P. 459 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45801-45900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	41prothprm 45801	41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
⊢ 𝑃 = ;41    ⇒   ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ)
21.43.12.4  Solutions of quadratic equations
Theorem	quad1 45802*	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 45803*	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 45804*	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 45805*	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))
21.43.13  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 45808 and df-odd 45809. Alternate definitions resp. characterizations are provided in dfeven2 45831, dfeven3 45840, dfeven4 45820 and in dfodd2 45818, dfodd3 45832, dfodd4 45841, dfodd5 45842, dfodd6 45819. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 45819 in opoeALTV 45865 and dfodd3 45832 in oddprmALTV 45869. 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 45864 and divgcdodd 16586).
21.43.13.1  Definitions and basic properties
Syntax	ceven 45806	Extend the definition of a class to include the set of even numbers.
class Even
Syntax	codd 45807	Extend the definition of a class to include the set of odd numbers.
class Odd
Definition	df-even 45808	Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
Definition	df-odd 45809	Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
Theorem	iseven 45810	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 45811	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 45812	An even number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)
Theorem	oddz 45813	An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)
Theorem	evendiv2z 45814	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 45815	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 45816	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 45817	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 45818	Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ}
Theorem	dfodd6 45819*	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)}
Theorem	dfeven4 45820*	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)}
Theorem	evenm1odd 45821	The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 − 1) ∈ Odd )
Theorem	evenp1odd 45822	The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd )
Theorem	oddp1eveni 45823	The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 + 1) ∈ Even )
Theorem	oddm1eveni 45824	The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 − 1) ∈ Even )
Theorem	evennodd 45825	An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → ¬ 𝑍 ∈ Odd )
Theorem	oddneven 45826	An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )
Theorem	enege 45827	The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Even → -𝐴 ∈ Even )
Theorem	onego 45828	The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd )
Theorem	m1expevenALTV 45829	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 45830	Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
⊢ (𝑁 ∈ Odd → (-1↑𝑁) = -1)
21.43.13.2  Alternate definitions using the "divides" relation
Theorem	dfeven2 45831	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧}
Theorem	dfodd3 45832	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
Theorem	iseven2 45833	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 45834	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 45835	2 divides an even number. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Even → 2 ∥ 𝑍)
Theorem	m2even 45836	A multiple of 2 is an even number. (Contributed by AV, 5-Jun-2023.)
⊢ (𝑍 ∈ ℤ → (2 · 𝑍) ∈ Even )
Theorem	2ndvdsodd 45837	2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → ¬ 2 ∥ 𝑍)
Theorem	2dvdsoddp1 45838	2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 + 1))
Theorem	2dvdsoddm1 45839	2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 − 1))
21.43.13.3  Alternate definitions using the "modulo" operation
Theorem	dfeven3 45840	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0}
Theorem	dfodd4 45841	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1}
Theorem	dfodd5 45842	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0}
Theorem	zefldiv2ALTV 45843	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 45844	The floor of an odd numer 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 45845	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))
21.43.13.4  Alternate definitions using the "gcd" operation
Theorem	iseven5 45846	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 45847	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 45848	Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 2}
Theorem	dfodd7 45849	Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 1}
Theorem	gcd2odd1 45850	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 45849 is longer (see proof in comment)! (Contributed by AV, 5-Jun-2023.)
⊢ (𝑍 ∈ Odd → (𝑍 gcd 2) = 1)
21.43.13.5  Theorems of part 5 revised
Theorem	zneoALTV 45851	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 45852	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 45853	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 45854	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 45855	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 )
21.43.13.6  Theorems of part 6 revised
Theorem	odd2np1ALTV 45856*	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 45857	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 45858	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 45859	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 45860	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 45861	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 45862	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 45863	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 45864	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 45865	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 45866	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 45867	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 45868	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 45869	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 )
21.43.13.7  Theorems of AV's mathbox revised
Theorem	0evenALTV 45870	0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
⊢ 0 ∈ Even
Theorem	0noddALTV 45871	0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
⊢ 0 ∉ Odd
Theorem	1oddALTV 45872	1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 1 ∈ Odd
Theorem	1nevenALTV 45873	1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 1 ∉ Even
Theorem	2evenALTV 45874	2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 2 ∈ Even
Theorem	2noddALTV 45875	2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
⊢ 2 ∉ Odd
Theorem	nn0o1gt2ALTV 45876	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 45877	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 45878	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 45879	An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ0)
Theorem	nneven 45880	An alternate characterization of an even positive integer. (Contributed by AV, 5-Jun-2023.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ)
Theorem	nn0onn0exALTV 45881*	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 45882*	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 45883*	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 45884	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 )
21.43.13.8  Additional theorems
Theorem	epoo 45885	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 45886	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 45887	The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even )
Theorem	emee 45888	The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 − 𝐵) ∈ Even )
Theorem	evensumeven 45889	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 45890	3 is an odd number. (Contributed by AV, 20-Jul-2020.)
⊢ 3 ∈ Odd
Theorem	4even 45891	4 is an even number. (Contributed by AV, 23-Jul-2020.)
⊢ 4 ∈ Even
Theorem	5odd 45892	5 is an odd number. (Contributed by AV, 23-Jul-2020.)
⊢ 5 ∈ Odd
Theorem	6even 45893	6 is an even number. (Contributed by AV, 20-Jul-2020.)
⊢ 6 ∈ Even
Theorem	7odd 45894	7 is an odd number. (Contributed by AV, 20-Jul-2020.)
⊢ 7 ∈ Odd
Theorem	8even 45895	8 is an even number. (Contributed by AV, 23-Jul-2020.)
⊢ 8 ∈ Even
Theorem	evenprm2 45896	A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)
⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
Theorem	oddprmne2 45897	Every prime number not being 2 is an odd prime number. (Contributed by AV, 21-Aug-2021.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) ↔ 𝑃 ∈ (ℙ ∖ {2}))
Theorem	oddprmuzge3 45898	A prime number which is odd is an integer greater than or equal to 3. (Contributed by AV, 20-Jul-2020.) (Proof shortened by AV, 21-Aug-2021.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) → 𝑃 ∈ (ℤ≥‘3))
Theorem	evenltle 45899	If an even number is greater than another even number, then it is greater than or equal to the other even number plus 2. (Contributed by AV, 25-Dec-2021.)
⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ∧ 𝑀 < 𝑁) → (𝑀 + 2) ≤ 𝑁)
Theorem	odd2prm2 45900	If an odd number is the sum of two prime numbers, one of the prime numbers must be 2. (Contributed by AV, 26-Dec-2021.)
⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))