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Theorem List for Intuitionistic Logic Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvdsmul1 11701 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmul2 11702 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremiddvdsexp 11703 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremmuldvds1 11704 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremmuldvds2 11705 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmul 11706 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.)

Theoremdvdsmulc 11707 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmulr 11708 Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmulcr 11709 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremsummodnegmod 11710 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.)

Theoremmodmulconst 11711 Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)

Theoremdvds2ln 11712 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.)

Theoremdvds2add 11713 If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2sub 11714 If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2subd 11715 Deduction form of dvds2sub 11714. (Contributed by Stanislas Polu, 9-Mar-2020.)

Theoremdvdstr 11716 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.)

Theoremdvdsmultr1 11717 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremdvdsmultr1d 11718 Natural deduction form of dvdsmultr1 11717. (Contributed by Stanislas Polu, 9-Mar-2020.)

Theoremdvdsmultr2 11719 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremordvdsmul 11720 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.)

Theoremdvdssub2 11721 If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsadd 11722 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsaddr 11723 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssub 11724 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssubr 11725 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdsadd2b 11726 Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremdvdslelemd 11727 Lemma for dvdsle 11728. (Contributed by Jim Kingdon, 8-Nov-2021.)

Theoremdvdsle 11728 The divisors of a positive integer are bounded by it. The proof does not use . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsleabs 11729 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.)

Theoremdvdsleabs2 11730 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremdvdsabseq 11731 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.)

Theoremdvdseq 11732 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.)

Theoremdivconjdvds 11733 If a nonzero integer divides another integer , the other integer divided by the nonzero integer (i.e. the divisor conjugate of to ) divides the other integer . Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)

Theoremdvdsdivcl 11734* The complement of a divisor of is also a divisor of . (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)

Theoremdvdsflip 11735* An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)

Theoremdvdsssfz1 11736* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremdvds1 11737 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremalzdvds 11738* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsext 11739* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremfzm1ndvds 11740 No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremfzo0dvdseq 11741 Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremfzocongeq 11742 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^

TheoremaddmodlteqALT 11743 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 10290 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
..^ ..^

Theoremdvdsfac 11744 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremdvdsexp 11745 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdsmod 11746 Any number whose mod base is divisible by a divisor of the base is also divisible by . This means that primes will also be relatively prime to the base when reduced for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremmulmoddvds 11747 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.)

Theorem3dvdsdec 11748 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 and actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers and , especially if is itself a decimal number, e.g., ;. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
;

Theorem3dvds2dec 11749 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 , and actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers , and . (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
;;

5.1.2  Even and odd numbers

The set of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 11753. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom to say that " is even" (which implies , see evenelz 11750) and to say that " is odd" (under the assumption that ). The previously proven theorems about even and odd numbers, like zneo 9259, zeo 9263, zeo2 9264, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 11772 and oddp1d2 11773. The corresponding theorems are zeneo 11754, zeo3 11751 and zeo4 11753.

Theoremevenelz 11750 An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11681. (Contributed by AV, 22-Jun-2021.)

Theoremzeo3 11751 An integer is even or odd. (Contributed by AV, 17-Jun-2021.)

Theoremzeoxor 11752 An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)

Theoremzeo4 11753 An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)

Theoremzeneo 11754 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 9259 follows immediately from the fact that a contradiction implies anything, see pm2.21i 636. (Contributed by AV, 22-Jun-2021.)

Theoremodd2np1lem 11755* Lemma for odd2np1 11756. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremodd2np1 11756* 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.)

Theoremeven2n 11757* An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)

Theoremoddm1even 11758 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoddp1even 11759 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoexpneg 11760 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)

Theoremmod2eq0even 11761 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.)

Theoremmod2eq1n2dvds 11762 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.)

Theoremoddnn02np1 11763* A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)

Theoremoddge22np1 11764* An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)

Theoremevennn02n 11765* A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)

Theoremevennn2n 11766* A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)

Theorem2tp1odd 11767 A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)

Theoremmulsucdiv2z 11768 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.)

Theoremsqoddm1div8z 11769 A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)

Theorem2teven 11770 A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)

Theoremzeo5 11771 An integer is either even or odd, version of zeo3 11751 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.)

Theoremevend2 11772 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 9263 and zeo2 9264. (Contributed by AV, 22-Jun-2021.)

Theoremoddp1d2 11773 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 9263 and zeo2 9264. (Contributed by AV, 22-Jun-2021.)

Theoremzob 11774 Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)

Theoremoddm1d2 11775 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.)

Theoremltoddhalfle 11776 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.)

Theoremhalfleoddlt 11777 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.)

Theoremopoe 11778 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomoe 11779 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopeo 11780 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomeo 11781 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremm1expe 11782 Exponentiation of -1 by an even power. Variant of m1expeven 10459. (Contributed by AV, 25-Jun-2021.)

Theoremm1expo 11783 Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)

Theoremm1exp1 11784 Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)

Theoremnn0enne 11785 A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)

Theoremnn0ehalf 11786 The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.)

Theoremnnehalf 11787 The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.)

Theoremnn0o1gt2 11788 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)

Theoremnno 11789 An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)

Theoremnn0o 11790 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)

Theoremnn0ob 11791 Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)

Theoremnn0oddm1d2 11792 A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)

Theoremnnoddm1d2 11793 A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)

Theoremz0even 11794 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)

Theoremn2dvds1 11795 2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremn2dvdsm1 11796 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)

Theoremz2even 11797 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)

Theoremn2dvds3 11798 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)

Theoremz4even 11799 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)

Theorem4dvdseven 11800 An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)

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