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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dvdsext 16301* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥))) | ||
Theorem | fzm1ndvds 16302 | No number between 1 and 𝑀 − 1 divides 𝑀. (Contributed by Mario Carneiro, 24-Jan-2015.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → ¬ 𝑀 ∥ 𝑁) | ||
Theorem | fzo0dvdseq 16303 | Zero is the only one of the first 𝐴 nonnegative integers that is divisible by 𝐴. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 ↔ 𝐵 = 0)) | ||
Theorem | fzocongeq 16304 | Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷 − 𝐶) ∥ (𝐴 − 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | addmodlteqALT 16305 | 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 13947 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽)) | ||
Theorem | dvdsfac 16306 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝐾 ∥ (!‘𝑁)) | ||
Theorem | dvdsexp2im 16307 | If an integer divides another integer, then it also divides any of its powers. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐾 ∥ 𝑀 → 𝐾 ∥ (𝑀↑𝑁))) | ||
Theorem | dvdsexp 16308 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∥ (𝐴↑𝑁)) | ||
Theorem | dvdsmod 16309 | Any number 𝐾 whose mod base 𝑁 is divisible by a divisor 𝑃 of the base is also divisible by 𝑃. This means that primes will also be relatively prime to the base when reduced mod 𝑁 for any base. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) ∧ 𝑃 ∥ 𝑁) → (𝑃 ∥ (𝐾 mod 𝑁) ↔ 𝑃 ∥ 𝐾)) | ||
Theorem | mulmoddvds 16310 | 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.) (Proof shortened by AV, 18-Mar-2022.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑁 ∥ 𝐴 → ((𝐴 · 𝐵) mod 𝑁) = 0)) | ||
Theorem | 3dvds 16311* | A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(0...𝑁)⟶ℤ) → (3 ∥ Σ𝑘 ∈ (0...𝑁)((𝐹‘𝑘) · (;10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘))) | ||
Theorem | 3dvdsdec 16312 | 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 𝐴 and 𝐵 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴 and 𝐵, especially if 𝐴 is itself a decimal number, e.g., 𝐴 = ;𝐶𝐷. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) | ||
Theorem | 3dvds2dec 16313 | 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 𝐴, 𝐵 and 𝐶 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴, 𝐵 and 𝐶. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 ⇒ ⊢ (3 ∥ ;;𝐴𝐵𝐶 ↔ 3 ∥ ((𝐴 + 𝐵) + 𝐶)) | ||
Theorem | fprodfvdvdsd 16314* | A finite product of integers is divisible by any of its factors being function values. (Contributed by AV, 1-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹:𝐵⟶ℤ) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∥ ∏𝑘 ∈ 𝐴 (𝐹‘𝑘)) | ||
Theorem | fproddvdsd 16315* | A finite product of integers is divisible by any of its factors. (Contributed by AV, 14-Aug-2020.) (Proof shortened by AV, 2-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℤ) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∥ ∏𝑘 ∈ 𝐴 𝑘) | ||
The set ℤ of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 16318. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom 2 ∥ 𝑁 to say that "𝑁 is even" (which implies 𝑁 ∈ ℤ, see evenelz 16316) and ¬ 2 ∥ 𝑁 to say that "𝑁 is odd" (under the assumption that 𝑁 ∈ ℤ). The previously proven theorems about even and odd numbers, like zneo 12678, zeo 12681, zeo2 12682, etc. use different representations, which are equivalent to the representations using the divides relation, see evend2 16337 and oddp1d2 16338. The corresponding theorems are zeneo 16319, zeo3 16317 and zeo4 16318. | ||
Theorem | evenelz 16316 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16239. (Contributed by AV, 22-Jun-2021.) |
⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ) | ||
Theorem | zeo3 16317 | An integer is even or odd. With this representation of even and odd integers, this variant of zeo 12681 follows immediately from the law of excluded middle, see exmidd 893. (Contributed by AV, 17-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ ¬ 2 ∥ 𝑁)) | ||
Theorem | zeo4 16318 | An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 12682 follows immediately from the principle of double negation, see notnotb 314. (Contributed by AV, 17-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁)) | ||
Theorem | zeneo 16319 | 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 12678 follows immediately from the fact that a contradiction implies anything, see pm2.21i 119. (Contributed by AV, 22-Jun-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)) | ||
Theorem | odd2np1lem 16320* | Lemma for odd2np1 16321. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝑁 ∈ ℕ0 → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑁)) | ||
Theorem | odd2np1 16321* | 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.) |
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁)) | ||
Theorem | even2n 16322* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) | ||
Theorem | oddm1even 16323 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) | ||
Theorem | oddp1even 16324 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1))) | ||
Theorem | oexpneg 16325 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (-𝐴↑𝑁) = -(𝐴↑𝑁)) | ||
Theorem | mod2eq0even 16326 | 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.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ↔ 2 ∥ 𝑁)) | ||
Theorem | mod2eq1n2dvds 16327 | 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.) (Proof shortened by AV, 5-Jul-2020.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) | ||
Theorem | oddnn02np1 16328* | A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.) |
⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 ((2 · 𝑛) + 1) = 𝑁)) | ||
Theorem | oddge22np1 16329* | An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 9-Jul-2022.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ ((2 · 𝑛) + 1) = 𝑁)) | ||
Theorem | evennn02n 16330* | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ (𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁)) | ||
Theorem | evennn2n 16331* | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁)) | ||
Theorem | 2tp1odd 16332 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → ¬ 2 ∥ 𝐵) | ||
Theorem | mulsucdiv2z 16333 | 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.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ) | ||
Theorem | sqoddm1div8z 16334 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (((𝑁↑2) − 1) / 8) ∈ ℤ) | ||
Theorem | 2teven 16335 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = (2 · 𝐴)) → 2 ∥ 𝐵) | ||
Theorem | zeo5 16336 | An integer is either even or odd, version of zeo3 16317 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ 2 ∥ (𝑁 + 1))) | ||
Theorem | evend2 16337 | 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 12681 and zeo2 12682. (Contributed by AV, 22-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℤ)) | ||
Theorem | oddp1d2 16338 | 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 12681 and zeo2 12682. (Contributed by AV, 22-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | ||
Theorem | zob 16339 | Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.) |
⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | ||
Theorem | oddm1d2 16340 | 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.) |
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | ||
Theorem | ltoddhalfle 16341 | 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.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → (𝑀 < (𝑁 / 2) ↔ 𝑀 ≤ ((𝑁 − 1) / 2))) | ||
Theorem | halfleoddlt 16342 | 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.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → ((𝑁 / 2) ≤ 𝑀 ↔ (𝑁 / 2) < 𝑀)) | ||
Theorem | opoe 16343 | The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ ¬ 2 ∥ 𝐵)) → 2 ∥ (𝐴 + 𝐵)) | ||
Theorem | omoe 16344 | The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ ¬ 2 ∥ 𝐵)) → 2 ∥ (𝐴 − 𝐵)) | ||
Theorem | opeo 16345 | The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 + 𝐵)) | ||
Theorem | omeo 16346 | The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) ∧ (𝐵 ∈ ℤ ∧ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 − 𝐵)) | ||
Theorem | z0even 16347 | 2 divides 0. That means 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
⊢ 2 ∥ 0 | ||
Theorem | n2dvds1 16348 | 2 does not divide 1. That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.) (Proof shortened by Steven Nguyen, 3-May-2023.) |
⊢ ¬ 2 ∥ 1 | ||
Theorem | n2dvdsm1 16349 | 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.) |
⊢ ¬ 2 ∥ -1 | ||
Theorem | z2even 16350 | 2 divides 2. That means 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
⊢ 2 ∥ 2 | ||
Theorem | n2dvds3 16351 | 2 does not divide 3. That means 3 is odd. (Contributed by AV, 28-Feb-2021.) (Proof shortened by Steven Nguyen, 3-May-2023.) |
⊢ ¬ 2 ∥ 3 | ||
Theorem | z4even 16352 | 2 divides 4. That means 4 is even. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.) |
⊢ 2 ∥ 4 | ||
Theorem | 4dvdseven 16353 | An integer which is divisible by 4 is divisible by 2, that is, is even. (Contributed by AV, 4-Jul-2021.) |
⊢ (4 ∥ 𝑁 → 2 ∥ 𝑁) | ||
Theorem | m1expe 16354 | Exponentiation of -1 by an even power. Variant of m1expeven 14110. (Contributed by AV, 25-Jun-2021.) |
⊢ (2 ∥ 𝑁 → (-1↑𝑁) = 1) | ||
Theorem | m1expo 16355 | Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1) | ||
Theorem | m1exp1 16356 | Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) = 1 ↔ 2 ∥ 𝑁)) | ||
Theorem | nn0enne 16357 | A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.) |
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ0 ↔ (𝑁 / 2) ∈ ℕ)) | ||
Theorem | nn0ehalf 16358 | The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ0) | ||
Theorem | nnehalf 16359 | The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ) | ||
Theorem | nn0onn 16360 | An odd nonnegative integer is positive. (Contributed by Steven Nguyen, 25-Mar-2023.) |
⊢ ((𝑁 ∈ ℕ0 ∧ ¬ 2 ∥ 𝑁) → 𝑁 ∈ ℕ) | ||
Theorem | nn0o1gt2 16361 | An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (𝑁 = 1 ∨ 2 < 𝑁)) | ||
Theorem | nno 16362 | An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ) | ||
Theorem | nn0o 16363 | An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) | ||
Theorem | nn0ob 16364 | Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.) |
⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℕ0 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
Theorem | nn0oddm1d2 16365 | A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
Theorem | nnoddm1d2 16366 | A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) | ||
Theorem | sumeven 16367* | If every term in a sum is even, then so is the sum. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 2 ∥ 𝐵) ⇒ ⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | sumodd 16368* | If every term in a sum is odd, then the sum is even iff the number of terms in the sum is even. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) ⇒ ⊢ (𝜑 → (2 ∥ (♯‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵)) | ||
Theorem | evensumodd 16369* | If every term in a sum with an even number of terms is odd, then the sum is even. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → 2 ∥ (♯‘𝐴)) ⇒ ⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | oddsumodd 16370* | If every term in a sum with an odd number of terms is odd, then the sum is odd. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → ¬ 2 ∥ (♯‘𝐴)) ⇒ ⊢ (𝜑 → ¬ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | pwp1fsum 16371* | The n-th power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) | ||
Theorem | oddpwp1fsum 16372* | An odd power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑁) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) | ||
Theorem | divalglem0 16373 | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ ⇒ ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))))) | ||
Theorem | divalglem1 16374 | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 ⇒ ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷))) | ||
Theorem | divalglem2 16375* | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ inf(𝑆, ℝ, < ) ∈ 𝑆 | ||
Theorem | divalglem4 16376* | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)} | ||
Theorem | divalglem5 16377* | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} & ⊢ 𝑅 = inf(𝑆, ℝ, < ) ⇒ ⊢ (0 ≤ 𝑅 ∧ 𝑅 < (abs‘𝐷)) | ||
Theorem | divalglem6 16378 | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝑋 ∈ (0...(𝐴 − 1)) & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · 𝐴)) ∈ (0...(𝐴 − 1))) | ||
Theorem | divalglem7 16379 | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 ⇒ ⊢ ((𝑋 ∈ (0...((abs‘𝐷) − 1)) ∧ 𝐾 ∈ ℤ) → (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1)))) | ||
Theorem | divalglem8 16380* | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑋 < (abs‘𝐷) ∧ 𝑌 < (abs‘𝐷))) → (𝐾 ∈ ℤ → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝑋 = 𝑌))) | ||
Theorem | divalglem9 16381* | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} & ⊢ 𝑅 = inf(𝑆, ℝ, < ) ⇒ ⊢ ∃!𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷) | ||
Theorem | divalglem10 16382* | Lemma for divalg 16383. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by AV, 2-Oct-2020.) |
⊢ 𝑁 ∈ ℤ & ⊢ 𝐷 ∈ ℤ & ⊢ 𝐷 ≠ 0 & ⊢ 𝑆 = {𝑟 ∈ ℕ0 ∣ 𝐷 ∥ (𝑁 − 𝑟)} ⇒ ⊢ ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) | ||
Theorem | divalg 16383* | The division algorithm (theorem). Dividing an integer 𝑁 by a nonzero integer 𝐷 produces a (unique) quotient 𝑞 and a unique remainder 0 ≤ 𝑟 < (abs‘𝐷). Theorem 1.14 in [ApostolNT] p. 19. The proof does not use / or ⌊ or mod. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) | ||
Theorem | divalgb 16384* | Express the division algorithm as stated in divalg 16383 in terms of ∥. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃!𝑟 ∈ ℕ0 (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))) | ||
Theorem | divalg2 16385* | The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃!𝑟 ∈ ℕ0 (𝑟 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑟))) | ||
Theorem | divalgmod 16386 | The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor (compare divalg2 16385 and divalgb 16384). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 ∈ ℕ0 ∧ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅))))) | ||
Theorem | divalgmodcl 16387 | The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor. Variant of divalgmod 16386. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ 𝑅 ∈ ℕ0) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) | ||
Theorem | modremain 16388* | The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → ((𝑁 mod 𝐷) = 𝑅 ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) | ||
Theorem | ndvdssub 16389 | Corollary of the division algorithm. If an integer 𝐷 greater than 1 divides 𝑁, then it does not divide any of 𝑁 − 1, 𝑁 − 2... 𝑁 − (𝐷 − 1). (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐾 < 𝐷)) → (𝐷 ∥ 𝑁 → ¬ 𝐷 ∥ (𝑁 − 𝐾))) | ||
Theorem | ndvdsadd 16390 | Corollary of the division algorithm. If an integer 𝐷 greater than 1 divides 𝑁, then it does not divide any of 𝑁 + 1, 𝑁 + 2... 𝑁 + (𝐷 − 1). (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐾 < 𝐷)) → (𝐷 ∥ 𝑁 → ¬ 𝐷 ∥ (𝑁 + 𝐾))) | ||
Theorem | ndvdsp1 16391 | Special case of ndvdsadd 16390. If an integer 𝐷 greater than 1 divides 𝑁, it does not divide 𝑁 + 1. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ 1 < 𝐷) → (𝐷 ∥ 𝑁 → ¬ 𝐷 ∥ (𝑁 + 1))) | ||
Theorem | ndvdsi 16392 | A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝑄 ∈ ℕ0 & ⊢ 𝑅 ∈ ℕ & ⊢ ((𝐴 · 𝑄) + 𝑅) = 𝐵 & ⊢ 𝑅 < 𝐴 ⇒ ⊢ ¬ 𝐴 ∥ 𝐵 | ||
Theorem | flodddiv4 16393 | The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = ((2 · 𝑀) + 1)) → (⌊‘(𝑁 / 4)) = if(2 ∥ 𝑀, (𝑀 / 2), ((𝑀 − 1) / 2))) | ||
Theorem | fldivndvdslt 16394 | 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.) |
⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℤ ∧ 𝐿 ≠ 0) ∧ ¬ 𝐿 ∥ 𝐾) → (⌊‘(𝐾 / 𝐿)) < (𝐾 / 𝐿)) | ||
Theorem | flodddiv4lt 16395 | The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | ||
Theorem | flodddiv4t2lthalf 16396 | 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.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) | ||
Syntax | cbits 16397 | Define the binary bits of an integer. |
class bits | ||
Syntax | csad 16398 | Define the sequence addition on bit sequences. |
class sadd | ||
Syntax | csmu 16399 | Define the sequence multiplication on bit sequences. |
class smul | ||
Definition | df-bits 16400* | Define the binary bits of an integer. The expression 𝑀 ∈ (bits‘𝑁) means that the 𝑀-th bit of 𝑁 is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))}) |
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