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Theorem List for Intuitionistic Logic Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprm2orodd 11201 A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
 |-  ( P  e.  Prime  ->  ( P  =  2  \/  -.  2  ||  P ) )
 
Theorem2prm 11202 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  2  e.  Prime
 
Theorem3prm 11203 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  3  e.  Prime
 
Theorem4nprm 11204 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  4  e.  Prime
 
Theoremprmuz2 11205 A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 ) )
 
Theoremprmgt1 11206 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( P  e.  Prime  -> 
 1  <  P )
 
Theoremprmm2nn0 11207 Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
 |-  ( P  e.  Prime  ->  ( P  -  2
 )  e.  NN0 )
 
Theoremoddprmgt2 11208 An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
 |-  ( P  e.  ( Prime  \  { 2 } )  ->  2  <  P )
 
Theoremoddprmge3 11209 An odd prime is greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.)
 |-  ( P  e.  ( Prime  \  { 2 } )  ->  P  e.  ( ZZ>= `  3 )
 )
 
Theoremsqnprm 11210 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( A  e.  ZZ  ->  -.  ( A ^
 2 )  e.  Prime )
 
Theoremdvdsprm 11211 An integer greater than or equal to 2 divides a prime number iff it is equal to it. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( N  e.  ( ZZ>= `  2 )  /\  P  e.  Prime )  ->  ( N  ||  P  <->  N  =  P ) )
 
Theoremexprmfct 11212* Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  E. p  e.  Prime  p 
 ||  N )
 
Theoremprmdvdsfz 11213* Each integer greater than 1 and less then or equal to a fixed number is divisible by a prime less then or equal to this fixed number. (Contributed by AV, 15-Aug-2020.)
 |-  ( ( N  e.  NN  /\  I  e.  (
 2 ... N ) ) 
 ->  E. p  e.  Prime  ( p  <_  N  /\  p  ||  I ) )
 
Theoremnprmdvds1 11214 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
 |-  ( P  e.  Prime  ->  -.  P  ||  1 )
 
Theoremdivgcdodd 11215 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 ) ) ) )
 
4.2.2  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 11218.

 
Theoremcoprm 11216 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 11217 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 11218 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 11219* A number is prime iff it satisfies Euclid's lemma euclemma 11218. (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 11220 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 11221 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 11222 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 11223 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 11224 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 11225 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 11226 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 11227 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 11228 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 11229 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 11230 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 11231 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 11232 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
 
4.2.3  Non-rationality of square root of 2
 
Theoremsqrt2irrlem 11233 Lemma for sqrt2irr 11234. 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 11234 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 11251 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 11233, 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 11235 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
 |-  ( sqr `  2
 )  e.  RR
 
Theorempw2dvdslemn 11236* Lemma for pw2dvds 11237. 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 11237* 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 11238 Lemma for pw2dvdseu 11239. 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 11239* 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 11240* Lemma for oddpwdc 11245. 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 11241* Lemma for oddpwdc 11245. 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 11242* Lemma for oddpwdc 11245. 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 11243* Lemma for oddpwdc 11245. 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 11244* Lemma for oddpwdc 11245. 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 11245* 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 11246* 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 11247* 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 11248 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 11249 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 11250 Lemma for sqrt2irrap 11251. 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 11251 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 11234. (Contributed by Jim Kingdon, 2-Oct-2021.)
 |-  ( Q  e.  QQ  ->  ( sqr `  2
 ) #  Q )
 
4.2.4  Properties of the canonical representation of a rational
 
Syntaxcnumer 11252 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 11253 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 11254* 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 11255* 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 11256* 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 11257* 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 11258 Lemma for qnumcl 11259 and qdencl 11260. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 11259 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 11260 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 11261 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 11262 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 11263 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 11264 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 11265 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 11266 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 11267 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 11268 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 11269 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 11270 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 11271 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 11272 nn0gcdsq 11271 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 11273 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 11274 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 ) )
 
Theoremdensq 11275 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) )
 
Theoremqden1elz 11276 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 11277 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 11278 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 )
 
4.2.5  Euler's theorem
 
Syntaxcphi 11279 Extend class notation with the Euler phi function.
 class  phi
 
Definitiondf-phi 11280* 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 11281* 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 11282* 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 11283 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 11284 Closure for the value of the Euler 
phi function. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
 
Theoremphibndlem 11285* Lemma for phibnd 11286. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  { x  e.  (
 1 ... N )  |  ( x  gcd  N )  =  1 }  C_  ( 1 ... ( N  -  1 ) ) )
 
Theoremphibnd 11286 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 11287 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 11288 Value of the Euler  phi function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( phi `  1
 )  =  1
 
Theoremdfphi2 11289* 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 11290* 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 11291 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 11292 Value of the Euler  phi function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1
 ) )
 
Theoremcrth 11293* 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 11294* Lemma for phimul 11295. (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 11295 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 ) ) )
 
Theoremhashgcdlem 11296* 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 11297* 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 ) )
 
4.3  Cardinality of real and complex number subsets
 
4.3.1  Countability of integers and rationals
 
Theoremoddennn 11298 There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.)
 |- 
 { z  e.  NN  |  -.  2  ||  z }  ~~  NN
 
Theoremevenennn 11299 There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.)
 |- 
 { z  e.  NN  |  2  ||  z }  ~~  NN
 
Theoremxpnnen 11300 The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.)
 |-  ( NN  X.  NN )  ~~  NN
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