P. 124 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
4.10.1.1  The circle constant (tau = 2 pi)
Syntax	ctau 12301	Extend class notation to include the constant tau, tau = 6.28318....
class tau
Definition	df-tau 12302	Define the circle constant tau, tau = 6.28318..., which is the smallest positive real number whose cosine is one. Various notations have been used or proposed for this number including tau, a three-legged variant of pi, or 2 pi. Note the difference between this constant tau and the formula variable ta. Following our convention, the constant is displayed in upright font while the variable is in italic font; furthermore, the colors are different. (Contributed by Jim Kingdon, 9-Apr-2018.) (Revised by AV, 1-Oct-2020.)
|- tau = inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < )
4.10.2  _e is irrational
Theorem	eirraplem 12303*	Lemma for eirrap 12304. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
|- F = ( n e. NN0 |-> ( 1 / ( ! ` n ) ) )   &    |- ( ph -> P e. ZZ )   &    |- ( ph -> Q e. NN )   =>    |- ( ph -> _e # ( P / Q ) )
Theorem	eirrap 12304	_e is irrational. That is, for any rational number, _e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that _e is not rational, which is eirr 12305. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( Q e. QQ -> _e # Q )
Theorem	eirr 12305	_e is not rational. In the absence of excluded middle, we can distinguish between this and saying that _e is irrational in the sense of being apart from any rational number, which is eirrap 12304. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
|- _e e/ QQ
Theorem	egt2lt3 12306	Euler's constant _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
|- ( 2 < _e /\ _e < 3 )
Theorem	epos 12307	Euler's constant _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- 0 < _e
Theorem	epr 12308	Euler's constant _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- _e e. RR+
Theorem	ene0 12309	_e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 0
Theorem	eap0 12310	_e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 0
Theorem	ene1 12311	_e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 1
Theorem	eap1 12312	_e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 1
PART 5  ELEMENTARY NUMBER THEORY This part introduces elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.
5.1  Elementary properties of divisibility
5.1.1  The divides relation
Syntax	cdvds 12313	Extend the definition of a class to include the divides relation. See df-dvds 12314.
class ||
Definition	df-dvds 12314*	Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) }
Theorem	divides 12315*	Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 16149). As proven in dvdsval3 12317, M || N <-> ( N mod M ) = 0. See divides 12315 and dvdsval2 12316 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. n e. ZZ ( n x. M ) = N ) )
Theorem	dvdsval2 12316	One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
|- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> ( N / M ) e. ZZ ) )
Theorem	dvdsval3 12317	One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.)
|- ( ( M e. NN /\ N e. ZZ ) -> ( M || N <-> ( N mod M ) = 0 ) )
Theorem	dvdszrcl 12318	Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- ( X || Y -> ( X e. ZZ /\ Y e. ZZ ) )
Theorem	dvdsmod0 12319	If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.)
|- ( ( M e. NN /\ M || N ) -> ( N mod M ) = 0 )
Theorem	p1modz1 12320	If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.)
|- ( ( M || A /\ 1 < M ) -> ( ( A + 1 ) mod M ) = 1 )
Theorem	dvdsmodexp 12321	If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 12771). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.)
|- ( ( N e. NN /\ B e. NN /\ N || A ) -> ( ( A ^ B ) mod N ) = ( A mod N ) )
Theorem	nndivdvds 12322	Strong form of dvdsval2 12316 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( B || A <-> ( A / B ) e. NN ) )
Theorem	nndivides 12323*	Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
|- ( ( M e. NN /\ N e. NN ) -> ( M || N <-> E. n e. NN ( n x. M ) = N ) )
Theorem	dvdsdc 12324	Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
|- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N )
Theorem	moddvds 12325	Two ways to say A == B (mod N), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod N ) = ( B mod N ) <-> N || ( A - B ) ) )
Theorem	modm1div 12326	An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.)
|- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ZZ ) -> ( ( A mod N ) = 1 <-> N || ( A - 1 ) ) )
Theorem	dvds0lem 12327	A lemma to assist theorems of || with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K x. M ) = N ) -> M || N )
Theorem	dvds1lem 12328*	A lemma to assist theorems of || with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ph -> ( J e. ZZ /\ K e. ZZ ) )   &    |- ( ph -> ( M e. ZZ /\ N e. ZZ ) )   &    |- ( ( ph /\ x e. ZZ ) -> Z e. ZZ )   &    |- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> ( Z x. M ) = N ) )   =>    |- ( ph -> ( J || K -> M || N ) )
Theorem	dvds2lem 12329*	A lemma to assist theorems of || with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ph -> ( I e. ZZ /\ J e. ZZ ) )   &    |- ( ph -> ( K e. ZZ /\ L e. ZZ ) )   &    |- ( ph -> ( M e. ZZ /\ N e. ZZ ) )   &    |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> Z e. ZZ )   &    |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. I ) = J /\ ( y x. K ) = L ) -> ( Z x. M ) = N ) )   =>    |- ( ph -> ( ( I || J /\ K || L ) -> M || N ) )
Theorem	iddvds 12330	An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> N || N )
Theorem	1dvds 12331	1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> 1 || N )
Theorem	dvds0 12332	Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> N || 0 )
Theorem	negdvdsb 12333	An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) )
Theorem	dvdsnegb 12334	An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || -u N ) )
Theorem	absdvdsb 12335	An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
Theorem	dvdsabsb 12336	An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( abs ` N ) ) )
Theorem	0dvds 12337	Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
Theorem	zdvdsdc 12338	Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N )
Theorem	dvdsmul1 12339	An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
Theorem	dvdsmul2 12340	An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) )
Theorem	iddvdsexp 12341	An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)
|- ( ( M e. ZZ /\ N e. NN ) -> M || ( M ^ N ) )
Theorem	muldvds1 12342	If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> K || N ) )
Theorem	muldvds2 12343	If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> M || N ) )
Theorem	dvdscmul 12344	Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
Theorem	dvdsmulc 12345	Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
Theorem	dvdscmulr 12346	Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) || ( K x. N ) <-> M || N ) )
Theorem	dvdsmulcr 12347	Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) || ( N x. K ) <-> M || N ) )
Theorem	summodnegmod 12348	The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( ( A + B ) mod N ) = 0 <-> ( A mod N ) = ( -u B mod N ) ) )
Theorem	modmulconst 12349	Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ M e. NN ) -> ( ( A mod M ) = ( B mod M ) <-> ( ( C x. A ) mod ( C x. M ) ) = ( ( C x. B ) mod ( C x. M ) ) ) )
Theorem	dvds2ln 12350	If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ) -> ( ( K || M /\ K || N ) -> K || ( ( I x. M ) + ( J x. N ) ) ) )
Theorem	dvds2add 12351	If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M + N ) ) )
Theorem	dvds2sub 12352	If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M - N ) ) )
Theorem	dvds2subd 12353	Deduction form of dvds2sub 12352. (Contributed by Stanislas Polu, 9-Mar-2020.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K || M )   &    |- ( ph -> K || N )   =>    |- ( ph -> K || ( M - N ) )
Theorem	dvdstr 12354	The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || N ) -> K || N ) )
Theorem	dvds2addd 12355	Deduction form of dvds2add 12351. (Contributed by SN, 21-Aug-2024.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K || M )   &    |- ( ph -> K || N )   =>    |- ( ph -> K || ( M + N ) )
Theorem	dvdstrd 12356	The divides relation is transitive, a deduction version of dvdstr 12354. (Contributed by metakunt, 12-May-2024.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K || M )   &    |- ( ph -> M || N )   =>    |- ( ph -> K || N )
Theorem	dvdsmultr1 12357	If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || M -> K || ( M x. N ) ) )
Theorem	dvdsmultr1d 12358	Natural deduction form of dvdsmultr1 12357. (Contributed by Stanislas Polu, 9-Mar-2020.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K || M )   =>    |- ( ph -> K || ( M x. N ) )
Theorem	dvdsmultr2 12359	If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || N -> K || ( M x. N ) ) )
Theorem	ordvdsmul 12360	If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M \/ K || N ) -> K || ( M x. N ) ) )
Theorem	dvdssub2 12361	If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)
|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K || ( M - N ) ) -> ( K || M <-> K || N ) )
Theorem	dvdsadd 12362	An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( M + N ) ) )
Theorem	dvdsaddr 12363	An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( N + M ) ) )
Theorem	dvdssub 12364	An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( M - N ) ) )
Theorem	dvdssubr 12365	An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( N - M ) ) )
Theorem	dvdsadd2b 12366	Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )
Theorem	dvdsaddre2b 12367	Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12366 only requiring B to be a real number (not necessarily an integer). (Contributed by AV, 19-Jul-2021.)
|- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )
Theorem	fsumdvds 12368*	If every term in a sum is divisible by N, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)
|- ( ph -> A e. Fin )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ k e. A ) -> B e. ZZ )   &    |- ( ( ph /\ k e. A ) -> N || B )   =>    |- ( ph -> N || sum_ k e. A B )
Theorem	dvdslelemd 12369	Lemma for dvdsle 12370. (Contributed by Jim Kingdon, 8-Nov-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> N < M )   =>    |- ( ph -> ( K x. M ) =/= N )
Theorem	dvdsle 12370	The divisors of a positive integer are bounded by it. The proof does not use /. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( M || N -> M <_ N ) )
Theorem	dvdsleabs 12371	The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> M <_ ( abs ` N ) ) )
Theorem	dvdsleabs2 12372	Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)
|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> ( abs ` M ) <_ ( abs ` N ) ) )
Theorem	dvdsabseq 12373	If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)
|- ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) )
Theorem	dvdseq 12374	If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.)
|- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( M || N /\ N || M ) ) -> M = N )
Theorem	divconjdvds 12375	If a nonzero integer M divides another integer N, the other integer N divided by the nonzero integer M (i.e. the divisor conjugate of N to M) divides the other integer N. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
|- ( ( M || N /\ M =/= 0 ) -> ( N / M ) || N )
Theorem	dvdsdivcl 12376*	The complement of a divisor of N is also a divisor of N. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)
|- ( ( N e. NN /\ A e. { x e. NN | x || N } ) -> ( N / A ) e. { x e. NN | x || N } )
Theorem	dvdsflip 12377*	An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)
|- A = { x e. NN | x || N }   &    |- F = ( y e. A |-> ( N / y ) )   =>    |- ( N e. NN -> F : A -1-1-onto-> A )
Theorem	dvdsssfz1 12378*	The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)
|- ( A e. NN -> { p e. NN | p || A } C_ ( 1 ... A ) )
Theorem	dvds1 12379	The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
|- ( M e. NN0 -> ( M || 1 <-> M = 1 ) )
Theorem	alzdvds 12380*	Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> ( A. x e. ZZ x || N <-> N = 0 ) )
Theorem	dvdsext 12381*	Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. x e. NN0 ( A || x <-> B || x ) ) )
Theorem	fzm1ndvds 12382	No number between 1 and M - 1 divides M. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> -. M || N )
Theorem	fzo0dvdseq 12383	Zero is the only one of the first A nonnegative integers that is divisible by A. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( B e. ( 0..^ A ) -> ( A || B <-> B = 0 ) )
Theorem	fzocongeq 12384	Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. ( C..^ D ) /\ B e. ( C..^ D ) ) -> ( ( D - C ) || ( A - B ) <-> A = B ) )
Theorem	addmodlteqALT 12385	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 10632 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) /\ S e. ZZ ) -> ( ( ( I + S ) mod N ) = ( ( J + S ) mod N ) <-> I = J ) )
Theorem	dvdsfac 12386	A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
|- ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) -> K || ( ! ` N ) )
Theorem	dvdsexp 12387	A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( A e. ZZ /\ M e. NN0 /\ N e. ( ZZ>= ` M ) ) -> ( A ^ M ) || ( A ^ N ) )
Theorem	dvdsmod 12388	Any number K whose mod base N is divisible by a divisor P of the base is also divisible by P. This means that primes will also be relatively prime to the base when reduced mod N for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || K ) )
Theorem	mulmoddvds 12389	If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
|- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A -> ( ( A x. B ) mod N ) = 0 ) )
Theorem	3dvds 12390*	A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.)
|- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )
Theorem	3dvdsdec 12391	A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if A and B actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A and B, especially if A is itself a decimal number, e.g., A = ; C D. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
|- A e. NN0   &    |- B e. NN0   =>    |- ( 3 || ; A B <-> 3 || ( A + B ) )
Theorem	3dvds2dec 12392	A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if A, B and C actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A, B and C. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
|- A e. NN0   &    |- B e. NN0   &    |- C e. NN0   =>    |- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )
5.1.2  Even and odd numbers The set ZZ of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 12396. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom 2 || N to say that " N is even" (which implies N e. ZZ, see evenelz 12393) and -. 2 || N to say that " N is odd" (under the assumption that N e. ZZ). The previously proven theorems about even and odd numbers, like zneo 9559, zeo 9563, zeo2 9564, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 12415 and oddp1d2 12416. The corresponding theorems are zeneo 12397, zeo3 12394 and zeo4 12396.
Theorem	evenelz 12393	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 12318. (Contributed by AV, 22-Jun-2021.)
|- ( 2 || N -> N e. ZZ )
Theorem	zeo3 12394	An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
Theorem	zeoxor 12395	An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)
|- ( N e. ZZ -> ( 2 || N \/_ -. 2 || N ) )
Theorem	zeo4 12396	An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> -. -. 2 || N ) )
Theorem	zeneo 12397	No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 9559 follows immediately from the fact that a contradiction implies anything, see pm2.21i 649. (Contributed by AV, 22-Jun-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) )
Theorem	odd2np1lem 12398*	Lemma for odd2np1 12399. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( N e. NN0 -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
Theorem	odd2np1 12399*	An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
Theorem	even2n 12400*	An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)
|- ( 2 || N <-> E. n e. ZZ ( 2 x. n ) = N )