P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvdsgcd 10401	An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M gcd N ) ) )
Theorem	dvdsgcdb 10402	Biconditional form of dvdsgcd 10401. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) <-> K || ( M gcd N ) ) )
Theorem	dfgcd2 10403*	Alternate definition of the gcd operator, see definition in [ApostolNT] p. 15. (Contributed by AV, 8-Aug-2021.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( D = ( M gcd N ) <-> ( 0 <_ D /\ ( D || M /\ D || N ) /\ A. e e. ZZ ( ( e || M /\ e || N ) -> e || D ) ) ) )
Theorem	gcdass 10404	Associative law for gcd operator. Theorem 1.4(b) in [ApostolNT] p. 16. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) gcd P ) = ( N gcd ( M gcd P ) ) )
Theorem	mulgcd 10405	Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)
|- ( ( K e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
Theorem	absmulgcd 10406	Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( abs ` ( K x. ( M gcd N ) ) ) )
Theorem	mulgcdr 10407	Reverse distribution law for the gcd operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN0 ) -> ( ( A x. C ) gcd ( B x. C ) ) = ( ( A gcd B ) x. C ) )
Theorem	gcddiv 10408	Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ ( C || A /\ C || B ) ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) )
Theorem	gcdmultiple 10409	The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. NN ) -> ( M gcd ( M x. N ) ) = M )
Theorem	gcdmultiplez 10410	Extend gcdmultiple 10409 so N can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M )
Theorem	gcdzeq 10411	A positive integer A is equal to its gcd with an integer B if and only if A divides B. Generalization of gcdeq 10412. (Contributed by AV, 1-Jul-2020.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( ( A gcd B ) = A <-> A || B ) )
Theorem	gcdeq 10412	A is equal to its gcd with B if and only if A divides B. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by AV, 8-Aug-2021.)
|- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) = A <-> A || B ) )
Theorem	dvdssqim 10413	Unidirectional form of dvdssq 10420. (Contributed by Scott Fenton, 19-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M ^ 2 ) || ( N ^ 2 ) ) )
Theorem	dvdsmulgcd 10414	Relationship between the order of an element and that of a multiple. (a divisibility equivalent). (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( B e. ZZ /\ C e. ZZ ) -> ( A || ( B x. C ) <-> A || ( B x. ( C gcd A ) ) ) )
Theorem	rpmulgcd 10415	If K and M are relatively prime, then the GCD of K and M x. N is the GCD of K and N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( K gcd ( M x. N ) ) = ( K gcd N ) )
Theorem	rplpwr 10416	If A and B are relatively prime, then so are A ^ N and B. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) )
Theorem	rppwr 10417	If A and B are relatively prime, then so are A ^ N and B ^ N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd ( B ^ N ) ) = 1 ) )
Theorem	sqgcd 10418	Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. NN ) -> ( ( M gcd N ) ^ 2 ) = ( ( M ^ 2 ) gcd ( N ^ 2 ) ) )
Theorem	dvdssqlem 10419	Lemma for dvdssq 10420. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. NN ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )
Theorem	dvdssq 10420	Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )
Theorem	bezoutr 10421	Partial converse to bezout 10400. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( A gcd B ) || ( ( A x. X ) + ( B x. Y ) ) )
Theorem	bezoutr1 10422	Converse of bezout 10400 for when the greater common divisor is one (sufficient condition for relative primality). (Contributed by Stefan O'Rear, 23-Sep-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( ( ( A x. X ) + ( B x. Y ) ) = 1 -> ( A gcd B ) = 1 ) )
4.1.6  Algorithms
Theorem	nn0seqcvgd 10423*	A strictly-decreasing nonnegative integer sequence with initial term N reaches zero by the N th term. Deduction version. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ph -> F : NN0 --> NN0 )   &    |- ( ph -> N = ( F ` 0 ) )   &    |- ( ( ph /\ k e. NN0 ) -> ( ( F ` ( k + 1 ) ) =/= 0 -> ( F ` ( k + 1 ) ) < ( F ` k ) ) )   =>    |- ( ph -> ( F ` N ) = 0 )
Theorem	ialgrlem1st 10424	Lemma for ialgr0 10426. Expressing algrflemg 5871 in a form suitable for theorems such as iseq1 9442 or iseqfn 9441. (Contributed by Jim Kingdon, 22-Jul-2021.)
|- ( ph -> F : S --> S )   =>    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
Theorem	ialgrlemconst 10425	Lemma for ialgr0 10426. Closure of a constant function, in a form suitable for theorems such as iseq1 9442 or iseqfn 9441. (Contributed by Jim Kingdon, 22-Jul-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> A e. S )   =>    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )
Theorem	ialgr0 10426	The value of the algorithm iterator R at 0 is the initial state A. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) , S )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   &    |- ( ph -> S e. V )   =>    |- ( ph -> ( R ` M ) = A )
Theorem	ialgrf 10427	An algorithm is a step function F : S --> S on a state space S. An algorithm acts on an initial state A e. S by iteratively applying F to give A, ( F ` A ), ( F ` ( F ` A ) ) and so on. An algorithm is said to halt if a fixed point of F is reached after a finite number of iterations. The algorithm iterator R : NN0 --> S "runs" the algorithm F so that ( R ` k ) is the state after k iterations of F on the initial state A. Domain and codomain of the algorithm iterator R. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) , S )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   &    |- ( ph -> S e. V )   =>    |- ( ph -> R : Z --> S )
Theorem	ialgrp1 10428	The value of the algorithm iterator R at ( K + 1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) , S )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   &    |- ( ph -> S e. V )   =>    |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )
Theorem	ialginv 10429*	If I is an invariant of F, its value is unchanged after any number of iterations of F. (Contributed by Paul Chapman, 31-Mar-2011.)
|- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )   &    |- F : S --> S   &    |- I Fn S   &    |- ( x e. S -> ( I ` ( F ` x ) ) = ( I ` x ) )   &    |- S e. V   =>    |- ( ( A e. S /\ K e. NN0 ) -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) )
Theorem	ialgcvg 10430*	One way to prove that an algorithm halts is to construct a countdown function C : S --> NN0 whose value is guaranteed to decrease for each iteration of F until it reaches 0. That is, if X e. S is not a fixed point of F, then ( C ` ( F ` X ) ) < ( C ` X ). If C is a countdown function for algorithm F, the sequence ( C ` ( R ` k ) ) reaches 0 after at most N steps, where N is the value of C for the initial state A. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F : S --> S   &    |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   &    |- S e. V   =>    |- ( A e. S -> ( C ` ( R ` N ) ) = 0 )
Theorem	algcvgblem 10431	Lemma for algcvgb 10432. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) ) )
Theorem	algcvgb 10432	Two ways of expressing that C is a countdown function for algorithm F. The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently nonzero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.)
|- F : S --> S   &    |- C : S --> NN0   =>    |- ( X e. S -> ( ( ( C ` ( F ` X ) ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) <-> ( ( ( C ` X ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) /\ ( ( C ` X ) = 0 -> ( C ` ( F ` X ) ) = 0 ) ) ) )
Theorem	ialgcvga 10433*	The countdown function C remains 0 after N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F : S --> S   &    |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   &    |- S e. V   =>    |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( C ` ( R ` K ) ) = 0 ) )
Theorem	ialgfx 10434*	If F reaches a fixed point when the countdown function C reaches 0, F remains fixed after N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F : S --> S   &    |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   &    |- S e. V   &    |- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) )   =>    |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) )
4.1.7  Euclid's Algorithm
Theorem	eucalgval2 10435*	The value of the step function E for Euclid's Algorithm on an ordered pair. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M E N ) = if ( N = 0 , <. M , N >. , <. N , ( M mod N ) >. ) )
Theorem	eucalgval 10436*	Euclid's Algorithm eucialg 10441 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. The value of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
Theorem	eucalgf 10437*	Domain and codomain of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- E : ( NN0 X. NN0 ) --> ( NN0 X. NN0 )
Theorem	eucalginv 10438*	The invariant of the step function E for Euclid's Algorithm is the gcd operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` X ) )
Theorem	eucalglt 10439*	The second member of the state decreases with each iteration of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( X e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` X ) ) =/= 0 -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) ) )
Theorem	eucialgcvga 10440*	Once Euclid's Algorithm halts after N steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.)
|- 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 } ) , ( NN0 X. NN0 ) )   &    |- N = ( 2nd ` A )   =>    |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) )
Theorem	eucialg 10441*	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. Theorem 1.15 in [ApostolNT] p. 20. 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 Jim Kingdon, 11-Jan-2022.)
|- 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 } ) , ( NN0 X. NN0 ) )   &    |- N = ( 2nd ` A )   &    |- A = <. M , N >.   =>    |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( M gcd N ) )
4.1.8  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." In this section, an operation calculating the least common multiple of two integers (df-lcm 10443). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention.
Syntax	clcm 10442	Extend the definition of a class to include the least common multiple operator.
class lcm
Definition	df-lcm 10443*	Define the lcm operator. For example, ( 6 lcm 9 ) = 1 8. (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 , < ) ) )
Theorem	lcmmndc 10444	Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
Theorem	lcmval 10445*	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 10339 and gcdval 10351. (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	lcmcom 10446	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 10447	The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 10446 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 10448*	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	lcmcllem 10449*	Lemma for lcmn0cl 10450 and dvdslcm 10451. (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 10450	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 10451	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 10452	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 10453	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 10454	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 10455	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 10456	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 10457	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 10458	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 10459	Lemma for lcmgcd 10460 and lcmdvds 10461. 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 10460	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 or of Bézout's identity bezout 10400; 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 integers 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 10447 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 10461	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 10462	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 10463	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 10464	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 10465	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 10466	Biconditional form of lcmdvds 10461. (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 10467	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 10468	The least common multiple of three and two is six. This proof does not use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.)
|- ( 3 lcm 2 ) = 6
Theorem	6lcm4e12 10469	The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
|- ( 6 lcm 4 ) = ; 1 2
4.1.9  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 10470. The negation, i.e. two integers are not coprime, can be expressed either by ( A gcd B ) =/= 1, see ncoprmgcdne1b 10471, or equivalently by 1 < ( A gcd B ), see ncoprmgcdgt1b 10472. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 10474 (as opposed to Euclid's lemma for primes).
Theorem	coprmgcdb 10470*	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 ) )
Theorem	ncoprmgcdne1b 10471*	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 ) )
Theorem	ncoprmgcdgt1b 10472*	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 ) ) )
Theorem	coprmdvds1 10473	If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.)
|- ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) -> ( ( I e. NN /\ I || F /\ I || G ) -> I = 1 ) )
Theorem	coprmdvds 10474	Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M x. N ) /\ ( K gcd M ) = 1 ) -> K || N ) )
Theorem	coprmdvds2 10475	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 ) )
Theorem	mulgcddvds 10476	One half of rpmulgcd2 10477, 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 ) ) )
Theorem	rpmulgcd2 10477	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 ) ) )
Theorem	qredeq 10478	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 ) )
Theorem	qredeu 10479*	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 ) ) ) )
Theorem	rpmul 10480	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 ) )
Theorem	rpdvds 10481	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 )
4.1.10  Cancellability of congruences
Theorem	congr 10482*	Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer A is congruent to an integer B modulo M if their difference is a multiple of M. See also the definition in [ApostolNT] p. 104: "... a is congruent to b modulo m, and we write a == b (mod m) if m divides the difference a - b", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> E. n e. ZZ ( n x. M ) = ( A - B ) ) )
Theorem	divgcdcoprm0 10483	Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
Theorem	divgcdcoprmex 10484*	Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.)
|- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> E. a e. ZZ E. b e. ZZ ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) )
Theorem	cncongr1 10485	One direction of the bicondition in cncongr 10487. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) ) -> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) -> ( A mod M ) = ( B mod M ) ) )
Theorem	cncongr2 10486	The other direction of the bicondition in cncongr 10487. (Contributed by AV, 11-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) ) -> ( ( A mod M ) = ( B mod M ) -> ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) ) )
Theorem	cncongr 10487	Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ M = ( N / ( C gcd N ) ) ) ) -> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) <-> ( A mod M ) = ( B mod M ) ) )
Theorem	cncongrcoprm 10488	Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( N e. NN /\ ( C gcd N ) = 1 ) ) -> ( ( ( A x. C ) mod N ) = ( ( B x. C ) mod N ) <-> ( A mod N ) = ( B mod N ) ) )
4.2  Elementary prime number theory
4.2.1  Elementary properties Remark: to represent odd prime numbers, i.e., all prime numbers except 2, the idiom P e. ( Prime \ { 2 } ) is used. It is a little bit shorter than ( P e. Prime /\ P =/= 2 ). Both representations can be converted into each other by eldifsn 3517.
Syntax	cprime 10489	Extend the definition of a class to include the set of prime numbers.
class Prime
Definition	df-prm 10490*	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 10491*	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 10492	A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( P e. Prime -> P e. NN )
Theorem	prmz 10493	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 10494	The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
|- Prime C_ NN
Theorem	prmex 10495	The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
|- Prime e. _V
Theorem	1nprm 10496	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 10497*	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 10498*	Lemma for isprm2 10499. (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 10499*	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. Definition in [ApostolNT] p. 16. (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 10500*	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 ) )