P. 147 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eucalg 14601*	Euclid's Algorithm computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0. Upon halting, the 1st member of the final state ( R ` N ) is equal to the gcd of the values comprising the input state <. M , N >.. This is Metamath 100 proof #69 (greatest common divisor algorithm). (Contributed by Paul Chapman, 31-Mar-2011.) (Proof shortened by Mario Carneiro, 29-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   &    |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )   &    |- A = <. M , N >.   =>    |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( M gcd N ) )
6.1.10  The least common multiple According to Wikipedia ("Least common multiple", 27-Aug-2020, https://en.wikipedia.org/wiki/Least_common_multiple): "In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility. ... The lcm of more than two integers is also well-defined: it is the smallest positive integer that is divisible by each of them." In this section, an operation calculating the least common multiple of two integers (df-lcm 14606) as well as a function mapping a set of integers to their least common multiple (df-lcmf 14608) are provided. Both definitions are valid for all integers, including negative integers and 0, obeying the above mentioned convention. It is shown by lcmfpr 14655 that the two definitions are compatible. It seems a little confusing at the first glance that the least common multiple is defined by a supremum. Actually, it is an infimum, because the inverse of "less than" `' < is used to turn supremum into infimum - currently we don't have infimum defined separately...
Syntax	clcm 14602	Extend the definition of a class to include the least common multiple operator.
class lcm
Syntax	clcmf 14603	Extend the definition of a class to include the least common multiple function.
class lcm
Syntax	clcmold 14604	Extend the definition of a class to include the least common multiple operator (old version.
class lcm
Syntax	clcmfold 14605	Extend the definition of a class to include the least common multiple function (old version).
class lcm
Definition	df-lcm 14606*	Define the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
Definition	df-lcmOLD 14607*	Define the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) Obsolete version of df-lcm 14606 as of 16-Sep-2020. (New usage is discouraged.)
|- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , sup ( { n e. NN | ( x || n /\ y || n ) } , RR , `' < ) ) )
Definition	df-lcmf 14608*	Define the lcm function on a set of integers. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.)
|- lcm = ( z e. ~P ZZ |-> if ( 0 e. z , 0 , inf ( { n e. NN | A. m e. z m || n } , RR , < ) ) )
Definition	df-lcmfOLD 14609*	Define the lcm function on a set of integers. (Contributed by AV, 21-Aug-2020.) Obsolete version of df-lcmf 14608 as of 16-Sep-2020. (New usage is discouraged.)
|- lcm = ( z e. ~P ZZ |-> if ( 0 e. z , 0 , sup ( { n e. NN | A. m e. z m || n } , RR , `' < ) ) )
Theorem	lcmval 14610*	Value of the lcm operator. ( M lcm N ) is the least common multiple of M and N. If either M or N is 0, the result is defined conventionally as 0. Contrast with df-gcd 14524 and gcdval 14525. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
Theorem	lcmvalOLD 14611*	Value of the lcm operator. ( M lcm N ) is the least common multiple of M and N. If either M or N is 0, the result is defined conventionally as 0. Contrast with df-gcd 14524 and gcdval 14525. (Contributed by Steve Rodriguez, 20-Jan-2020.) Obsolete version of lcmval 14610 as of 16-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , sup ( { n e. NN | ( M || n /\ N || n ) } , RR , `' < ) ) )
Theorem	lcmcom 14612	The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
Theorem	lcm0val 14613	The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 14612 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( M e. ZZ -> ( M lcm 0 ) = 0 )
Theorem	lcmn0val 14614*	The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
Theorem	lcmn0valOLD 14615*	The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) Obsolete version of lcmn0val 14614 as of 16-Sep-2020. ( (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = sup ( { n e. NN | ( M || n /\ N || n ) } , RR , `' < ) )
Theorem	lcmcllem 14616*	Lemma for lcmn0cl 14617 and dvdslcm 14618. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } )
Theorem	lcmn0cl 14617	Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
Theorem	dvdslcm 14618	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 ) ) )
Theorem	lcmledvds 14619	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 ) )
Theorem	lcmeq0 14620	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 ) ) )
Theorem	lcmcl 14621	Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
Theorem	gcddvdslcm 14622	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 ) )
Theorem	lcmneg 14623	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 ) )
Theorem	neglcm 14624	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 ) )
Theorem	lcmabs 14625	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 ) )
Theorem	lcmgcdlem 14626	Lemma for lcmgcd 14627 and lcmdvds 14628. 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 ) ) )
Theorem	lcmgcd 14627	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 14926 or of Bézout's identity bezout 14565; 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 14613 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 ) ) )
Theorem	lcmdvds 14628	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 ) )
Theorem	lcmid 14629	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 ) )
Theorem	lcm1 14630	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 ) )
Theorem	lcmgcdnn 14631	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 ) )
Theorem	lcmgcdeq 14632	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 ) ) )
Theorem	lcmdvdsb 14633	Biconditional form of lcmdvds 14628. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K ) <-> ( M lcm N ) || K ) )
Theorem	lcmass 14634	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 ) ) )
Theorem	3lcm2e6woprm 14635	The least common multiple of three and two is six. In contrast to 3lcm2e6 14736, 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
Theorem	6lcm4e12 14636	The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
|- ( 6 lcm 4 ) = ; 1 2
Theorem	lcmscllemOLD 14637*	Lemma for lcmsOLD 14639, analogous to lcmcllem 14616. (Contributed by AV, 14-Aug-2020.) Obsolete version of lcmfcllem 14653 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 } )
Theorem	lcmsnnOLD 14638*	The least common multiple of a finite set of positive integers is a positive integer, analogous to lcmn0cl 14617. (Contributed by AV, 14-Aug-2020.) Obsolete version of lcmfn0cl 14654 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 )
Theorem	lcmsOLD 14639*	The least common multiple of a finite set of positive integers is divisible by each element of the set, analogous to dvdslcm 14618. (Contributed by AV, 14-Aug-2020.) Obsolete version of dvdslcmf 14659 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 )
Theorem	lcmsledvdsOLD 14640*	The least common multiple of a set of integers dividing an integer is less than or equal to this integer. Analogous to lcmledvds 14619. (Contributed by AV, 16-Aug-2020.) Obsolete version of lcmfledvds 14660 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 ) )
Theorem	lcmslefacOLD 14641*	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 14676 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 ) )
Theorem	absproddvds 14642*	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 )
Theorem	absprodnn 14643*	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 )
Theorem	fissn0dvds 14644*	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 )
Theorem	fissn0dvdsn0 14645*	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 } =/= (/) )
Theorem	lcmfval 14646*	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 , < ) ) )
Theorem	lcmf0val 14647	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 )
Theorem	lcmfn0val 14648*	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 , < ) )
Theorem	lcmfnnval 14649*	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 , < ) )
Theorem	lcmfvalOLD 14650*	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 14646 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 , `' < ) ) )
Theorem	lcmfn0valOLD 14651*	The value of the lcm function for a set without 0. (Contributed by AV, 21-Aug-2020.) Obsolete version of lcmfn0val 14648 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 , `' < ) )
Theorem	lcmfnnvalOLD 14652*	The value of the lcm function for a subset of the positive integers. (Contributed by AV, 21-Aug-2020.) Obsolete version of lcmfnnval 14649 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 , `' < ) )
Theorem	lcmfcllem 14653*	Lemma for lcmfn0cl 14654 and dvdslcmf 14659. (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 } )
Theorem	lcmfn0cl 14654	Closure of the lcm function. (Contributed by AV, 21-Aug-2020.)
|- ( ( Z C_ ZZ /\ Z e. Fin /\ 0 e/ Z ) -> (lcm ` Z ) e. NN )
Theorem	lcmfpr 14655	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 ) )
Theorem	lcmfcl 14656	Closure of the lcm function. (Contributed by AV, 21-Aug-2020.)
|- ( ( Z C_ ZZ /\ Z e. Fin ) -> (lcm ` Z ) e. NN0 )
Theorem	lcmfnncl 14657	Closure of the lcm function. (Contributed by AV, 20-Apr-2020.)
|- ( ( Z C_ NN /\ Z e. Fin ) -> (lcm ` Z ) e. NN )
Theorem	lcmfeq0b 14658	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 ) )
Theorem	dvdslcmf 14659*	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 ) )
Theorem	lcmfledvds 14660*	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 ) )
Theorem	lcmf 14661*	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 ) ) ) )
Theorem	lcmf0 14662	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
Theorem	lcmfsn 14663	The least common multiple of a singleton is its absolute value. (Contributed by AV, 22-Aug-2020.)
|- ( M e. ZZ -> (lcm ` { M } ) = ( abs ` M ) )
Theorem	lcmftp 14664	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 14672, 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 ) )
Theorem	lcmfunsnlem1 14665*	Lemma for lcmfdvds 14670 and lcmfunsnlem 14669 (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 ) )
Theorem	lcmfunsnlem2lem1 14666*	Lemma 1 for lcmfunsnlem2 14668. (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 ) )
Theorem	lcmfunsnlem2lem2 14667*	Lemma 2 for lcmfunsnlem2 14668. (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 ) )
Theorem	lcmfunsnlem2 14668*	Lemma for lcmfunsn 14672 and lcmfunsnlem 14669 (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 ) )
Theorem	lcmfunsnlem 14669*	Lemma for lcmfdvds 14670 and lcmfunsn 14672. 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 14665 and lcmfunsnlem2 14668 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 ) ) )
Theorem	lcmfdvds 14670*	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 ) )
Theorem	lcmfdvdsb 14671*	Biconditional form of lcmfdvds 14670. (Contributed by AV, 26-Aug-2020.)
|- ( ( K e. ZZ /\ Z C_ ZZ /\ Z e. Fin ) -> ( A. m e. Z m || K <-> (lcm ` Z ) || K ) )
Theorem	lcmfunsn 14672	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 ) )
Theorem	lcmfun 14673	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 ) ) )
Theorem	lcmfass 14674	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 ) } ) ) )
Theorem	lcmf2a3a4e12 14675	The least common multiple of 2 , 3 and 4 is 12. (Contributed by AV, 27-Aug-2020.)
|- (lcm ` { 2 , 3 , 4 } ) = ; 1 2
Theorem	lcmflefac 14676	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
Syntax	cprime 14677	Extend the definition of a class to include the set of prime numbers.
class Prime
Definition	df-prm 14678*	Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
|- Prime = { p e. NN | { n e. NN | n || p } ~~ 2o }
Theorem	isprm 14679*	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 ) )
Theorem	prmnn 14680	A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( P e. Prime -> P e. NN )
Theorem	prmz 14681	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 )
Theorem	prmssnn 14682	The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
|- Prime C_ NN
Theorem	prmex 14683	The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
|- Prime e. _V
Theorem	1nprm 14684	1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
|- -. 1 e. Prime
Theorem	1idssfct 14685*	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 } )
Theorem	isprm2lem 14686*	Lemma for isprm2 14687. (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 } ) )
Theorem	isprm2 14687*	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 ) ) ) )
Theorem	isprm3 14688*	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 ) )
Theorem	isprm4 14689*	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 ) ) )
Theorem	prmind2 14690*	A variation on prmind 14691 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 )
Theorem	prmind 14691*	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 )
Theorem	dvdsprime 14692	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 ) ) )
Theorem	nprm 14693	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 )
Theorem	nprmi 14694	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
Theorem	2prm 14695	2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
|- 2 e. Prime
Theorem	3prm 14696	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|- 3 e. Prime
Theorem	prmuz2 14697	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 ) )
Theorem	prmgt1 14698	A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( P e. Prime -> 1 < P )
Theorem	prmn2uzge3 14699	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 ) )
Theorem	prmm2nn0 14700	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 )