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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprmdiv 13101 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 13102 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 13103 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 13104* 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 13105 - Lemma for odzcl 13106, 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 13106 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 13107 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 13108 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 13109 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 13110 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 13111 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 13112 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 13113 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 13114 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 13115 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 13116 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 13117* Lemma for pythagtrip 13135. 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 13118* Lemma for pythagtrip 13135. 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 13119 Lemma for pythagtrip 13135. 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 13120 Lemma for pythagtrip 13135. 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 13121 Lemma for pythagtrip 13135. 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 13122 Lemma for pythagtrip 13135. 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 13123 Lemma for pythagtrip 13135. 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 13124 Lemma for pythagtrip 13135. 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 13125 Lemma for pythagtrip 13135. 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 13126 Lemma for pythagtrip 13135. 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 13127 Lemma for pythagtrip 13135. 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 13128 Lemma for pythagtrip 13135. 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 13129 Lemma for pythagtrip 13135. 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 13130 Lemma for pythagtrip 13135. 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 13131 Lemma for pythagtrip 13135. 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 13132 Lemma for pythagtrip 13135. 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 13133* Lemma for pythagtrip 13135. 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 13134* Lemma for pythagtrip 13135. 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 13135* 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 13136* 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 13137 Extend class notation with the prime count function.
 class  pCnt
 
Definitiondf-pc 13138* 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 13139* - 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 13140* 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 13141* 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 13142* 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 13143* 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 13144* 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 13145* 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 13146* Lemma for pceu 13147. (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 13147* 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 13148* 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 13149 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 13150 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 13151 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 13152 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 13153 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 13154 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 13155 The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( P  e.  Prime  ->  ( P  pCnt  0 )  =  +oo )
 
Theorempc1 13156 Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( P  pCnt  1 )  =  0 )
 
Theorempcqcl 13157 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 13158 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 13159 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 13160 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 13161 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 13162 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 13163 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 13164 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 13165 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 13166 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 13167 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 13168 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 13169  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 13170 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 13171 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 13172 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 13173 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 13174 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 )
 )
 
Theorempcabs 13175 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 ) )
 
Theorempcdvdstr 13176 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 ) )
 
Theorempcgcd1 13177 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 ) )
 
Theorempcgcd 13178 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 ) ) )
 
Theorempc2dvds 13179* 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 ) ) )
 
Theorempc11 13180* 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 ) ) )
 
Theorempcz 13181* 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 ) ) )
 
Theorempcprmpw2 13182* 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 ) ) ) )
 
Theorempcprmpw 13183* 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 ) ) ) )
 
Theorempcaddlem 13184 Lemma for pcadd 13185. The original numbers  A and  B have been decomposed using the prime count function as  ( P ^ M )  x.  ( R  /  S ) where  R ,  S are both not divisible by  P and  M  =  ( P  pCnt  A ), and similarly for  B. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  =  ( ( P ^ M )  x.  ( R  /  S ) ) )   &    |-  ( ph  ->  B  =  ( ( P ^ N )  x.  ( T  /  U ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  ( R  e.  ZZ  /\  -.  P  ||  R ) )   &    |-  ( ph  ->  ( S  e.  NN  /\  -.  P  ||  S ) )   &    |-  ( ph  ->  ( T  e.  ZZ  /\  -.  P  ||  T ) )   &    |-  ( ph  ->  ( U  e.  NN  /\  -.  P  ||  U ) )   =>    |-  ( ph  ->  M 
 <_  ( P  pCnt  ( A  +  B )
 ) )
 
Theorempcadd 13185 An inequality for the prime count of a sum. This is the source of the ultrametric inequality for the p-adic metric. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )   =>    |-  ( ph  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( A  +  B ) ) )
 
Theorempcadd2 13186 The inequality of pcadd 13185 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  ( P  pCnt  A )  < 
 ( P  pCnt  B ) )   =>    |-  ( ph  ->  ( P  pCnt  A )  =  ( P  pCnt  ( A  +  B )
 ) )
 
Theorempcmptcl 13187 Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   =>    |-  ( ph  ->  ( F : NN --> NN  /\  seq  1 (  x.  ,  F ) : NN --> NN ) )
 
Theorempcmpt 13188* Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( n  =  P  ->  A  =  B )   =>    |-  ( ph  ->  ( P  pCnt  (  seq  1 (  x.  ,  F ) `
  N ) )  =  if ( P 
 <_  N ,  B , 
 0 ) )
 
Theorempcmpt2 13189* Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( n  =  P  ->  A  =  B )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  N )
 )   =>    |-  ( ph  ->  ( P  pCnt  ( (  seq  1 (  x.  ,  F ) `  M )  /  (  seq  1 (  x. 
 ,  F ) `  N ) ) )  =  if ( ( P  <_  M  /\  -.  P  <_  N ) ,  B ,  0 ) )
 
Theorempcmptdvds 13190 The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ( ZZ>=
 `  N ) )   =>    |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `  N )  ||  (  seq  1 (  x. 
 ,  F ) `  M ) )
 
Theorempcprod 13191* The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( N  e.  NN  ->  (  seq  1 (  x.  ,  F ) `
  N )  =  N )
 
Theoremsumhash 13192* The sum of 1 over a set is the size of the set. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 20-May-2014.)
 |-  ( ( B  e.  Fin  /\  A  C_  B )  -> 
 sum_ k  e.  B  if ( k  e.  A ,  1 ,  0 )  =  ( # `  A ) )
 
Theoremfldivp1 13193 The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( |_ `  ( ( M  +  1 )  /  N ) )  -  ( |_ `  ( M  /  N ) ) )  =  if ( N  ||  ( M  +  1
 ) ,  1 ,  0 ) )
 
Theorempcfaclem 13194 Lemma for pcfac 13195. (Contributed by Mario Carneiro, 20-May-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( |_ `  ( N  /  ( P ^ M ) ) )  =  0 )
 
Theorempcfac 13195* Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P  pCnt  ( ! `  N ) )  =  sum_ k  e.  ( 1 ...
 M ) ( |_ `  ( N  /  ( P ^ k ) ) ) )
 
Theorempcbc 13196* Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
 |-  ( ( N  e.  NN  /\  K  e.  (
 0 ... N )  /\  P  e.  Prime )  ->  ( P  pCnt  ( N  _C  K ) )  =  sum_ k  e.  (
 1 ... N ) ( ( |_ `  ( N  /  ( P ^
 k ) ) )  -  ( ( |_ `  ( ( N  -  K )  /  ( P ^ k ) ) )  +  ( |_ `  ( K  /  ( P ^ k ) ) ) ) ) )
 
Theoremqexpz 13197 If a power of a rational number is an integer, then the number is an integer. In other words, all n-th roots are irrational unless they are integers (so that the original number is an n-th power). (Contributed by Mario Carneiro, 10-Aug-2015.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  A  e.  ZZ )
 
Theoremexpnprm 13198 A second or higher power of a rational number is not a prime number. Or by contraposition, the n-th root of a prime number is irrational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)
 |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>=
 `  2 ) ) 
 ->  -.  ( A ^ N )  e.  Prime )
 
6.2.6  Pocklington's theorem
 
Theoremprmpwdvds 13199 A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)
 |-  ( ( ( K  e.  ZZ  /\  D  e.  ZZ )  /\  ( P  e.  Prime  /\  N  e.  NN )  /\  ( D  ||  ( K  x.  ( P ^ N ) )  /\  -.  D  ||  ( K  x.  ( P ^ ( N  -  1 ) ) ) ) )  ->  ( P ^ N )  ||  D )
 
Theorempockthlem 13200 Lemma for pockthg 13201. (Contributed by Mario Carneiro, 2-Mar-2014.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  B  <  A )   &    |-  ( ph  ->  N  =  ( ( A  x.  B )  +  1
 ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  P 
 ||  N )   &    |-  ( ph  ->  Q  e.  Prime )   &    |-  ( ph  ->  ( Q  pCnt  A )  e.  NN )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  ( ( C ^ ( N  -  1 ) ) 
 mod  N )  =  1 )   &    |-  ( ph  ->  ( ( ( C ^
 ( ( N  -  1 )  /  Q ) )  -  1 ) 
 gcd  N )  =  1 )   =>    |-  ( ph  ->  ( Q  pCnt  A )  <_  ( Q  pCnt  ( P  -  1 ) ) )
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