P. 126 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	avglt1 12501	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
Theorem	avglt2 12502	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
Theorem	avgle1 12503	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ((𝐴 + 𝐵) / 2)))
Theorem	avgle2 12504	Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵))
Theorem	avgle 12505	The average of two numbers is less than or equal to at least one of them. (Contributed by NM, 9-Dec-2005.) (Revised by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵))
Theorem	2timesd 12506	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))
Theorem	times2d 12507	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	halfcld 12508	Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℂ)
Theorem	2halvesd 12509	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	rehalfcld 12510	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ)
Theorem	lt2halvesd 12511	A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < (𝐶 / 2))    &   ⊢ (𝜑 → 𝐵 < (𝐶 / 2))    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < 𝐶)
Theorem	rehalfcli 12512	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 / 2) ∈ ℝ
Theorem	lt2addmuld 12513	If two real numbers are less than a third real number, the sum of the two real numbers is less than twice the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < (2 · 𝐶))
Theorem	add1p1 12514	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
Theorem	sub1m1 12515	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
Theorem	cnm2m1cnm3 12516	Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ (𝐴 ∈ ℂ → ((𝐴 − 2) − 1) = (𝐴 − 3))
Theorem	xp1d2m1eqxm1d2 12517	A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2))
Theorem	div4p1lem1div2 12518	An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
5.4.6  The Archimedean property
Theorem	nnunb 12519*	The set of positive integers is unbounded above. Theorem I.28 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
⊢ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)
Theorem	arch 12520*	Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Theorem	nnrecl 12521*	There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)
Theorem	bndndx 12522*	A bounded real sequence 𝐴(𝑘) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘)
5.4.7  Nonnegative integers (as a subset of complex numbers)
Syntax	cn0 12523	Extend class notation to include the class of nonnegative integers.
class ℕ0
Definition	df-n0 12524	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 = (ℕ ∪ {0})
Theorem	elnn0 12525	Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
Theorem	nnssnn0 12526	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ ⊆ ℕ0
Theorem	nn0ssre 12527	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 ⊆ ℝ
Theorem	nn0sscn 12528	Nonnegative integers are a subset of the complex numbers. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
⊢ ℕ0 ⊆ ℂ
Theorem	nn0ex 12529	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
⊢ ℕ0 ∈ V
Theorem	nnnn0 12530	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Theorem	nnnn0i 12531	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 12532	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
Theorem	nn0cn 12533	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
Theorem	nn0rei 12534	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nn0cni 12535	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	dfn2 12536	The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
⊢ ℕ = (ℕ0 ∖ {0})
Theorem	elnnne0 12537	The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
Theorem	0nn0 12538	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 12539	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 12540	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 12541	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 12542	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 12543	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 12544	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0
Theorem	7nn0 12545	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 7 ∈ ℕ0
Theorem	8nn0 12546	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 8 ∈ ℕ0
Theorem	9nn0 12547	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 9 ∈ ℕ0
Theorem	nn0ge0 12548	A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
Theorem	nn0nlt0 12549	A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℕ0 → ¬ 𝐴 < 0)
Theorem	nn0ge0i 12550	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 0 ≤ 𝑁
Theorem	nn0le0eq0 12551	A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 0 ↔ 𝑁 = 0))
Theorem	nn0p1gt0 12552	A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
Theorem	nnnn0addcl 12553	A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ)
Theorem	nn0nnaddcl 12554	A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
Theorem	0mnnnnn0 12555	The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0)
Theorem	un0addcl 12556	If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ 𝑇 = (𝑆 ∪ {0})    &   ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 + 𝑁) ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 + 𝑁) ∈ 𝑇)
Theorem	un0mulcl 12557	If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ 𝑇 = (𝑆 ∪ {0})    &   ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 · 𝑁) ∈ 𝑇)
Theorem	nn0addcl 12558	Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
Theorem	nn0mulcl 12559	Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0)
Theorem	nn0addcli 12560	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 + 𝑁) ∈ ℕ0
Theorem	nn0mulcli 12561	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 · 𝑁) ∈ ℕ0
Theorem	nn0p1nn 12562	A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 12275. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
Theorem	peano2nn0 12563	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
Theorem	nnm1nn0 12564	A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
Theorem	elnn0nn 12565	The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ))
Theorem	elnnnn0 12566	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0))
Theorem	elnnnn0b 12567	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
Theorem	elnnnn0c 12568	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
Theorem	nn0addge1 12569	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝑁))
Theorem	nn0addge2 12570	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝑁 + 𝐴))
Theorem	nn0addge1i 12571	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 𝐴 ≤ (𝐴 + 𝑁)
Theorem	nn0addge2i 12572	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 𝐴 ≤ (𝑁 + 𝐴)
Theorem	nn0sub 12573	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0))
Theorem	ltsubnn0 12574	Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐵 < 𝐴 → (𝐴 − 𝐵) ∈ ℕ0))
Theorem	nn0negleid 12575	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴)
Theorem	difgtsumgt 12576	If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵)))
Theorem	nn0le2x 12577	A nonnegative integer is less than or equal to twice itself. Generalization of nn0le2xi 12578. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by AV, 9-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ (2 · 𝑁))
Theorem	nn0le2xi 12578	A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by AV, 9-Sep-2025.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 𝑁 ≤ (2 · 𝑁)
Theorem	nn0lele2xi 12579	'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ (2 · 𝑀))
Theorem	fcdmnn0supp 12580	Two ways to write the support of a function into ℕ0. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
Theorem	fcdmnn0fsupp 12581	A function into ℕ0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin))
Theorem	fcdmnn0suppg 12582	Version of fcdmnn0supp 12580 avoiding ax-rep 5284 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
Theorem	fcdmnn0fsuppg 12583	Version of fcdmnn0fsupp 12581 avoiding ax-rep 5284 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin))
Theorem	nnnn0d 12584	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ0)
Theorem	nn0red 12585	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nn0cnd 12586	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	nn0ge0d 12587	A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 0 ≤ 𝐴)
Theorem	nn0addcld 12588	Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0)
Theorem	nn0mulcld 12589	Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0)
Theorem	nn0readdcl 12590	Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	nn0n0n1ge2 12591	A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁)
Theorem	nn0n0n1ge2b 12592	A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))
Theorem	nn0ge2m1nn 12593	If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
Theorem	nn0ge2m1nn0 12594	If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ0)
Theorem	nn0nndivcl 12595	Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ)
5.4.8  Extended nonnegative integers The function values of the hash (set size) function are either nonnegative integers or positive infinity, see hashf 14373. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers ℝ*, see df-xr 11296. The definition of extended nonnegative integers can be used in Ramsey theory, because the Ramsey number is either a nonnegative integer or plus infinity, see ramcl2 17049, or for the degree of polynomials, see mdegcl 26122, or for the degree of vertices in graph theory, see vtxdgf 29503.
Syntax	cxnn0 12596	The set of extended nonnegative integers.
class ℕ0*
Definition	df-xnn0 12597	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 11296. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0* = (ℕ0 ∪ {+∞})
Theorem	elxnn0 12598	An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
Theorem	nn0ssxnn0 12599	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0 ⊆ ℕ0*
Theorem	nn0xnn0 12600	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)