HomeHome Metamath Proof Explorer
Theorem List (p. 147 of 402)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26569)
  Hilbert Space Explorer  Hilbert Space Explorer
(26570-28092)
  Users' Mathboxes  Users' Mathboxes
(28093-40161)
 

Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlcmfnncl 14601 Closure of the lcm function. (Contributed by AV, 20-Apr-2020.)
 |-  ( ( Z  C_  NN  /\  Z  e.  Fin )  ->  (lcm `  Z )  e. 
 NN )
 
Theoremlcmfeq0b 14602 The least common multiple of a set of integers is 0 iff at least one of its element is 0. (Contributed by AV, 21-Aug-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin )  ->  ( (lcm `  Z )  =  0  <->  0  e.  Z ) )
 
Theoremdvdslcmf 14603* The least common multiple of a set of integers is divisible by each of its elements. (Contributed by AV, 22-Aug-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin )  ->  A. x  e.  Z  x  ||  (lcm `  Z ) )
 
Theoremlcmfledvds 14604* A positive integer which is divisible by all elements of a set of integers bounds the least common multiple of the set. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) 
 ->  ( ( K  e.  NN  /\  A. m  e.  Z  m  ||  K )  ->  (lcm `  Z )  <_  K ) )
 
Theoremlcmf 14605* Characterization of the least common multiple of a set of integers (without 0): A positiven integer is the least common multiple of a set of integers iff it divides each of the elements of the set and every integer which divides each of the elements of the set is greater than or equal to this integer. (Contributed by AV, 22-Aug-2020.)
 |-  ( ( K  e.  NN  /\  ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) )  ->  ( K  =  (lcm `  Z )  <->  ( A. m  e.  Z  m  ||  K  /\  A. k  e.  NN  ( A. m  e.  Z  m  ||  k  ->  K  <_  k ) ) ) )
 
Theoremlcmf0 14606 The least common multiple of the empty set is 1. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  (lcm `  (/) )  =  1
 
Theoremlcmfsn 14607 The least common multiple of a singleton is its absolute value. (Contributed by AV, 22-Aug-2020.)
 |-  ( M  e.  ZZ  ->  (lcm `  { M }
 )  =  ( abs `  M ) )
 
Theoremlcmftp 14608 The least common multiple of a triple of integers is the least common multiple of the third integer and the the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 14616, this explicit proof (not based on induction) should be kept. (Proof modification is discouraged.) (Contributed by AV, 23-Aug-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (lcm `  { A ,  B ,  C }
 )  =  ( ( A lcm  B ) lcm  C ) )
 
Theoremlcmfunsnlem1 14609* Lemma for lcmfdvds 14614 and lcmfunsnlem 14613 (Induction step part 1). (Contributed by AV, 25-Aug-2020.)
 |-  ( ( ( z  e.  ZZ  /\  y  C_ 
 ZZ  /\  y  e.  Fin )  /\  ( A. k  e.  ZZ  ( A. m  e.  y  m  ||  k  ->  (lcm `  y )  ||  k
 )  /\  A. n  e. 
 ZZ  (lcm `  ( y  u. 
 { n } )
 )  =  ( (lcm `  y ) lcm  n ) ) )  ->  A. k  e.  ZZ  ( A. m  e.  ( y  u.  {
 z } ) m 
 ||  k  ->  (lcm `  ( y  u.  {
 z } ) ) 
 ||  k ) )
 
Theoremlcmfunsnlem2lem1 14610* Lemma 1 for lcmfunsnlem2 14612. (Contributed by AV, 26-Aug-2020.)
 |-  ( ( ( 0 
 e/  y  /\  z  =/=  0  /\  n  =/=  0 )  /\  ( n  e.  ZZ  /\  (
 ( z  e.  ZZ  /\  y  C_  ZZ  /\  y  e.  Fin )  /\  ( A. k  e.  ZZ  ( A. m  e.  y  m  ||  k  ->  (lcm `  y )  ||  k
 )  /\  A. n  e. 
 ZZ  (lcm `  ( y  u. 
 { n } )
 )  =  ( (lcm `  y ) lcm  n ) ) ) ) ) 
 ->  A. k  e.  NN  ( A. i  e.  (
 ( y  u.  {
 z } )  u. 
 { n } )
 i  ||  k  ->  ( (lcm `  ( y  u. 
 { z } )
 ) lcm  n )  <_  k ) )
 
Theoremlcmfunsnlem2lem2 14611* Lemma 2 for lcmfunsnlem2 14612. (Contributed by AV, 26-Aug-2020.)
 |-  ( ( ( 0 
 e/  y  /\  z  =/=  0  /\  n  =/=  0 )  /\  ( n  e.  ZZ  /\  (
 ( z  e.  ZZ  /\  y  C_  ZZ  /\  y  e.  Fin )  /\  ( A. k  e.  ZZ  ( A. m  e.  y  m  ||  k  ->  (lcm `  y )  ||  k
 )  /\  A. n  e. 
 ZZ  (lcm `  ( y  u. 
 { n } )
 )  =  ( (lcm `  y ) lcm  n ) ) ) ) ) 
 ->  (lcm `  ( ( y  u.  { z }
 )  u.  { n } ) )  =  ( (lcm `  ( y  u. 
 { z } )
 ) lcm  n ) )
 
Theoremlcmfunsnlem2 14612* Lemma for lcmfunsn 14616 and lcmfunsnlem 14613 (Induction step part 2). (Contributed by AV, 26-Aug-2020.)
 |-  ( ( ( z  e.  ZZ  /\  y  C_ 
 ZZ  /\  y  e.  Fin )  /\  ( A. k  e.  ZZ  ( A. m  e.  y  m  ||  k  ->  (lcm `  y )  ||  k
 )  /\  A. n  e. 
 ZZ  (lcm `  ( y  u. 
 { n } )
 )  =  ( (lcm `  y ) lcm  n ) ) )  ->  A. n  e.  ZZ  (lcm `  ( ( y  u.  { z }
 )  u.  { n } ) )  =  ( (lcm `  ( y  u. 
 { z } )
 ) lcm  n ) )
 
Theoremlcmfunsnlem 14613* Lemma for lcmfdvds 14614 and lcmfunsn 14616. These two theorems must be proven simultaneously by induction on the cardinality of a finite set  Y, because they depend on each other. This can be seen by the two parts lcmfunsnlem1 14609 and lcmfunsnlem2 14612 of the induction step, each of them using both induction hypotheses. (Contributed by AV, 26-Aug-2020.)
 |-  ( ( Y  C_  ZZ  /\  Y  e.  Fin )  ->  ( A. k  e.  ZZ  ( A. m  e.  Y  m  ||  k  ->  (lcm `  Y )  ||  k )  /\  A. n  e.  ZZ  (lcm `  ( Y  u.  { n } ) )  =  ( (lcm `  Y ) lcm  n ) ) )
 
Theoremlcmfdvds 14614* The least common multiple of a set of integers divides any integer which is divisible by all elements of the set. (Contributed by AV, 26-Aug-2020.)
 |-  ( ( K  e.  ZZ  /\  Z  C_  ZZ  /\  Z  e.  Fin )  ->  ( A. m  e.  Z  m  ||  K  ->  (lcm `  Z )  ||  K ) )
 
Theoremlcmfdvdsb 14615* Biconditional form of lcmfdvds 14614. (Contributed by AV, 26-Aug-2020.)
 |-  ( ( K  e.  ZZ  /\  Z  C_  ZZ  /\  Z  e.  Fin )  ->  ( A. m  e.  Z  m  ||  K  <->  (lcm `  Z )  ||  K ) )
 
Theoremlcmfunsn 14616 The lcm function for a union of a set of integer and a singleton. (Contributed by AV, 26-Aug-2020.)
 |-  ( ( Y  C_  ZZ  /\  Y  e.  Fin  /\  N  e.  ZZ )  ->  (lcm `  ( Y  u.  { N } ) )  =  ( (lcm `  Y ) lcm  N ) )
 
Theoremlcmfun 14617 The lcm function for a union of sets of integers. (Contributed by AV, 27-Aug-2020.)
 |-  ( ( ( Y 
 C_  ZZ  /\  Y  e.  Fin )  /\  ( Z 
 C_  ZZ  /\  Z  e.  Fin ) )  ->  (lcm `  ( Y  u.  Z ) )  =  (
 (lcm `  Y ) lcm  (lcm `  Z ) ) )
 
Theoremlcmfass 14618 Associative law for the lcm function. (Contributed by AV, 27-Aug-2020.)
 |-  ( ( ( Y 
 C_  ZZ  /\  Y  e.  Fin )  /\  ( Z 
 C_  ZZ  /\  Z  e.  Fin ) )  ->  (lcm `  ( { (lcm `  Y ) }  u.  Z ) )  =  (lcm `  ( Y  u.  {
 (lcm `  Z ) }
 ) ) )
 
Theoremlcmf2a3a4e12 14619 The least common multiple of 2 , 3 and 4 is 12. (Contributed by AV, 27-Aug-2020.)
 |-  (lcm `  { 2 ,  3 ,  4 } )  = ; 1 2
 
Theoremlcmflefac 14620 The least common multiple of all positive integers less than or equal to an integer is less than or equal to the factorial of the integer. (Contributed by AV, 16-Aug-2020.) (Revised by AV, 27-Aug-2020.)
 |-  ( N  e.  NN  ->  (lcm `  ( 1 ...
 N ) )  <_  ( ! `  N ) )
 
6.2  Elementary prime number theory
 
6.2.1  Elementary properties
 
Syntaxcprime 14621 Extend the definition of a class to include the set of prime numbers.
 class  Prime
 
Definitiondf-prm 14622* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |- 
 Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
 
Theoremisprm 14623* The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
 
Theoremprmnn 14624 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  ->  P  e.  NN )
 
Theoremprmz 14625 A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.)
 |-  ( P  e.  Prime  ->  P  e.  ZZ )
 
Theoremprmssnn 14626 The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
 |- 
 Prime  C_  NN
 
Theoremprmex 14627 The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
 |- 
 Prime  e.  _V
 
Theorem1nprm 14628 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
 |- 
 -.  1  e.  Prime
 
Theorem1idssfct 14629* The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN  ->  { 1 ,  N }  C_  { n  e. 
 NN  |  n  ||  N } )
 
Theoremisprm2lem 14630* Lemma for isprm2 14631. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( P  e.  NN  /\  P  =/=  1
 )  ->  ( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  =  {
 1 ,  P }
 ) )
 
Theoremisprm2 14631* The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
 
Theoremisprm3 14632* The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  ( 2 ... ( P  -  1 ) )  -.  z  ||  P ) )
 
Theoremisprm4 14633* The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  ( ZZ>= `  2 )
 ( z  ||  P  ->  z  =  P ) ) )
 
Theoremprmind2 14634* A variation on prmind 14635 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  z  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  ( y  x.  z )  ->  ( ph 
 <->  ta ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ps   &    |-  (
 ( x  e.  Prime  /\ 
 A. y  e.  (
 1 ... ( x  -  1 ) ) ch )  ->  ph )   &    |-  (
 ( y  e.  ( ZZ>=
 `  2 )  /\  z  e.  ( ZZ>= `  2 ) )  ->  ( ( ch  /\  th )  ->  ta )
 )   =>    |-  ( A  e.  NN  ->  et )
 
Theoremprmind 14635* Perform induction over the multiplicative structure of  NN. If a property  ph ( x ) holds for the primes and  1 and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  z  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  ( y  x.  z )  ->  ( ph 
 <->  ta ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ps   &    |-  ( x  e.  Prime  ->  ph )   &    |-  (
 ( y  e.  ( ZZ>=
 `  2 )  /\  z  e.  ( ZZ>= `  2 ) )  ->  ( ( ch  /\  th )  ->  ta )
 )   =>    |-  ( A  e.  NN  ->  et )
 
Theoremdvdsprime 14636 If  M divides a prime, then  M is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)
 |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P 
 <->  ( M  =  P  \/  M  =  1 ) ) )
 
Theoremnprm 14637 A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( A  e.  ( ZZ>= `  2 )  /\  B  e.  ( ZZ>= `  2 ) )  ->  -.  ( A  x.  B )  e.  Prime )
 
Theoremnprmi 14638 An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  A  e.  NN   &    |-  B  e.  NN   &    |-  1  <  A   &    |-  1  <  B   &    |-  ( A  x.  B )  =  N   =>    |-  -.  N  e.  Prime
 
Theorem2prm 14639 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  2  e.  Prime
 
Theorem3prm 14640 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  3  e.  Prime
 
Theoremprmuz2 14641 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 14642 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( P  e.  Prime  -> 
 1  <  P )
 
Theoremprmn2uzge3 14643 A prime number which is not 2 is an integer greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  ( ( P  e.  Prime  /\  P  =/=  2
 )  ->  P  e.  ( ZZ>= `  3 )
 )
 
Theoremprmm2nn0 14644 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 )
 
Theoremsqnprm 14645 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( A  e.  ZZ  ->  -.  ( A ^
 2 )  e.  Prime )
 
Theoremdvdsprm 14646 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 14647* 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 14648* 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 14649 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 )
 
Theoremisprm5 14650* One need only check prime divisors of  P up to  sqr P in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  Prime  ( ( z ^ 2 )  <_  P  ->  -.  z  ||  P ) ) )
 
Theoremmaxprmfct 14651* The set of prime factors of an integer greater than or equal to 2 satisfies the conditions to have a supremum, and that supremum is a member of the set. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  S  =  { z  e.  Prime  |  z  ||  N }   =>    |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( ( S  C_  ZZ  /\  S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x )  /\  sup ( S ,  RR ,  <  )  e.  S ) )
 
Theoremdivgcdodd 14652 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 ) ) ) )
 
6.2.2  Coprimality and Euclid's lemma

According to Wikipedia "Coprime integers", see https://en.wikipedia.org/wiki/Coprime_integers (16-Aug-2020) "[...] two integers a and b are said to be relatively prime, mutually prime, or coprime [...] if the only positive integer (factor) that divides both of them is 1. Consequently, any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.". In the following, we use this equivalent characterization to say that  A  e.  ZZ and  B  e.  ZZ are coprime (or relatively prime) if  ( A  gcd  B )  =  1. The equivalence of the definitions is shown by coprmgcdb 14653. The negation, i.e. two integers are not coprime, can be expressed either by  ( A  gcd  B )  =/=  1, see ncoprmgcdne1b 14654, or equivalently by  1  <  ( A  gcd  B ), see ncoprmgcdgt1b 14655.

The proof of Euclid's lemma, see euclemma 14664, is based on theorems about coprimality (e.g. on coprmdvds 14658).

 
Theoremcoprmgcdb 14653* Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A. i  e.  NN  ( ( i 
 ||  A  /\  i  ||  B )  ->  i  =  1 )  <->  ( A  gcd  B )  =  1 ) )
 
Theoremncoprmgcdne1b 14654* Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
 ( i  ||  A  /\  i  ||  B )  <-> 
 ( A  gcd  B )  =/=  1 ) )
 
Theoremncoprmgcdgt1b 14655* Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
 ( i  ||  A  /\  i  ||  B )  <-> 
 1  <  ( A  gcd  B ) ) )
 
Theoremcoprm 14656 A prime number either divides an integer or is coprime to it, but not both. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
 
Theoremprmrp 14657 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 ) )
 
Theoremcoprmdvds 14658 If an integer divides the product of two integers and is coprime to one of them, then it divides the other. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  ( M  x.  N )  /\  ( K  gcd  M )  =  1 ) 
 ->  K  ||  N )
 )
 
Theoremcoprmdvds2 14659 If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 gcd  N )  =  1 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M  x.  N )  ||  K ) )
 
Theoremmulgcddvds 14660 One half of rpmulgcd2 14661, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) ) 
 ||  ( ( K 
 gcd  M )  x.  ( K  gcd  N ) ) )
 
Theoremrpmulgcd2 14661 If  M is relatively prime to  N, then the GCD of  K with  M  x.  N is the product of the GCDs with  M and  N respectively. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M 
 gcd  N )  =  1 )  ->  ( K  gcd  ( M  x.  N ) )  =  (
 ( K  gcd  M )  x.  ( K  gcd  N ) ) )
 
Theoremqredeq 14662 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M 
 gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P 
 gcd  Q )  =  1 )  /\  ( M 
 /  N )  =  ( P  /  Q ) )  ->  ( M  =  P  /\  N  =  Q ) )
 
Theoremqredeu 14663* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( A  e.  QQ  ->  E! x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) )
 
Theoremeuclemma 14664 Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. (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 14665* A number is prime iff it satisfies Euclid's lemma euclemma 14664. (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 14666 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 14667 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 14668 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 14669 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 14670 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 14671 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 14672 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 14673 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 ) )
 
Theoremrpmul 14674 If  K is relatively prime to  M and to  N, it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( K 
 gcd  M )  =  1 
 /\  ( K  gcd  N )  =  1 ) 
 ->  ( K  gcd  ( M  x.  N ) )  =  1 ) )
 
Theoremrpdvds 14675 If  K is relatively prime to  N then it is also relatively prime to any divisor  M of  N. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  =  1 )
 
Theoremncoprmlnprm 14676 If two positive integers are not coprime, the larger of them is not a prime number. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  NN  /\  A  <  B ) 
 ->  ( 1  <  ( A  gcd  B )  ->  B  e/  Prime ) )
 
Theoremcoprmprod 14677* The product of the elements of a sequence of pairwise coprime positive integers is coprime to a positive integer which is coprime to all integers of the sequence. (Contributed by AV, 18-Aug-2020.)
 |-  ( ( ( M  e.  Fin  /\  M  C_  NN  /\  N  e.  NN )  /\  F : NN --> NN  /\  A. m  e.  M  ( ( F `
  m )  gcd  N )  =  1 ) 
 ->  ( A. m  e.  M  A. n  e.  ( M  \  { m } ) ( ( F `  m ) 
 gcd  ( F `  n ) )  =  1  ->  ( prod_ m  e.  M  ( F `
  m )  gcd  N )  =  1 ) )
 
Theoremcoprmproddvdslem 14678* Lemma for coprmproddvds 14679: Induction step. (Contributed by AV, 19-Aug-2020.)
 |-  ( ( y  e. 
 Fin  /\  -.  z  e.  y )  ->  (
 ( ( ( y 
 C_  NN  /\  ( K  e.  NN  /\  F : NN --> NN ) ) 
 /\  ( A. m  e.  y  A. n  e.  ( y  \  { m } ) ( ( F `  m ) 
 gcd  ( F `  n ) )  =  1  /\  A. m  e.  y  ( F `  m )  ||  K ) )  ->  prod_ m  e.  y  ( F `  m )  ||  K )  ->  ( ( ( ( y  u.  {
 z } )  C_  NN  /\  ( K  e.  NN  /\  F : NN --> NN ) )  /\  ( A. m  e.  (
 y  u.  { z } ) A. n  e.  ( ( y  u. 
 { z } )  \  { m } )
 ( ( F `  m )  gcd  ( F `
  n ) )  =  1  /\  A. m  e.  ( y  u.  { z } )
 ( F `  m )  ||  K ) ) 
 ->  prod_ m  e.  (
 y  u.  { z } ) ( F `
  m )  ||  K ) ) )
 
Theoremcoprmproddvds 14679* If a positive integer is divisible by each element of a set of pairwise coprime positive integers, then it is divisible by their product. (Contributed by AV, 19-Aug-2020.)
 |-  ( ( ( M 
 C_  NN  /\  M  e.  Fin )  /\  ( K  e.  NN  /\  F : NN --> NN )  /\  ( A. m  e.  M  A. n  e.  ( M 
 \  { m }
 ) ( ( F `
  m )  gcd  ( F `  n ) )  =  1  /\  A. m  e.  M  ( F `  m ) 
 ||  K ) ) 
 ->  prod_ m  e.  M  ( F `  m ) 
 ||  K )
 
Theorem3lcm2e6 14680 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
 
6.2.3  Properties of the canonical representation of a rational
 
Syntaxcnumer 14681 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 14682 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 14683* 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 14684* 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 14685* 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 14686* 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 14687 Lemma for qnumcl 14688 and qdencl 14689. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 14688 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 14689 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 14690 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 14691 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 14692 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 14693 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 14694 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 14695 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 14696 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 14697 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 14698 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 14699 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 14700 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 ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40161
  Copyright terms: Public domain < Previous  Next >