P. 163 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvdsmultr2 16201	If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ 𝑁 → 𝐾 ∥ (𝑀 · 𝑁)))
Theorem	dvdsmultr2d 16202	Deduction form of dvdsmultr2 16201. (Contributed by SN, 23-Aug-2024.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ∥ 𝑁)    ⇒   ⊢ (𝜑 → 𝐾 ∥ (𝑀 · 𝑁))
Theorem	ordvdsmul 16203	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.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ 𝑀 ∨ 𝐾 ∥ 𝑁) → 𝐾 ∥ (𝑀 · 𝑁)))
Theorem	dvdssub2 16204	If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∥ (𝑀 − 𝑁)) → (𝐾 ∥ 𝑀 ↔ 𝐾 ∥ 𝑁))
Theorem	dvdsadd 16205	An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ 𝑀 ∥ (𝑀 + 𝑁)))
Theorem	dvdsaddr 16206	An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ 𝑀 ∥ (𝑁 + 𝑀)))
Theorem	dvdssub 16207	An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ 𝑀 ∥ (𝑀 − 𝑁)))
Theorem	dvdssubr 16208	An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ 𝑀 ∥ (𝑁 − 𝑀)))
Theorem	dvdsadd2b 16209	Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶)) → (𝐴 ∥ 𝐵 ↔ 𝐴 ∥ (𝐶 + 𝐵)))
Theorem	dvdsaddre2b 16210	Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 16209 only requiring 𝐵 to be a real number (not necessarily an integer). (Contributed by AV, 19-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶)) → (𝐴 ∥ 𝐵 ↔ 𝐴 ∥ (𝐶 + 𝐵)))
Theorem	fsumdvds 16211*	If every term in a sum is divisible by 𝑁, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵)    ⇒   ⊢ (𝜑 → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵)
Theorem	dvdslelem 16212	Lemma for dvdsle 16213. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐾 ∈ ℤ    ⇒   ⊢ (𝑁 < 𝑀 → (𝐾 · 𝑀) ≠ 𝑁)
Theorem	dvdsle 16213	The divisors of a positive integer are bounded by it. The proof does not use /. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 → 𝑀 ≤ 𝑁))
Theorem	dvdsleabs 16214	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.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀 ∥ 𝑁 → 𝑀 ≤ (abs‘𝑁)))
Theorem	dvdsleabs2 16215	Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀 ∥ 𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
Theorem	dvdsabseq 16216	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.)
⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → (abs‘𝑀) = (abs‘𝑁))
Theorem	dvdseq 16217	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.)
⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀)) → 𝑀 = 𝑁)
Theorem	divconjdvds 16218	If a nonzero integer 𝑀 divides another integer 𝑁, the other integer 𝑁 divided by the nonzero integer 𝑀 (i.e. the divisor conjugate of 𝑁 to 𝑀) divides the other integer 𝑁. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
⊢ ((𝑀 ∥ 𝑁 ∧ 𝑀 ≠ 0) → (𝑁 / 𝑀) ∥ 𝑁)
Theorem	dvdsdivcl 16219*	The complement of a divisor of 𝑁 is also a divisor of 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
Theorem	dvdsflip 16220*	An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)
⊢ 𝐴 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}    &   ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝑁 / 𝑦))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐴–1-1-onto→𝐴)
Theorem	dvdsssfz1 16221*	The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴))
Theorem	dvds1 16222	The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
⊢ (𝑀 ∈ ℕ0 → (𝑀 ∥ 1 ↔ 𝑀 = 1))
Theorem	alzdvds 16223*	Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝑁 ∈ ℤ → (∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 ↔ 𝑁 = 0))
Theorem	dvdsext 16224*	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 16225	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 16226	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 16227	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 16228	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 13845 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 16229	A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝐾 ∥ (!‘𝑁))
Theorem	dvdsexp2im 16230	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 16231	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 16232	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 16233	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 16234*	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 16235	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 16236	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 16237*	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 16238*	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)
6.1.4  Even and odd numbers The set ℤ of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 16241. 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 16239) 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 12548, zeo 12551, zeo2 12552, etc. use different representations, which are equivalent to the representations using the divides relation, see evend2 16260 and oddp1d2 16261. The corresponding theorems are zeneo 16242, zeo3 16240 and zeo4 16241.
Theorem	evenelz 16239	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16160. (Contributed by AV, 22-Jun-2021.)
⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ)
Theorem	zeo3 16240	An integer is even or odd. With this representation of even and odd integers, this variant of zeo 12551 follows immediately from the law of excluded middle, see exmidd 895. (Contributed by AV, 17-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ ¬ 2 ∥ 𝑁))
Theorem	zeo4 16241	An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 12552 follows immediately from the principle of double negation, see notnotb 315. (Contributed by AV, 17-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))
Theorem	zeneo 16242	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 12548 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 16243*	Lemma for odd2np1 16244. (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 16244*	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 16245*	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 16246	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 16247	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 16248	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 16249	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 16250	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 16251*	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 16252*	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 16253*	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 16254*	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 16255	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 16256	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 16257	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 16258	A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = (2 · 𝐴)) → 2 ∥ 𝐵)
Theorem	zeo5 16259	An integer is either even or odd, version of zeo3 16240 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 16260	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 12551 and zeo2 12552. (Contributed by AV, 22-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℤ))
Theorem	oddp1d2 16261	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 12551 and zeo2 12552. (Contributed by AV, 22-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ))
Theorem	zob 16262	Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ))
Theorem	oddm1d2 16263	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 16264	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 16265	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 16266	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 16267	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 16268	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 16269	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 16270	2 divides 0. That means 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
⊢ 2 ∥ 0
Theorem	n2dvds1 16271	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 16272	2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
⊢ ¬ 2 ∥ -1
Theorem	z2even 16273	2 divides 2. That means 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
⊢ 2 ∥ 2
Theorem	n2dvds3 16274	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 16275	2 divides 4. That means 4 is even. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
⊢ 2 ∥ 4
Theorem	4dvdseven 16276	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 16277	Exponentiation of -1 by an even power. Variant of m1expeven 14008. (Contributed by AV, 25-Jun-2021.)
⊢ (2 ∥ 𝑁 → (-1↑𝑁) = 1)
Theorem	m1expo 16278	Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1)
Theorem	m1exp1 16279	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 16280	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 16281	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 16282	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 16283	An odd nonnegative integer is positive. (Contributed by Steven Nguyen, 25-Mar-2023.)
⊢ ((𝑁 ∈ ℕ0 ∧ ¬ 2 ∥ 𝑁) → 𝑁 ∈ ℕ)
Theorem	nn0o1gt2 16284	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 16285	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 16286	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 16287	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 16288	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 16289	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 16290*	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 16291*	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 16292*	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 16293*	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 16294*	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 16295*	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))))
6.1.5  The division algorithm
Theorem	divalglem0 16296	Lemma for divalg 16306. (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 16297	Lemma for divalg 16306. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ 𝑁 ∈ ℤ    &   ⊢ 𝐷 ∈ ℤ    &   ⊢ 𝐷 ≠ 0    ⇒   ⊢ 0 ≤ (𝑁 + (abs‘(𝑁 · 𝐷)))
Theorem	divalglem2 16298*	Lemma for divalg 16306. (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 16299*	Lemma for divalg 16306. (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 16300*	Lemma for divalg 16306. (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))