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Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclimcaucn 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.)

Theoremserif0 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.)

3.8.2  Finite and infinite sums

Syntaxcsu 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.)

Definitiondf-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.)

Theoremsumeq1 10105 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Theoremnfsum1 10106 Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Theoremnfsum 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.)

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

Syntaxcdvds 10108 Extend the definition of a class to include the divides relation. See df-dvds 10109.

Definitiondf-dvds 10109* Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivides 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.)

Theoremdvdsval2 10111 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremdvdsval3 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.)

Theoremdvdszrcl 10113 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremnndivdvds 10114 Strong form of dvdsval2 10111 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremnndivides 10115* Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)

Theoremdvdsdc 10116 Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
DECID

Theoremmoddvds 10117 Two ways to say (mod ), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdvds0lem 10118 A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds1lem 10119* A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2lem 10120* A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremiddvds 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.)

Theorem1dvds 10122 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds0 10123 Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegdvdsb 10124 An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsnegb 10125 An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremabsdvdsb 10126 An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsabsb 10127 An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Theorem0dvds 10128 Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmul1 10129 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmul2 10130 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremiddvdsexp 10131 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremmuldvds1 10132 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremmuldvds2 10133 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmul 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.)

Theoremdvdsmulc 10135 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmulr 10136 Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmulcr 10137 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremsummodnegmod 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.)

Theoremmodmulconst 10139 Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)

Theoremdvds2ln 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.)

Theoremdvds2add 10141 If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2sub 10142 If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2subd 10143 Natural deduction form of dvds2sub 10142. (Contributed by Stanislas Polu, 9-Mar-2020.)

Theoremdvdstr 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.)

Theoremdvdsmultr1 10145 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremdvdsmultr1d 10146 Natural deduction form of dvdsmultr1 10145. (Contributed by Stanislas Polu, 9-Mar-2020.)

Theoremdvdsmultr2 10147 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremordvdsmul 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.)

Theoremdvdssub2 10149 If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsadd 10150 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsaddr 10151 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssub 10152 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssubr 10153 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdsadd2b 10154 Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremdvdslelemd 10155 Lemma for dvdsle 10156. (Contributed by Jim Kingdon, 8-Nov-2021.)

Theoremdvdsle 10156 The divisors of a positive integer are bounded by it. The proof does not use . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsleabs 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.)

Theoremdvdsleabs2 10158 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremdvdsabseq 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.)

Theoremdvdseq 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.)

Theoremdivconjdvds 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.)

Theoremdvdsdivcl 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.)

Theoremdvdsflip 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.)

Theoremdvdsssfz1 10164* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremdvds1 10165 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremalzdvds 10166* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsext 10167* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremfzm1ndvds 10168 No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremfzo0dvdseq 10169 Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremfzocongeq 10170 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^

TheoremaddmodlteqALT 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.)
..^ ..^

Theoremdvdsfac 10172 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremdvdsexp 10173 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdsmod 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.)

Theoremmulmoddvds 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.)

Theorem3dvdsdec 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.)
;

Theorem3dvds2dec 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.)
;;

4.1.2  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 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.

Theoremevenelz 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.)

Theoremzeo3 10179 An integer is even or odd. (Contributed by AV, 17-Jun-2021.)

Theoremzeoxor 10180 An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)

Theoremzeo4 10181 An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)

Theoremzeneo 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.)

Theoremodd2np1lem 10183* Lemma for odd2np1 10184. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremodd2np1 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.)

Theoremeven2n 10185* An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)

Theoremoddm1even 10186 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoddp1even 10187 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoexpneg 10188 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)

Theoremmod2eq0even 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.)

Theoremfz01or 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.)

Theoremmod2eq1n2dvds 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.)

Theoremoddnn02np1 10192* A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)

Theoremoddge22np1 10193* An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)

Theoremevennn02n 10194* A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)

Theoremevennn2n 10195* A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)

Theorem2tp1odd 10196 A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)

Theoremmulsucdiv2z 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.)

Theoremsqoddm1div8z 10198 A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)

Theorem2teven 10199 A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)

Theoremzeo5 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.)

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