P. 127 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12601-12700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvdssq 12601	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 12602	Partial converse to bezout 12581. 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 12603	Converse of bezout 12581 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 ) )
5.1.7  Decidable sets of integers
Theorem	nnmindc 12604*	An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ E. y y e. A ) -> inf ( A , RR , < ) e. A )
Theorem	nnminle 12605*	The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12604. (Contributed by Jim Kingdon, 26-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ B e. A ) -> inf ( A , RR , < ) <_ B )
Theorem	nnwodc 12606*	Well-ordering principle: any inhabited decidable set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 23-Oct-2024.)
|- ( ( A C_ NN /\ E. w w e. A /\ A. j e. NN DECID j e. A ) -> E. x e. A A. y e. A x <_ y )
Theorem	uzwodc 12607*	Well-ordering principle: any inhabited decidable subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) (Revised by Jim Kingdon, 22-Oct-2024.)
|- ( ( S C_ ( ZZ>= ` M ) /\ E. x x e. S /\ A. x e. ( ZZ>= ` M )DECID x e. S ) -> E. j e. S A. k e. S j <_ k )
Theorem	nnwofdc 12608*	Well-ordering principle: any inhabited decidable set of positive integers has a least element. This version allows x and y to be present in A as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   =>    |- ( ( A C_ NN /\ E. z z e. A /\ A. j e. NN DECID j e. A ) -> E. x e. A A. y e. A x <_ y )
Theorem	nnwosdc 12609*	Well-ordering principle: any inhabited decidable set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 25-Oct-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( ( E. x e. NN ph /\ A. x e. NN DECID ph ) -> E. x e. NN ( ph /\ A. y e. NN ( ps -> x <_ y ) ) )
Theorem	nninfctlemfo 12610*	Lemma for nninfct 12611. (Contributed by Jim Kingdon, 10-Jul-2025.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( om e. Omni -> I :NN0* -onto->ℕ∞ )
Theorem	nninfct 12611	The limited principle of omniscience (LPO) implies that ℕ∞ is countable. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- ( om e. Omni -> E. f f : om -onto-> (ℕ∞ ⊔ 1o ) )
5.1.8  Algorithms
Theorem	nn0seqcvgd 12612*	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 12613	Lemma for ialgr0 12615. Expressing algrflemg 6394 in a form suitable for theorems such as seq3-1 10723 or seqf 10725. (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 12614	Lemma for ialgr0 12615. Closure of a constant function, in a form suitable for theorems such as seq3-1 10723 or seqf 10725. (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 12615	The value of the algorithm iterator R at 0 is the initial state A. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   =>    |- ( ph -> ( R ` M ) = A )
Theorem	algrf 12616	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 } ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   =>    |- ( ph -> R : Z --> S )
Theorem	algrp1 12617	The value of the algorithm iterator R at ( K + 1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   =>    |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )
Theorem	alginv 12618*	If I is an invariant of F, then 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 } ) )   &    |- F : S --> S   &    |- ( x e. S -> ( I ` ( F ` x ) ) = ( I ` x ) )   =>    |- ( ( A e. S /\ K e. NN0 ) -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) )
Theorem	algcvg 12619*	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 } ) )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   =>    |- ( A e. S -> ( C ` ( R ` N ) ) = 0 )
Theorem	algcvgblem 12620	Lemma for algcvgb 12621. (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 12621	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	algcvga 12622*	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 } ) )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   =>    |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( C ` ( R ` K ) ) = 0 ) )
Theorem	algfx 12623*	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 } ) )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   &    |- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) )   =>    |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) )
5.1.9  Euclid's Algorithm
Theorem	eucalgval2 12624*	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 12625*	Euclid's Algorithm eucalg 12630 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 12626*	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 12627*	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 12628*	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	eucalgcvga 12629*	Once Euclid's Algorithm halts after N steps, the second element of the state remains 0 . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised 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 } ) )   &    |- N = ( 2nd ` A )   =>    |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) )
Theorem	eucalg 12630*	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 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 ) )
5.1.10  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." In this section, an operation calculating the least common multiple of two integers (df-lcm 12632). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention.
Syntax	clcm 12631	Extend the definition of a class to include the least common multiple operator.
class lcm
Definition	df-lcm 12632*	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 12633	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 12634*	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 12524 and gcdval 12529. (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 12635	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 12636	The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 12635 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 12637*	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 12638*	Lemma for lcmn0cl 12639 and dvdslcm 12640. (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 12639	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 12640	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 12641	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 12642	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 12643	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 12644	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 12645	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 12646	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 12647	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 12648	Lemma for lcmgcd 12649 and lcmdvds 12650. 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 12649	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 12581; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12581 and https://math.stackexchange.com/a/470827 12581. 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 12636 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 12650	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 12651	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 12652	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 12653	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 12654	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 12655	Biconditional form of lcmdvds 12650. (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 12656	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 12657	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 12658	The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
|- ( 6 lcm 4 ) = ; 1 2
5.1.11  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 12659. The negation, i.e. two integers are not coprime, can be expressed either by ( A gcd B ) =/= 1, see ncoprmgcdne1b 12660, or equivalently by 1 < ( A gcd B ), see ncoprmgcdgt1b 12661. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 12663 (as opposed to Euclid's lemma for primes).
Theorem	coprmgcdb 12659*	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 12660*	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 12661*	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 12662	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 12663	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 12664	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 12665	One half of rpmulgcd2 12666, 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 12666	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 12667	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 12668*	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 12669	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 12670	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 )
5.1.12  Cancellability of congruences
Theorem	congr 12671*	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 12672	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 12673*	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 12674	One direction of the bicondition in cncongr 12676. 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 12675	The other direction of the bicondition in cncongr 12676. (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 12676	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 12677	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 ) ) )
5.2  Elementary prime number theory
5.2.1  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 3800.
Syntax	cprime 12678	Extend the definition of a class to include the set of prime numbers.
class Prime
Definition	df-prm 12679*	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 12680*	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 12681	A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( P e. Prime -> P e. NN )
Theorem	prmz 12682	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 12683	The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
|- Prime C_ NN
Theorem	prmex 12684	The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
|- Prime e. _V
Theorem	1nprm 12685	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 12686*	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 12687*	Lemma for isprm2 12688. (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 12688*	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 12689*	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 12690*	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 12691*	A variation on prmind 12692 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 12692*	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 12693	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 12694	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 12695	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	dvdsnprmd 12696	If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.)
|- ( ph -> 1 < A )   &    |- ( ph -> A < N )   &    |- ( ph -> A || N )   =>    |- ( ph -> -. N e. Prime )
Theorem	prm2orodd 12697	A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
|- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
Theorem	2prm 12698	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 12699	3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
|- 3 e. Prime
Theorem	4nprm 12700	4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
|- -. 4 e. Prime