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Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivgcdodd 12101 Either  A  /  ( A  gcd  B ) is odd or  B  /  ( A  gcd  B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( -.  2  ||  ( A  /  ( A  gcd  B ) )  \/  -.  2  ||  ( B  /  ( A  gcd  B ) ) ) )
 
5.2.2  Coprimality and Euclid's lemma (cont.)

This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 12104.

 
Theoremcoprm 12102 A prime number either divides an integer or is coprime to it, but not both. Theorem 1.8 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
 
Theoremprmrp 12103 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime ) 
 ->  ( ( P  gcd  Q )  =  1  <->  P  =/=  Q ) )
 
Theoremeuclemma 12104 Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. Theorem 1.9 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( ( P  e.  Prime  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( P  ||  ( M  x.  N )  <->  ( P  ||  M  \/  P  ||  N ) ) )
 
Theoremisprm6 12105* A number is prime iff it satisfies Euclid's lemma euclemma 12104. (Contributed by Mario Carneiro, 6-Sep-2015.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. x  e.  ZZ  A. y  e. 
 ZZ  ( P  ||  ( x  x.  y
 )  ->  ( P  ||  x  \/  P  ||  y ) ) ) )
 
Theoremprmdvdsexp 12106 A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <->  P  ||  A ) )
 
Theoremprmdvdsexpb 12107 A prime divides a positive power of another iff they are equal. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime  /\  N  e.  NN )  ->  ( P  ||  ( Q ^ N )  <->  P  =  Q ) )
 
Theoremprmdvdsexpr 12108 If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime  /\  N  e.  NN0 )  ->  ( P  ||  ( Q ^ N )  ->  P  =  Q )
 )
 
Theoremprmexpb 12109 Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  ( ( P ^ M )  =  ( Q ^ N )  <->  ( P  =  Q  /\  M  =  N ) ) )
 
Theoremprmfac1 12110 The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  P  e.  Prime  /\  P  ||  ( ! `  N ) )  ->  P  <_  N )
 
Theoremrpexp 12111 If two numbers  A and  B are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) )
 
Theoremrpexp1i 12112 Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ M )  gcd  B )  =  1 ) )
 
Theoremrpexp12i 12113 Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 )
 )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ M )  gcd  ( B ^ N ) )  =  1 ) )
 
Theoremprmndvdsfaclt 12114 A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN0 )  ->  ( N  <  P 
 ->  -.  P  ||  ( ! `  N ) ) )
 
Theoremcncongrprm 12115 Corollary 2 of Cancellability of Congruences: Two products with a common factor are congruent modulo a prime number not dividing the common factor iff the other factors are congruent modulo the prime number. (Contributed by AV, 13-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( P  e.  Prime  /\  -.  P  ||  C ) )  ->  ( ( ( A  x.  C )  mod  P )  =  ( ( B  x.  C ) 
 mod  P )  <->  ( A  mod  P )  =  ( B 
 mod  P ) ) )
 
Theoremisevengcd2 12116 The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
 |-  ( Z  e.  ZZ  ->  ( 2  ||  Z  <->  ( 2  gcd  Z )  =  2 ) )
 
Theoremisoddgcd1 12117 The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
 |-  ( Z  e.  ZZ  ->  ( -.  2  ||  Z 
 <->  ( 2  gcd  Z )  =  1 )
 )
 
Theorem3lcm2e6 12118 The least common multiple of three and two is six. The operands are unequal primes and thus coprime, so the result is (the absolute value of) their product. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 27-Aug-2020.)
 |-  ( 3 lcm  2 )  =  6
 
5.2.3  Non-rationality of square root of 2
 
Theoremsqrt2irrlem 12119 Lemma for sqrt2irr 12120. This is the core of the proof: - if  A  /  B  =  sqr ( 2 ), then 
A and  B are even, so  A  /  2 and  B  /  2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  ( sqr `  2
 )  =  ( A 
 /  B ) )   =>    |-  ( ph  ->  ( ( A  /  2 )  e. 
 ZZ  /\  ( B  /  2 )  e.  NN ) )
 
Theoremsqrt2irr 12120 The square root of 2 is not rational. That is, for any rational number,  ( sqr `  2
) does not equal it. However, if we were to say "the square root of 2 is irrational" that would mean something stronger: "for any rational number, 
( sqr `  2
) is apart from it" (the two statements are equivalent given excluded middle). See sqrt2irrap 12138 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12119, which shows that if  A  /  B  =  sqr ( 2 ), then 
A and  B are even, so  A  /  2 and  B  /  2 are smaller representatives, which is absurd. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

 |-  ( sqr `  2
 )  e/  QQ
 
Theoremsqrt2re 12121 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
 |-  ( sqr `  2
 )  e.  RR
 
Theoremsqrt2irr0 12122 The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
 |-  ( sqr `  2
 )  e.  ( RR  \  QQ )
 
Theorempw2dvdslemn 12123* Lemma for pw2dvds 12124. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( ( N  e.  NN  /\  A  e.  NN  /\ 
 -.  ( 2 ^ A )  ||  N ) 
 ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 )
 )  ||  N )
 )
 
Theorempw2dvds 12124* A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( N  e.  NN  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 )
 )  ||  N )
 )
 
Theorempw2dvdseulemle 12125 Lemma for pw2dvdseu 12126. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  ( 2 ^ A ) 
 ||  N )   &    |-  ( ph  ->  -.  ( 2 ^ ( B  +  1 ) )  ||  N )   =>    |-  ( ph  ->  A  <_  B )
 
Theorempw2dvdseu 12126* A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.)
 |-  ( N  e.  NN  ->  E! m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 )
 )  ||  N )
 )
 
Theoremoddpwdclemxy 12127* Lemma for oddpwdc 12132. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.)
 |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) ) 
 ->  ( X  =  ( A  /  ( 2 ^ ( iota_ z  e. 
 NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) ) ) )  /\  Y  =  ( iota_ z  e.  NN0  ( ( 2 ^
 z )  ||  A  /\  -.  ( 2 ^
 ( z  +  1 ) )  ||  A ) ) ) )
 
Theoremoddpwdclemdvds 12128* Lemma for oddpwdc 12132. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  ( A  e.  NN  ->  ( 2 ^ ( iota_
 z  e.  NN0  (
 ( 2 ^ z
 )  ||  A  /\  -.  ( 2 ^ (
 z  +  1 ) )  ||  A )
 ) )  ||  A )
 
Theoremoddpwdclemndvds 12129* Lemma for oddpwdc 12132. A natural number is not divisible by one more than the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  ( A  e.  NN  ->  -.  ( 2 ^
 ( ( iota_ z  e. 
 NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) )  +  1 ) )  ||  A )
 
Theoremoddpwdclemodd 12130* Lemma for oddpwdc 12132. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.)
 |-  ( A  e.  NN  ->  -.  2  ||  ( A  /  ( 2 ^
 ( iota_ z  e.  NN0  ( ( 2 ^
 z )  ||  A  /\  -.  ( 2 ^
 ( z  +  1 ) )  ||  A ) ) ) ) )
 
Theoremoddpwdclemdc 12131* Lemma for oddpwdc 12132. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.)
 |-  ( ( ( ( X  e.  NN  /\  -.  2  ||  X )  /\  Y  e.  NN0 )  /\  A  =  ( ( 2 ^ Y )  x.  X ) )  <-> 
 ( A  e.  NN  /\  ( X  =  ( A  /  ( 2 ^ ( iota_ z  e. 
 NN0  ( ( 2 ^ z )  ||  A  /\  -.  ( 2 ^ ( z  +  1 ) )  ||  A ) ) ) )  /\  Y  =  ( iota_ z  e.  NN0  ( ( 2 ^
 z )  ||  A  /\  -.  ( 2 ^
 ( z  +  1 ) )  ||  A ) ) ) ) )
 
Theoremoddpwdc 12132* The function  F that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.)
 |-  J  =  { z  e.  NN  |  -.  2  ||  z }   &    |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x ) )   =>    |-  F : ( J  X.  NN0 ) -1-1-onto-> NN
 
Theoremsqpweven 12133* The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  J  =  { z  e.  NN  |  -.  2  ||  z }   &    |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x ) )   =>    |-  ( A  e.  NN  ->  2  ||  ( 2nd `  ( `' F `  ( A ^ 2 ) ) ) )
 
Theorem2sqpwodd 12134* The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  J  =  { z  e.  NN  |  -.  2  ||  z }   &    |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x ) )   =>    |-  ( A  e.  NN  ->  -.  2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) ) )
 
Theoremsqne2sq 12135 The square of a natural number can never be equal to two times the square of a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A ^
 2 )  =/=  (
 2  x.  ( B ^ 2 ) ) )
 
Theoremznege1 12136 The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B ) 
 ->  1  <_  ( abs `  ( A  -  B ) ) )
 
Theoremsqrt2irraplemnn 12137 Lemma for sqrt2irrap 12138. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( sqr `  2
 ) #  ( A  /  B ) )
 
Theoremsqrt2irrap 12138 The square root of 2 is irrational. That is, for any rational number,  ( sqr `  2
) is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 12120. (Contributed by Jim Kingdon, 2-Oct-2021.)
 |-  ( Q  e.  QQ  ->  ( sqr `  2
 ) #  Q )
 
5.2.4  Properties of the canonical representation of a rational
 
Syntaxcnumer 12139 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 12140 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 12141* The canonical numerator of a rational is the numerator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer  =  ( y  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Definitiondf-denom 12142* The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom  =  ( y  e.  QQ  |->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumval 12143* Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqdenval 12144* Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumdencl 12145 Lemma for qnumcl 12146 and qdencl 12147. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 12146 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 12147 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 12148 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 12149 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 12150 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 12151 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 12152 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 12153 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 12154 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 12155 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 12156 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 12157 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 12158 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 12159 nn0gcdsq 12158 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 12160 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 12161 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 ) )
 
Theoremdensq 12162 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) )
 
Theoremqden1elz 12163 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 12164 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 12165 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 12166 Extend class notation with the order function on the class of integers modulo N.
 class  odZ
 
Syntaxcphi 12167 Extend class notation with the Euler phi function.
 class  phi
 
Definitiondf-odz 12168* 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 12169* 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 12170* 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 12171* 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 12172 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 12173 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
 
Theoremphibndlem 12174* Lemma for phibnd 12175. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  C_  ( 1 ... ( N  -  1 ) ) )
 
Theoremphibnd 12175 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 12176 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 12177 Value of the Euler  phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( phi `  1
 )  =  1
 
Theoremdfphi2 12178* 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 12179* 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 12180 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 12181 Value of the Euler  phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1
 ) )
 
Theoremcrth 12182* 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 12183* Lemma for phimul 12184. (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 12184 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 12185* Lemma for eulerth 12191. (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 12186* Lemma for eulerth 12191. 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 12187* Lemma for eulerth 12191. 
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 12188* Lemma for eulerth 12191. (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 12189* Lemma for eulerth 12191. 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 12190* Lemma for eulerth 12191. 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 12191 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 12192 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 12193 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 12194 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 12195 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 12196* 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 12197* 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 12198* 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 12199* 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 12200 - Lemma for odzcl 12201, 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 ) ) )
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