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Type | Label | Description |
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Statement | ||
Theorem | fsump1 10701* | The addition of the next term in a finite sum of is the current term plus i.e. . (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.) |
Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Syntax | cdvds 10702 | Extend the definition of a class to include the divides relation. See df-dvds 10703. |
Definition | df-dvds 10703* | Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | divides 10704* | Define the divides relation. means divides into with no remainder. For example, (ex-dvds 11126). As proven in dvdsval3 10706, . See divides 10704 and dvdsval2 10705 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.) |
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.) |
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.) |
Theorem | dvdszrcl 10707 | Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Theorem | nndivdvds 10708 | Strong form of dvdsval2 10705 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Theorem | nndivides 10709* | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
Theorem | dvdsdc 10710 | Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) |
DECID | ||
Theorem | moddvds 10711 | Two ways to say (mod ), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | dvds0lem 10712 | A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds1lem 10713* | A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2lem 10714* | A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
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.) |
Theorem | 1dvds 10716 | 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds0 10717 | Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | negdvdsb 10718 | An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsnegb 10719 | An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | absdvdsb 10720 | An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsabsb 10721 | An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | 0dvds 10722 | Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | zdvdsdc 10723 | Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
DECID | ||
Theorem | dvdsmul1 10724 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmul2 10725 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | iddvdsexp 10726 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | muldvds1 10727 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | muldvds2 10728 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
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.) |
Theorem | dvdsmulc 10730 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdscmulr 10731 | Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmulcr 10732 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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.) |
Theorem | modmulconst 10734 | Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
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.) |
Theorem | dvds2add 10736 | If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2sub 10737 | If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2subd 10738 | Natural deduction form of dvds2sub 10737. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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.) |
Theorem | dvdsmultr1 10740 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | dvdsmultr1d 10741 | Natural deduction form of dvdsmultr1 10740. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Theorem | dvdsmultr2 10742 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
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.) |
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.) |
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.) |
Theorem | dvdsaddr 10746 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssub 10747 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssubr 10748 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdsadd2b 10749 | Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | dvdslelemd 10750 | Lemma for dvdsle 10751. (Contributed by Jim Kingdon, 8-Nov-2021.) |
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.) |
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.) |
Theorem | dvdsleabs2 10753 | Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
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.) |
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.) |
Theorem | divconjdvds 10756 | 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.) |
Theorem | dvdsdivcl 10757* | 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.) |
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.) |
Theorem | dvdsssfz1 10759* | The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Theorem | dvds1 10760 | The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | alzdvds 10761* | Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsext 10762* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Theorem | fzm1ndvds 10763 | No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.) |
Theorem | fzo0dvdseq 10764 | Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ | ||
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.) |
..^ ..^ | ||
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.) |
..^ ..^ | ||
Theorem | dvdsfac 10767 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
Theorem | dvdsexp 10768 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dvdsmod 10769 | 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.) |
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.) |
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 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.) |
; | ||
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 , 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.) |
;; | ||
The set 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 to say that " is even" (which implies , see evenelz 10773) and to say that " is odd" (under the assumption that ). 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.) |
Theorem | zeo3 10774 | An integer is even or odd. (Contributed by AV, 17-Jun-2021.) |
Theorem | zeoxor 10775 | An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.) |
Theorem | zeo4 10776 | An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.) |
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.) |
Theorem | odd2np1lem 10778* | Lemma for odd2np1 10779. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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.) |
Theorem | even2n 10780* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
Theorem | oddm1even 10781 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oddp1even 10782 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oexpneg 10783 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) |
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.) |
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.) |
Theorem | oddnn02np1 10786* | A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.) |
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.) |
Theorem | evennn02n 10788* | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | evennn2n 10789* | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | 2tp1odd 10790 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
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.) |
Theorem | sqoddm1div8z 10792 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
Theorem | 2teven 10793 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
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.) |
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.) |
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.) |
Theorem | zob 10797 | Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.) |
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.) |
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.) |
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.) |
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