P. 485 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48401-48500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nprmdvdsfacm1lem4 48401	Lemma 4 for nprmdvdsfacm1 48402. (Contributed by AV, 7-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))
Theorem	nprmdvdsfacm1 48402	A non-prime integer greater than 5 divides the factorial of the integer decreased by 1 (see remark in [Ribenboim] p. 181). Note: not valid for 𝑁 = 4, but for 𝑁 = 1! (Contributed by AV, 7-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ≥‘6) ∧ 𝑁 ∉ ℙ) → 𝑁 ∥ (!‘(𝑁 − 1)))
Theorem	ppivalnnprm 48403	Value of a term of the prime-counting function pi for positive integers, according to Ján Mináč, for a prime number. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑃 ∈ ℙ → (⌊‘((((!‘(𝑃 − 1)) + 1) / 𝑃) − (⌊‘((!‘(𝑃 − 1)) / 𝑃)))) = 1)
Theorem	ppivalnnnprmge6 48404	Value of a term of the prime-counting function pi for positive integers, according to Ján Mináč, for a non-prime number greater than 4. (Contributed by AV, 4-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ≥‘6) ∧ 𝑁 ∉ ℙ) → (⌊‘((((!‘(𝑁 − 1)) + 1) / 𝑁) − (⌊‘((!‘(𝑁 − 1)) / 𝑁)))) = 0)
Theorem	ppivalnn4 48405	Value of the term of the prime-counting function pi for positive integers, according to Ján Mináč, for 4. (Contributed by AV, 8-Apr-2026.)
⊢ (⌊‘((((!‘(4 − 1)) + 1) / 4) − (⌊‘((!‘(4 − 1)) / 4)))) = 0
Theorem	ppivalnnnprm 48406	Value of a term of the prime-counting function pi for positive integers, according to Ján Miná&ccaron, for a non-prime number greater than 1. (Contributed by AV, 8-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ∉ ℙ) → (⌊‘((((!‘(𝑁 − 1)) + 1) / 𝑁) − (⌊‘((!‘(𝑁 − 1)) / 𝑁)))) = 0)
Theorem	indprm 48407	An indicator function for prime numbers, according to Ján Mináč. (Contributed by AV, 4-Apr-2026.)
⊢ ((𝟭‘(ℤ≥‘2))‘ℙ) = (𝑘 ∈ (ℤ≥‘2) ↦ (⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘)))))
Theorem	indprmfz 48408*	An indicator function for prime numbers in a finite interval of integers, according to Ján Mináč. (Contributed by AV, 4-Apr-2026.)
⊢ 𝐼 = (2...𝐴)    ⇒   ⊢ ((𝟭‘𝐼)‘(𝐼 ∩ ℙ)) = (𝑘 ∈ 𝐼 ↦ (⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘)))))
Theorem	ppi1sum 48409	Value of the prime-counting function pi for 1, according to Ján Mináč. (Contributed by AV, 4-Apr-2026.)
⊢ (π‘1) = Σ𝑘 ∈ ∅ (⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘))))
Theorem	ppivalnn 48410*	Value of the prime-counting function pi for positive integers, according to Ján Mináč, see statement in [Ribenboim], p. 181. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑁 ∈ ℕ → (π‘𝑁) = Σ𝑘 ∈ (2...𝑁)(⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘)))))
21.49.12.6  Solutions of quadratic equations
Theorem	quad1 48411*	A condition for a quadratic equation with complex coefficients to have (exactly) one complex solution. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ ℂ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
Theorem	requad01 48412*	A condition for a quadratic equation with real coefficients to have (at least) one real solution. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 0 ≤ 𝐷))
Theorem	requad1 48413*	A condition for a quadratic equation with real coefficients to have (exactly) one real solution. (Contributed by AV, 26-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
Theorem	requad2 48414*	A condition for a quadratic equation with real coefficients to have (exactly) two different real solutions. (Contributed by AV, 28-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃!𝑝 ∈ 𝒫 ℝ((♯‘𝑝) = 2 ∧ ∀𝑥 ∈ 𝑝 ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0) ↔ 0 < 𝐷))
21.49.13  Even and odd numbers Even and odd numbers can be characterized in many different ways. In the following, the definition of even and odd numbers is based on the fact that dividing an even number (resp. an odd number increased by 1) by 2 is an integer, see df-even 48417 and df-odd 48418. Alternate definitions resp. characterizations are provided in dfeven2 48440, dfeven3 48449, dfeven4 48429 and in dfodd2 48427, dfodd3 48441, dfodd4 48450, dfodd5 48451, dfodd6 48428. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 48428 in opoeALTV 48474 and dfodd3 48441 in oddprmALTV 48478. Having a fixed definition for even and odd numbers, and alternate characterizations as theorems, advanced theorems about even and/or odd numbers can be expressed more explicitly, and the appropriate characterization can be chosen for their proof, which may become clearer and sometimes also shorter (see, for example, divgcdoddALTV 48473 and divgcdodd 16773).
21.49.13.1  Definitions and basic properties
Syntax	ceven 48415	Extend the definition of a class to include the set of even numbers.
class Even
Syntax	codd 48416	Extend the definition of a class to include the set of odd numbers.
class Odd
Definition	df-even 48417	Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
Definition	df-odd 48418	Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
Theorem	iseven 48419	The predicate "is an even number". An even number is an integer which is divisible by 2, i.e. the result of dividing the even integer by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
Theorem	isodd 48420	The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
Theorem	evenz 48421	An even number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)
Theorem	oddz 48422	An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)
Theorem	evendiv2z 48423	The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)
Theorem	oddp1div2z 48424	The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)
Theorem	oddm1div2z 48425	The result of dividing an odd number decreased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Odd → ((𝑍 − 1) / 2) ∈ ℤ)
Theorem	isodd2 48426	The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd number decreased by 1 and then divided by 2 is still an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 − 1) / 2) ∈ ℤ))
Theorem	dfodd2 48427	Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ}
Theorem	dfodd6 48428*	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)}
Theorem	dfeven4 48429*	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)}
Theorem	evenm1odd 48430	The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 − 1) ∈ Odd )
Theorem	evenp1odd 48431	The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd )
Theorem	oddp1eveni 48432	The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 + 1) ∈ Even )
Theorem	oddm1eveni 48433	The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 − 1) ∈ Even )
Theorem	evennodd 48434	An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → ¬ 𝑍 ∈ Odd )
Theorem	oddneven 48435	An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )
Theorem	enege 48436	The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Even → -𝐴 ∈ Even )
Theorem	onego 48437	The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd )
Theorem	m1expevenALTV 48438	Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.)
⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1)
Theorem	m1expoddALTV 48439	Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
⊢ (𝑁 ∈ Odd → (-1↑𝑁) = -1)
21.49.13.2  Alternate definitions using the "divides" relation
Theorem	dfeven2 48440	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧}
Theorem	dfodd3 48441	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
Theorem	iseven2 48442	The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ 2 ∥ 𝑍))
Theorem	isodd3 48443	The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍))
Theorem	2dvdseven 48444	2 divides an even number. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Even → 2 ∥ 𝑍)
Theorem	m2even 48445	A multiple of 2 is an even number. (Contributed by AV, 5-Jun-2023.)
⊢ (𝑍 ∈ ℤ → (2 · 𝑍) ∈ Even )
Theorem	2ndvdsodd 48446	2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → ¬ 2 ∥ 𝑍)
Theorem	2dvdsoddp1 48447	2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 + 1))
Theorem	2dvdsoddm1 48448	2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 − 1))
21.49.13.3  Alternate definitions using the "modulo" operation
Theorem	dfeven3 48449	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0}
Theorem	dfodd4 48450	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1}
Theorem	dfodd5 48451	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0}
Theorem	zefldiv2ALTV 48452	The floor of an even number divided by 2 is equal to the even number divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
⊢ (𝑁 ∈ Even → (⌊‘(𝑁 / 2)) = (𝑁 / 2))
Theorem	zofldiv2ALTV 48453	The floor of an odd number divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))
Theorem	oddflALTV 48454	Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 18-Jun-2020.)
⊢ (𝐾 ∈ Odd → 𝐾 = ((2 · (⌊‘(𝐾 / 2))) + 1))
21.49.13.4  Alternate definitions using the "gcd" operation
Theorem	iseven5 48455	The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 2))
Theorem	isodd7 48456	The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 1))
Theorem	dfeven5 48457	Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 2}
Theorem	dfodd7 48458	Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 1}
Theorem	gcd2odd1 48459	The greatest common divisor of an odd number and 2 is 1, i.e., 2 and any odd number are coprime. Remark: The proof using dfodd7 48458 is longer (see proof in comment)! (Contributed by AV, 5-Jun-2023.)
⊢ (𝑍 ∈ Odd → (𝑍 gcd 2) = 1)
21.49.13.5  Theorems of part 5 revised
Theorem	zneoALTV 48460	No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Revised by AV, 16-Jun-2020.)
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → 𝐴 ≠ 𝐵)
Theorem	zeoALTV 48461	An integer is even or odd. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ ℤ → (𝑍 ∈ Even ∨ 𝑍 ∈ Odd ))
Theorem	zeo2ALTV 48462	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) (Revised by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ ℤ → (𝑍 ∈ Even ↔ ¬ 𝑍 ∈ Odd ))
Theorem	nneoALTV 48463	A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 19-Jun-2020.)
⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd ))
Theorem	nneoiALTV 48464	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) (Revised by AV, 19-Jun-2020.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd )
21.49.13.6  Theorems of part 6 revised
Theorem	odd2np1ALTV 48465*	An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by AV, 19-Jun-2020.)
⊢ (𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁))
Theorem	oddm1evenALTV 48466	An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
⊢ (𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 − 1) ∈ Even ))
Theorem	oddp1evenALTV 48467	An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
⊢ (𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 + 1) ∈ Even ))
Theorem	oexpnegALTV 48468	The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Revised by AV, 19-Jun-2020.) (Proof shortened by AV, 10-Jul-2022.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∈ Odd ) → (-𝐴↑𝑁) = -(𝐴↑𝑁))
Theorem	oexpnegnz 48469	The exponential of the negative of a number not being 0, when the exponent is odd. (Contributed by AV, 19-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ Odd ) → (-𝐴↑𝑁) = -(𝐴↑𝑁))
Theorem	bits0ALTV 48470	Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
⊢ (𝑁 ∈ ℤ → (0 ∈ (bits‘𝑁) ↔ 𝑁 ∈ Odd ))
Theorem	bits0eALTV 48471	The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
⊢ (𝑁 ∈ Even → ¬ 0 ∈ (bits‘𝑁))
Theorem	bits0oALTV 48472	The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
⊢ (𝑁 ∈ Odd → 0 ∈ (bits‘𝑁))
Theorem	divgcdoddALTV 48473	Either 𝐴 / (𝐴 gcd 𝐵) is odd or 𝐵 / (𝐴 gcd 𝐵) is odd. (Contributed by Scott Fenton, 19-Apr-2014.) (Revised by AV, 21-Jun-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵)) ∈ Odd ∨ (𝐵 / (𝐴 gcd 𝐵)) ∈ Odd ))
Theorem	opoeALTV 48474	The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Even )
Theorem	opeoALTV 48475	The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Odd )
Theorem	omoeALTV 48476	The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴 − 𝐵) ∈ Even )
Theorem	omeoALTV 48477	The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴 − 𝐵) ∈ Odd )
Theorem	oddprmALTV 48478	A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.)
⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ Odd )
21.49.13.7  Theorems of AV's mathbox revised
Theorem	0evenALTV 48479	0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
⊢ 0 ∈ Even
Theorem	0noddALTV 48480	0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
⊢ 0 ∉ Odd
Theorem	1oddALTV 48481	1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 1 ∈ Odd
Theorem	1nevenALTV 48482	1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 1 ∉ Even
Theorem	2evenALTV 48483	2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 2 ∈ Even
Theorem	2noddALTV 48484	2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 2 ∉ Odd
Theorem	nn0o1gt2ALTV 48485	An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁))
Theorem	nnoALTV 48486	An alternate characterization of an odd number greater than 1. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ)
Theorem	nn0oALTV 48487	An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0)
Theorem	nn0e 48488	An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ0)
Theorem	nneven 48489	An alternate characterization of an even positive integer. (Contributed by AV, 5-Jun-2023.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ)
Theorem	nn0onn0exALTV 48490*	For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1))
Theorem	nn0enn0exALTV 48491*	For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Even ) → ∃𝑚 ∈ ℕ0 𝑁 = (2 · 𝑚))
Theorem	nnennexALTV 48492*	For each even positive integer there is a positive integer which, multiplied by 2, results in the even positive integer. (Contributed by AV, 5-Jun-2023.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → ∃𝑚 ∈ ℕ 𝑁 = (2 · 𝑚))
Theorem	nnpw2evenALTV 48493	2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 20-Jun-2020.)
⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ Even )
21.49.13.8  Additional theorems
Theorem	epoo 48494	The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Odd )
Theorem	emoo 48495	The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 − 𝐵) ∈ Odd )
Theorem	epee 48496	The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even )
Theorem	emee 48497	The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 − 𝐵) ∈ Even )
Theorem	evensumeven 48498	If a summand is even, the other summand is even iff the sum is even. (Contributed by AV, 21-Jul-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ Even ) → (𝐴 ∈ Even ↔ (𝐴 + 𝐵) ∈ Even ))
Theorem	3odd 48499	3 is an odd number. (Contributed by AV, 20-Jul-2020.)
⊢ 3 ∈ Odd
Theorem	4even 48500	4 is an even number. (Contributed by AV, 23-Jul-2020.)
⊢ 4 ∈ Even