P. 108 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fsump1 10701*	The addition of the next term in a finite sum of A ( k ) is the current term plus B i.e. A ( N + 1 ). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   &    |- ( k = ( N + 1 ) -> A = B )   =>    |- ( ph -> sum_ k e. ( M ... ( N + 1 ) ) A = ( sum_ k e. ( M ... N ) A + B ) )
PART 4  ELEMENTARY NUMBER THEORY Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.
4.1  Elementary properties of divisibility
4.1.1  The divides relation
Syntax	cdvds 10702	Extend the definition of a class to include the divides relation. See df-dvds 10703.
class ||
Definition	df-dvds 10703*	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 10704*	Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 11126). As proven in dvdsval3 10706, M || N <-> ( N mod M ) = 0. See divides 10704 and dvdsval2 10705 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 10705	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 10706	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 10707	Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- ( X || Y -> ( X e. ZZ /\ Y e. ZZ ) )
Theorem	nndivdvds 10708	Strong form of dvdsval2 10705 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 10709*	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 10710	Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
|- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N )
Theorem	moddvds 10711	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	dvds0lem 10712	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 10713*	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 10714*	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 10715	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 10716	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 10717	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 10718	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 10719	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 10720	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 10721	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 10722	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 10723	Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N )
Theorem	dvdsmul1 10724	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 10725	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 10726	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 10727	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 10728	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 10729	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 10730	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 10731	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 10732	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 10733	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 10734	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 10735	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 10736	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 10737	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 10738	Natural deduction form of dvds2sub 10737. (Contributed by Stanislas Polu, 9-Mar-2020.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> K || M )   &    |- ( ph -> K || N )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> K || ( M - N ) )
Theorem	dvdstr 10739	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	dvdsmultr1 10740	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 10741	Natural deduction form of dvdsmultr1 10740. (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 10742	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 10743	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 10744	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 10745	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 10746	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 10747	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 10748	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 10749	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	dvdslelemd 10750	Lemma for dvdsle 10751. (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 10751	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 10752	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 10753	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 10754	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 10755	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 10756	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 10757*	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 10758*	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 10759*	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 10760	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 10761*	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 10762*	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 10763	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 10764	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 10765	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 10766	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 9736 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 10767	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 10768	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 10769	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 10770	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	3dvdsdec 10771	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 10772	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 ) )
4.1.2  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 10776. 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 10773) 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 8783, zeo 8787, zeo2 8788, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 10795 and oddp1d2 10796. The corresponding theorems are zeneo 10777, zeo3 10774 and zeo4 10776.
Theorem	evenelz 10773	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 10707. (Contributed by AV, 22-Jun-2021.)
|- ( 2 || N -> N e. ZZ )
Theorem	zeo3 10774	An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
Theorem	zeoxor 10775	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 10776	An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> -. -. 2 || N ) )
Theorem	zeneo 10777	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 8783 follows immediately from the fact that a contradiction implies anything, see pm2.21i 608. (Contributed by AV, 22-Jun-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) )
Theorem	odd2np1lem 10778*	Lemma for odd2np1 10779. (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 10779*	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 10780*	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 )
Theorem	oddm1even 10781	An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N - 1 ) ) )
Theorem	oddp1even 10782	An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N + 1 ) ) )
Theorem	oexpneg 10783	The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)
|- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> ( -u A ^ N ) = -u ( A ^ N ) )
Theorem	mod2eq0even 10784	An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
|- ( N e. ZZ -> ( ( N mod 2 ) = 0 <-> 2 || N ) )
Theorem	mod2eq1n2dvds 10785	An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.)
|- ( N e. ZZ -> ( ( N mod 2 ) = 1 <-> -. 2 || N ) )
Theorem	oddnn02np1 10786*	A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)
|- ( N e. NN0 -> ( -. 2 || N <-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = N ) )
Theorem	oddge22np1 10787*	An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)
|- ( N e. ( ZZ>= ` 2 ) -> ( -. 2 || N <-> E. n e. NN ( ( 2 x. n ) + 1 ) = N ) )
Theorem	evennn02n 10788*	A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)
|- ( N e. NN0 -> ( 2 || N <-> E. n e. NN0 ( 2 x. n ) = N ) )
Theorem	evennn2n 10789*	A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)
|- ( N e. NN -> ( 2 || N <-> E. n e. NN ( 2 x. n ) = N ) )
Theorem	2tp1odd 10790	A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)
|- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )
Theorem	mulsucdiv2z 10791	An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.)
|- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
Theorem	sqoddm1div8z 10792	A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) e. ZZ )
Theorem	2teven 10793	A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
|- ( ( A e. ZZ /\ B = ( 2 x. A ) ) -> 2 || B )
Theorem	zeo5 10794	An integer is either even or odd, version of zeo3 10774 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.)
|- ( N e. ZZ -> ( 2 || N \/ 2 || ( N + 1 ) ) )
Theorem	evend2 10795	An integer is even iff its quotient with 2 is an integer. This is a representation of even numbers without using the divides relation, see zeo 8787 and zeo2 8788. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> ( N / 2 ) e. ZZ ) )
Theorem	oddp1d2 10796	An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 8787 and zeo2 8788. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	zob 10797	Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
|- ( N e. ZZ -> ( ( ( N + 1 ) / 2 ) e. ZZ <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
Theorem	oddm1d2 10798	An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
Theorem	ltoddhalfle 10799	An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.)
|- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( M < ( N / 2 ) <-> M <_ ( ( N - 1 ) / 2 ) ) )
Theorem	halfleoddlt 10800	An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) )