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Theorem List for Intuitionistic Logic Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvdsext 11401* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremfzm1ndvds 11402 No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremfzo0dvdseq 11403 Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremfzocongeq 11404 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^

TheoremaddmodlteqALT 11405 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 10064 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
..^ ..^

Theoremdvdsfac 11406 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremdvdsexp 11407 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdsmod 11408 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 11409 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 11410 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 11411 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 11415. 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 11412) and to say that " is odd" (under the assumption that ). The previously proven theorems about even and odd numbers, like zneo 9056, zeo 9060, zeo2 9061, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 11434 and oddp1d2 11435. The corresponding theorems are zeneo 11416, zeo3 11413 and zeo4 11415.

Theoremevenelz 11412 An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11346. (Contributed by AV, 22-Jun-2021.)

Theoremzeo3 11413 An integer is even or odd. (Contributed by AV, 17-Jun-2021.)

Theoremzeoxor 11414 An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)

Theoremzeo4 11415 An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)

Theoremzeneo 11416 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 9056 follows immediately from the fact that a contradiction implies anything, see pm2.21i 618. (Contributed by AV, 22-Jun-2021.)

Theoremodd2np1lem 11417* Lemma for odd2np1 11418. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremodd2np1 11418* 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 11419* An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)

Theoremoddm1even 11420 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoddp1even 11421 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoexpneg 11422 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)

Theoremmod2eq0even 11423 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 11424 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 11425* A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)

Theoremoddge22np1 11426* An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)

Theoremevennn02n 11427* A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)

Theoremevennn2n 11428* A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)

Theorem2tp1odd 11429 A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)

Theoremmulsucdiv2z 11430 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 11431 A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)

Theorem2teven 11432 A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)

Theoremzeo5 11433 An integer is either even or odd, version of zeo3 11413 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.)

Theoremevend2 11434 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 9060 and zeo2 9061. (Contributed by AV, 22-Jun-2021.)

Theoremoddp1d2 11435 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 9060 and zeo2 9061. (Contributed by AV, 22-Jun-2021.)

Theoremzob 11436 Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)

Theoremoddm1d2 11437 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 11438 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 11439 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 11440 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomoe 11441 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopeo 11442 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 11443 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 11444 Exponentiation of -1 by an even power. Variant of m1expeven 10233. (Contributed by AV, 25-Jun-2021.)

Theoremm1expo 11445 Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)

Theoremm1exp1 11446 Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)

Theoremnn0enne 11447 A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)

Theoremnn0ehalf 11448 The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.)

Theoremnnehalf 11449 The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.)

Theoremnn0o1gt2 11450 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)

Theoremnno 11451 An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)

Theoremnn0o 11452 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)

Theoremnn0ob 11453 Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)

Theoremnn0oddm1d2 11454 A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)

Theoremnnoddm1d2 11455 A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)

Theoremz0even 11456 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)

Theoremn2dvds1 11457 2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremn2dvdsm1 11458 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)

Theoremz2even 11459 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)

Theoremn2dvds3 11460 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)

Theoremz4even 11461 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)

Theorem4dvdseven 11462 An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)

5.1.3  The division algorithm

Theoremdivalglemnn 11463* Lemma for divalg 11469. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)

Theoremdivalglemqt 11464 Lemma for divalg 11469. The case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)

Theoremdivalglemnqt 11465 Lemma for divalg 11469. The case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)

Theoremdivalglemeunn 11466* Lemma for divalg 11469. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)

Theoremdivalglemex 11467* Lemma for divalg 11469. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.)

Theoremdivalglemeuneg 11468* Lemma for divalg 11469. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)

Theoremdivalg 11469* The division algorithm (theorem). Dividing an integer by a nonzero integer produces a (unique) quotient and a unique remainder . Theorem 1.14 in [ApostolNT] p. 19. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalgb 11470* Express the division algorithm as stated in divalg 11469 in terms of . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdivalg2 11471* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalgmod 11472 The result of the operator satisfies the requirements for the remainder in the division algorithm for a positive divisor (compare divalg2 11471 and divalgb 11470). This demonstration theorem justifies the use of to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)

Theoremdivalgmodcl 11473 The result of the operator satisfies the requirements for the remainder in the division algorithm for a positive divisor. Variant of divalgmod 11472. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.)

Theoremmodremain 11474* The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.)

Theoremndvdssub 11475 Corollary of the division algorithm. If an integer greater than divides , then it does not divide any of , ... . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsadd 11476 Corollary of the division algorithm. If an integer greater than divides , then it does not divide any of , ... . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsp1 11477 Special case of ndvdsadd 11476. If an integer greater than divides , it does not divide . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsi 11478 A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremflodddiv4 11479 The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.)

Theoremfldivndvdslt 11480 The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.)

Theoremflodddiv4lt 11481 The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.)

Theoremflodddiv4t2lthalf 11482 The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.)

5.1.4  The greatest common divisor operator

Syntaxcgcd 11483 Extend the definition of a class to include the greatest common divisor operator.

Definitiondf-gcd 11484* Define the operator. For example, (ex-gcd 12636). (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdmndc 11485 Decidablity lemma used in various proofs related to . (Contributed by Jim Kingdon, 12-Dec-2021.)
DECID

Theoremzsupcllemstep 11486* Lemma for zsupcl 11488. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)
DECID

Theoremzsupcllemex 11487* Lemma for zsupcl 11488. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
DECID

Theoremzsupcl 11488* Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at (which corresponds to the nonempty condition of classical supremum theorems), (b) decidable at each value after , and (c) be false after (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
DECID

Theoremzssinfcl 11489* The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
inf        inf

Theoreminfssuzex 11490* Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.)
DECID

Theoreminfssuzledc 11491* The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.)
DECID        inf

Theoreminfssuzcldc 11492* The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
DECID        inf

Theoremdvdsbnd 11493* There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.)

Theoremgcdsupex 11494* Existence of the supremum used in defining . (Contributed by Jim Kingdon, 12-Dec-2021.)

Theoremgcdsupcl 11495* Closure of the supremum used in defining . A lemma for gcdval 11496 and gcdn0cl 11499. (Contributed by Jim Kingdon, 11-Dec-2021.)

Theoremgcdval 11496* The value of the operator. is the greatest common divisor of and . If and are both , the result is defined conventionally as . (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremgcd0val 11497 The value, by convention, of the operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0val 11498* The value of the operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcdn0cl 11499 Closure of the operator. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremgcddvds 11500 The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)

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