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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvdslcm 14501 The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
 
Theoremlcmledvds 14502 A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 )
 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  <_  K ) )
 
Theoremlcmeq0 14503 The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm 
 N )  =  0  <-> 
 ( M  =  0  \/  N  =  0 ) ) )
 
Theoremlcmcl 14504 Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
 
Theoremgcddvdslcm 14505 The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  ( M lcm  N ) )
 
Theoremlcmneg 14506 Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N )  =  ( M lcm  N ) )
 
Theoremneglcm 14507 Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M lcm  N )  =  ( M lcm 
 N ) )
 
Theoremlcmabs 14508 The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M ) lcm  ( abs `  N ) )  =  ( M lcm  N ) )
 
Theoremlcmgcdlem 14509 Lemma for lcmgcd 14510 and lcmdvds 14511. Prove them for positive  M,  N, and  K. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( ( M lcm  N )  x.  ( M  gcd  N ) )  =  ( abs `  ( M  x.  N ) )  /\  ( ( K  e.  NN  /\  ( M  ||  K  /\  N  ||  K ) )  ->  ( M lcm 
 N )  ||  K ) ) )
 
Theoremlcmgcd 14510 The product of two numbers' least common multiple and greatest common divisor is the absolute value of the product of the two numbers. In particular, that absolute value is the least common multiple of two coprime numbers, for which  ( M  gcd  N
)  =  1.

Multiple methods exist for proving this, and it is often proven either as a consequence of the fundamental theorem of arithmetic 1arith 14809 or of Bézout's identity bezout 14448; see e.g. https://proofwiki.org/wiki/Product_of_GCD_and_LCM and https://math.stackexchange.com/a/470827. This proof uses the latter to first confirm it for positive integer  M and 
N (the "Second Proof" in the above Stack Exchange page), then shows that implies it for all nonzero integer inputs, then finally uses lcm0val 14496 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)

 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm 
 N )  x.  ( M  gcd  N ) )  =  ( abs `  ( M  x.  N ) ) )
 
Theoremlcmdvds 14511 The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K ) )
 
Theoremlcmid 14512 The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( M  e.  ZZ  ->  ( M lcm  M )  =  ( abs `  M ) )
 
Theoremlcm1 14513 The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.)
 |-  ( M  e.  ZZ  ->  ( M lcm  1 )  =  ( abs `  M ) )
 
Theoremlcmgcdnn 14514 The product of two positive integers' least common multiple and greatest common divisor is the product of the two integers. (Contributed by AV, 27-Aug-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( M lcm 
 N )  x.  ( M  gcd  N ) )  =  ( M  x.  N ) )
 
Theoremlcmgcdeq 14515 Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm 
 N )  =  ( M  gcd  N )  <-> 
 ( abs `  M )  =  ( abs `  N ) ) )
 
Theoremlcmdvdsb 14516 Biconditional form of lcmdvds 14511. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K ) 
 <->  ( M lcm  N ) 
 ||  K ) )
 
Theoremlcmass 14517 Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( N lcm  M ) lcm  P )  =  ( N lcm  ( M lcm  P ) ) )
 
Theorem3lcm2e6woprm 14518 The least common multiple of three and two is six. In contrast to 3lcm2e6 14619, this proof does not use the property of 2 and 3 being prime, therefore it is much longer. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 3 lcm  2 )  =  6
 
Theorem6lcm4e12 14519 The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
 |-  ( 6 lcm  4 )  = ; 1 2
 
TheoremlcmscllemOLD 14520* Lemma for lcmsOLD 14522, analogous to lcmcllem 14499. (Contributed by AV, 14-Aug-2020.) Obsolete version of lcmfcllem 14536 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  L  =  sup ( { n  e.  NN  |  A. m  e.  M  m  ||  n } ,  RR ,  `'  <  )   =>    |-  (
 ( M  C_  NN  /\  M  e.  Fin )  ->  L  e.  { n  e.  NN  |  A. m  e.  M  m  ||  n } )
 
TheoremlcmsnnOLD 14521* The least common multiple of a finite set of positive integers is a positive integer, analogous to lcmn0cl 14500. (Contributed by AV, 14-Aug-2020.) Obsolete version of lcmfn0cl 14537 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  L  =  sup ( { n  e.  NN  |  A. m  e.  M  m  ||  n } ,  RR ,  `'  <  )   =>    |-  (
 ( M  C_  NN  /\  M  e.  Fin )  ->  L  e.  NN )
 
TheoremlcmsOLD 14522* The least common multiple of a finite set of positive integers is divisible by each element of the set, analogous to dvdslcm 14501. (Contributed by AV, 14-Aug-2020.) Obsolete version of dvdslcmf 14542 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  L  =  sup ( { n  e.  NN  |  A. m  e.  M  m  ||  n } ,  RR ,  `'  <  )   =>    |-  (
 ( M  C_  NN  /\  M  e.  Fin )  ->  A. m  e.  M  m  ||  L )
 
TheoremlcmsledvdsOLD 14523* The least common multiple of a set of integers dividing an integer is less than or equal to this integer. Analogous to lcmledvds 14502. (Contributed by AV, 16-Aug-2020.) Obsolete version of lcmfledvds 14543 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  L  =  sup ( { n  e.  NN  |  A. m  e.  M  m  ||  n } ,  RR ,  `'  <  )   =>    |-  (
 ( K  e.  NN  /\  M  C_  NN  /\  M  e.  Fin )  ->  ( A. m  e.  M  m  ||  K  ->  L  <_  K ) )
 
TheoremlcmslefacOLD 14524* 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.) Obsolete version of lcmflefac 14559 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  F  =  ( x  e.  NN  |->  sup ( { n  e.  NN  |  A. m  e.  (
 1 ... x ) m 
 ||  n } ,  RR ,  `'  <  )
 )   =>    |-  ( N  e.  NN  ->  ( F `  N )  <_  ( ! `  N ) )
 
Theoremabsproddvds 14525* The absolute value of the product of the elements of a finite subset of the integers is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.)
 |-  ( ph  ->  Z  C_ 
 ZZ )   &    |-  ( ph  ->  Z  e.  Fin )   &    |-  P  =  ( abs `  prod_ z  e.  Z  z )   =>    |-  ( ph  ->  A. m  e.  Z  m  ||  P )
 
Theoremabsprodnn 14526* The absolute value of the product of the elements of a finite subset of the integers not containing 0 is a poitive integer. (Contributed by AV, 21-Aug-2020.)
 |-  ( ph  ->  Z  C_ 
 ZZ )   &    |-  ( ph  ->  Z  e.  Fin )   &    |-  P  =  ( abs `  prod_ z  e.  Z  z )   &    |-  ( ph  ->  0  e/  Z )   =>    |-  ( ph  ->  P  e.  NN )
 
Theoremfissn0dvds 14527* For each finite subset of the integers not containing 0 there is a positive integer which is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) 
 ->  E. n  e.  NN  A. m  e.  Z  m  ||  n )
 
Theoremfissn0dvdsn0 14528* For each finite subset of the integers not containing 0 there is a positive integer which is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) 
 ->  { n  e.  NN  |  A. m  e.  Z  m  ||  n }  =/=  (/) )
 
Theoremlcmfval 14529* Value of the lcm function.  (lcm `  Z ) is the least common multiple of the integers contained in the finite subset of integers  Z. If at least one of the elements of  Z is  0, the result is defined conventionally as  0. (Contributed by AV, 21-Apr-2020.) (Revised by AV, 16-Sep-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin )  ->  (lcm `  Z )  =  if ( 0  e.  Z ,  0 , inf ( { n  e. 
 NN  |  A. m  e.  Z  m  ||  n } ,  RR ,  <  ) ) )
 
Theoremlcmf0val 14530 The value, by convention, of the least common multiple for a set containing 0 is 0. (Contributed by AV, 21-Apr-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( Z  C_  ZZ  /\  0  e.  Z )  ->  (lcm `  Z )  =  0 )
 
Theoremlcmfn0val 14531* The value of the lcm function for a set without 0. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) 
 ->  (lcm `  Z )  = inf ( { n  e. 
 NN  |  A. m  e.  Z  m  ||  n } ,  RR ,  <  ) )
 
Theoremlcmfnnval 14532* The value of the lcm function for a subset of the positive integers. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.)
 |-  ( ( Z  C_  NN  /\  Z  e.  Fin )  ->  (lcm `  Z )  = inf ( { n  e. 
 NN  |  A. m  e.  Z  m  ||  n } ,  RR ,  <  ) )
 
TheoremlcmfvalOLD 14533* Value of the lcm function.  (lcm `  Z ) is the least common multiple of the integers contained in the finite subset of integers  Z. If at least one of the elements of  Z is  0, the result is defined conventionally as  0. (Contributed by AV, 21-Apr-2020.) Obsolete version of lcmfval 14529 as of 16-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin )  ->  (lcm `  Z )  =  if ( 0  e.  Z ,  0 , 
 sup ( { n  e.  NN  |  A. m  e.  Z  m  ||  n } ,  RR ,  `'  <  ) ) )
 
Theoremlcmfn0valOLD 14534* The value of the lcm function for a set without 0. (Contributed by AV, 21-Aug-2020.) Obsolete version of lcmfn0val 14531 as of 16-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) 
 ->  (lcm `  Z )  = 
 sup ( { n  e.  NN  |  A. m  e.  Z  m  ||  n } ,  RR ,  `'  <  ) )
 
TheoremlcmfnnvalOLD 14535* The value of the lcm function for a subset of the positive integers. (Contributed by AV, 21-Aug-2020.) Obsolete version of lcmfnnval 14532 as of 16-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( Z  C_  NN  /\  Z  e.  Fin )  ->  (lcm `  Z )  = 
 sup ( { n  e.  NN  |  A. m  e.  Z  m  ||  n } ,  RR ,  `'  <  ) )
 
Theoremlcmfcllem 14536* Lemma for lcmfn0cl 14537 and dvdslcmf 14542. (Contributed by AV, 21-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) 
 ->  (lcm `  Z )  e. 
 { n  e.  NN  |  A. m  e.  Z  m  ||  n } )
 
Theoremlcmfn0cl 14537 Closure of the lcm function. (Contributed by AV, 21-Aug-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin  /\  0  e/  Z ) 
 ->  (lcm `  Z )  e. 
 NN )
 
Theoremlcmfpr 14538 The value of the lcm function for an unordered pair is the value of the lcm operator for both elements. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  (lcm `  { M ,  N } )  =  ( M lcm  N ) )
 
Theoremlcmfcl 14539 Closure of the lcm function. (Contributed by AV, 21-Aug-2020.)
 |-  ( ( Z  C_  ZZ  /\  Z  e.  Fin )  ->  (lcm `  Z )  e. 
 NN0 )
 
Theoremlcmfnncl 14540 Closure of the lcm function. (Contributed by AV, 20-Apr-2020.)
 |-  ( ( Z  C_  NN  /\  Z  e.  Fin )  ->  (lcm `  Z )  e. 
 NN )
 
Theoremlcmfeq0b 14541 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 14542* 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 14543* 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 14544* 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 14545 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 14546 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 14547 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 14555, 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 14548* Lemma for lcmfdvds 14553 and lcmfunsnlem 14552 (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 14549* Lemma 1 for lcmfunsnlem2 14551. (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 14550* Lemma 2 for lcmfunsnlem2 14551. (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 14551* Lemma for lcmfunsn 14555 and lcmfunsnlem 14552 (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 14552* Lemma for lcmfdvds 14553 and lcmfunsn 14555. 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 14548 and lcmfunsnlem2 14551 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 14553* 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 14554* Biconditional form of lcmfdvds 14553. (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 14555 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 14556 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 14557 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 14558 The least common multiple of 2 , 3 and 4 is 12. (Contributed by AV, 27-Aug-2020.)
 |-  (lcm `  { 2 ,  3 ,  4 } )  = ; 1 2
 
Theoremlcmflefac 14559 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 14560 Extend the definition of a class to include the set of prime numbers.
 class  Prime
 
Definitiondf-prm 14561* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |- 
 Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
 
Theoremisprm 14562* 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 14563 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  ->  P  e.  NN )
 
Theoremprmz 14564 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 14565 The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
 |- 
 Prime  C_  NN
 
Theoremprmex 14566 The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
 |- 
 Prime  e.  _V
 
Theorem1nprm 14567 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 14568* 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 14569* Lemma for isprm2 14570. (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 14570* 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 14571* 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 14572* 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 14573* A variation on prmind 14574 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 14574* 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 14575 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 14576 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 14577 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 14578 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  2  e.  Prime
 
Theorem3prm 14579 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  3  e.  Prime
 
Theoremprmuz2 14580 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 14581 A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( P  e.  Prime  -> 
 1  <  P )
 
Theoremprmn2uzge3 14582 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 14583 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 14584 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( A  e.  ZZ  ->  -.  ( A ^
 2 )  e.  Prime )
 
Theoremdvdsprm 14585 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 14586* 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 14587* 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 14588 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 14589* 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 14590* 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 14591 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 14592. The negation, i.e. two integers are not coprime, can be expressed either by  ( A  gcd  B )  =/=  1, see ncoprmgcdne1b 14593, or equivalently by  1  <  ( A  gcd  B ), see ncoprmgcdgt1b 14594.

The proof of Euclid's lemma, see euclemma 14603, is based on theorems about coprimality (e.g. on coprmdvds 14597).

 
Theoremcoprmgcdb 14592* 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 14593* 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 14594* 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 14595 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 14596 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 14597 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 14598 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 14599 One half of rpmulgcd2 14600, 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 14600 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 ) ) )
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