Home | Intuitionistic Logic Explorer Theorem List (p. 102 of 106) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | climcaucn 10101* | A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 10097 but adds the part that is complex. (Contributed by Jim Kingdon, 24-Aug-2021.) |
Theorem | serif0 10102* | If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.) |
Syntax | csu 10103 | Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.) |
Definition | df-sum 10104* | Define the sum of a series with an index set of integers . is normally a free variable in , i.e. can be thought of as . This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers). Examples: means , and means 1/2 + 1/4 + 1/8 + ... = 1. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Theorem | sumeq1 10105 | Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Theorem | nfsum1 10106 | Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Theorem | nfsum 10107 | Bound-variable hypothesis builder for sum: if is (effectively) not free in and , it is not free in . (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Syntax | cdvds 10108 | Extend the definition of a class to include the divides relation. See df-dvds 10109. |
Definition | df-dvds 10109* | Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | divides 10110* | Define the divides relation. means divides into with no remainder. For example, (ex-dvds 10283). As proven in dvdsval3 10112, . See divides 10110 and dvdsval2 10111 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsval2 10111 | One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Theorem | dvdsval3 10112 | 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 10113 | Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Theorem | nndivdvds 10114 | Strong form of dvdsval2 10111 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Theorem | nndivides 10115* | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
Theorem | dvdsdc 10116 | Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) |
DECID | ||
Theorem | moddvds 10117 | Two ways to say (mod ), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | dvds0lem 10118 | A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds1lem 10119* | A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2lem 10120* | A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | iddvds 10121 | 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 10122 | 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds0 10123 | Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | negdvdsb 10124 | An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsnegb 10125 | An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | absdvdsb 10126 | An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsabsb 10127 | An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | 0dvds 10128 | Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmul1 10129 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmul2 10130 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | iddvdsexp 10131 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | muldvds1 10132 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | muldvds2 10133 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdscmul 10134 | 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 10135 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdscmulr 10136 | Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmulcr 10137 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | summodnegmod 10138 | 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 10139 | Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
Theorem | dvds2ln 10140 | 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 10141 | If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2sub 10142 | If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2subd 10143 | Natural deduction form of dvds2sub 10142. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Theorem | dvdstr 10144 | 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 10145 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | dvdsmultr1d 10146 | Natural deduction form of dvdsmultr1 10145. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Theorem | dvdsmultr2 10147 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | ordvdsmul 10148 | 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 10149 | 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 10150 | 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 10151 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssub 10152 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssubr 10153 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdsadd2b 10154 | Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | dvdslelemd 10155 | Lemma for dvdsle 10156. (Contributed by Jim Kingdon, 8-Nov-2021.) |
Theorem | dvdsle 10156 | The divisors of a positive integer are bounded by it. The proof does not use . (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsleabs 10157 | 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 10158 | Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
Theorem | dvdsabseq 10159 | 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 10160 | 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 10161 | 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 10162* | 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 10163* | 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 10164* | The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Theorem | dvds1 10165 | The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | alzdvds 10166* | Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsext 10167* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Theorem | fzm1ndvds 10168 | No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.) |
Theorem | fzo0dvdseq 10169 | Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ | ||
Theorem | fzocongeq 10170 | Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ ..^ | ||
Theorem | addmodlteqALT 10171 | 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 9348 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
..^ ..^ | ||
Theorem | dvdsfac 10172 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
Theorem | dvdsexp 10173 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dvdsmod 10174 | 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 10175 | 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 10176 | 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 10177 | 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 10181. 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 10178) and to say that " is odd" (under the assumption that ). The previously proven theorems about even and odd numbers, like zneo 8398, zeo 8402, zeo2 8403, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 10201 and oddp1d2 10202. The corresponding theorems are zeneo 10182, zeo3 10179 and zeo4 10181. | ||
Theorem | evenelz 10178 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 10113. (Contributed by AV, 22-Jun-2021.) |
Theorem | zeo3 10179 | An integer is even or odd. (Contributed by AV, 17-Jun-2021.) |
Theorem | zeoxor 10180 | An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.) |
Theorem | zeo4 10181 | An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.) |
Theorem | zeneo 10182 | 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 8398 follows immediately from the fact that a contradiction implies anything, see pm2.21i 585. (Contributed by AV, 22-Jun-2021.) |
Theorem | odd2np1lem 10183* | Lemma for odd2np1 10184. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | odd2np1 10184* | 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 10185* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
Theorem | oddm1even 10186 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oddp1even 10187 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oexpneg 10188 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) |
Theorem | mod2eq0even 10189 | 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 | fz01or 10190 | An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.) |
Theorem | mod2eq1n2dvds 10191 | 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 10192* | A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.) |
Theorem | oddge22np1 10193* | An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.) |
Theorem | evennn02n 10194* | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | evennn2n 10195* | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | 2tp1odd 10196 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
Theorem | mulsucdiv2z 10197 | 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 10198 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
Theorem | 2teven 10199 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
Theorem | zeo5 10200 | An integer is either even or odd, version of zeo3 10179 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |