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
Theorem	dvdsmodexp 12301	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 12751). (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 12302	Strong form of dvdsval2 12296 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 12303*	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 12304	Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
|- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N )
Theorem	moddvds 12305	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 12306	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 12307	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 12308*	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 12309*	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 12310	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 12311	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 12312	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 12313	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 12314	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 12315	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 12316	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 12317	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 12318	Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N )
Theorem	dvdsmul1 12319	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 12320	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 12321	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 12322	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 12323	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 12324	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 12325	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 12326	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 12327	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 12328	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 12329	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 12330	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 12331	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 12332	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 12333	Deduction form of dvds2sub 12332. (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 12334	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 12335	Deduction form of dvds2add 12331. (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 12336	The divides relation is transitive, a deduction version of dvdstr 12334. (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 12337	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 12338	Natural deduction form of dvdsmultr1 12337. (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 12339	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 12340	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 12341	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 12342	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 12343	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 12344	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 12345	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 12346	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 12347	Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12346 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 12348*	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 12349	Lemma for dvdsle 12350. (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 12350	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 12351	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 12352	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 12353	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 12354	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 12355	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 12356*	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 12357*	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 12358*	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 12359	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 12360*	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 12361*	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 12362	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 12363	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 12364	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 12365	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 10615 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 12366	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 12367	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 12368	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 12369	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 12370*	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 12371	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 12372	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 12376. 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 12373) 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 9544, zeo 9548, zeo2 9549, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 12395 and oddp1d2 12396. The corresponding theorems are zeneo 12377, zeo3 12374 and zeo4 12376.
Theorem	evenelz 12373	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 12298. (Contributed by AV, 22-Jun-2021.)
|- ( 2 || N -> N e. ZZ )
Theorem	zeo3 12374	An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
Theorem	zeoxor 12375	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 12376	An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> -. -. 2 || N ) )
Theorem	zeneo 12377	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 9544 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 12378*	Lemma for odd2np1 12379. (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 12379*	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 12380*	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 12381	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 12382	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 12383	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 12384	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 12385	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 12386*	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 12387*	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 12388*	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 12389*	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 12390	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 12391	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 12392	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 12393	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 12394	An integer is either even or odd, version of zeo3 12374 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 12395	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 9548 and zeo2 9549. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> ( N / 2 ) e. ZZ ) )
Theorem	oddp1d2 12396	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 9548 and zeo2 9549. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	zob 12397	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 12398	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 12399	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 12400	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 ) )