theorempcpremul 12770* 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 )
 
Theorempcval 12771* 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 ) ) ) )
 
Theorempceulem 12772* Lemma for pceu 12773. (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 ) )
 
Theorempceu 12773* 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 ) ) )
 
Theorempczpre 12774* 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 )
 
Theorempczcl 12775 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 )
 
Theorempccl 12776 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN )  ->  ( P  pCnt  N )  e.  NN0 )
 
Theorempccld 12777 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 )
 
Theorempcmul 12778 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 ) ) )
 
Theorempcdiv 12779 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 ) ) )
 
Theorempcqmul 12780 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 ) ) )
 
Theorempc0 12781 The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( P  e.  Prime  ->  ( P  pCnt  0 )  =  +oo )
 
Theorempc1 12782 Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( P  pCnt  1 )  =  0 )
 
Theorempcqcl 12783 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 )
 
Theorempcqdiv 12784 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 ) ) )
 
Theorempcrec 12785 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 ) )
 
Theorempcexp 12786 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 ) ) )
 
Theorempcxcl 12787 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* )
 
Theorempcge0 12788 The prime count of an integer is greater or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  0  <_  ( P  pCnt  N ) )
 
Theorempczdvds 12789 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 )
 
Theorempcdvds 12790 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 )
 
Theorempczndvds 12791 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 )
 
Theorempcndvds 12792 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 )
 
Theorempczndvds2 12793 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 ) ) ) )
 
Theorempcndvds2 12794 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 ) ) ) )
 
Theorempcdvdsb 12795  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 ) )
 
Theorempcelnn 12796 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 ) )
 
Theorempceq0 12797 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 )
 )
 
Theorempcidlem 12798 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 )
 
Theorempcid 12799 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 )
 
Theorempcneg 12800 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 )
 )
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