P. 93 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2rene0 9201	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 9202	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 9203	-1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 9204	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 9205	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 9206	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halfge0 9207	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
⊢ 0 ≤ (1 / 2)
Theorem	halflt1 9208	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 9209	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 9210	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 9211	One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
⊢ (((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))
Theorem	it0e0 9212	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0
Theorem	2mulicn 9213	(2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ∈ ℂ
Theorem	iap0 9214	The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ i # 0
Theorem	2muliap0 9215	2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ (2 · i) # 0
Theorem	2muline0 9216	(2 · i) ≠ 0. See also 2muliap0 9215. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ≠ 0
4.4.5  Simple number properties
Theorem	halfcl 9217	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)
Theorem	rehalfcl 9218	Real closure of half. (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
Theorem	half0 9219	Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) = 0 ↔ 𝐴 = 0))
Theorem	2halves 9220	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	halfpos2 9221	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
Theorem	halfpos 9222	A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (𝐴 / 2) < 𝐴))
Theorem	halfnneg2 9223	A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))
Theorem	halfaddsubcl 9224	Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) / 2) ∈ ℂ ∧ ((𝐴 − 𝐵) / 2) ∈ ℂ))
Theorem	halfaddsub 9225	Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐵) / 2) + ((𝐴 − 𝐵) / 2)) = 𝐴 ∧ (((𝐴 + 𝐵) / 2) − ((𝐴 − 𝐵) / 2)) = 𝐵))
Theorem	subhalfhalf 9226	Subtracting the half of a number from the number yields the half of the number. (Contributed by AV, 28-Jun-2021.)
⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2))
Theorem	lt2halves 9227	A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶))
Theorem	addltmul 9228	Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))
Theorem	nominpos 9229*	There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
⊢ ¬ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ¬ ∃𝑦 ∈ ℝ (0 < 𝑦 ∧ 𝑦 < 𝑥))
Theorem	avglt1 9230	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
Theorem	avglt2 9231	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
Theorem	avgle1 9232	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ((𝐴 + 𝐵) / 2)))
Theorem	avgle2 9233	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	2timesd 9234	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))
Theorem	times2d 9235	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	halfcld 9236	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 9237	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	rehalfcld 9238	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ)
Theorem	lt2halvesd 9239	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 9240	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 / 2) ∈ ℝ
Theorem	add1p1 9241	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
Theorem	sub1m1 9242	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
Theorem	cnm2m1cnm3 9243	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 9244	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 9245	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))
4.4.6  The Archimedean property
Theorem	arch 9246*	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 9247*	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 9248*	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)
4.4.7  Nonnegative integers (as a subset of complex numbers)
Syntax	cn0 9249	Extend class notation to include the class of nonnegative integers.
class ℕ0
Definition	df-n0 9250	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 = (ℕ ∪ {0})
Theorem	elnn0 9251	Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
Theorem	nnssnn0 9252	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ ⊆ ℕ0
Theorem	nn0ssre 9253	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 ⊆ ℝ
Theorem	nn0sscn 9254	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℂ
Theorem	nn0ex 9255	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
⊢ ℕ0 ∈ V
Theorem	nnnn0 9256	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Theorem	nnnn0i 9257	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 9258	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
Theorem	nn0cn 9259	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
Theorem	nn0rei 9260	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nn0cni 9261	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	dfn2 9262	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 9263	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 9264	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 9265	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 9266	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 9267	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 9268	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 9269	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 9270	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0
Theorem	7nn0 9271	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 7 ∈ ℕ0
Theorem	8nn0 9272	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 8 ∈ ℕ0
Theorem	9nn0 9273	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 9 ∈ ℕ0
Theorem	nn0ge0 9274	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 9275	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 9276	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 0 ≤ 𝑁
Theorem	nn0le0eq0 9277	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 9278	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 9279	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 9280	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 9281	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 9282	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 9283	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 9284	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 9285	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 9286	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 + 𝑁) ∈ ℕ0
Theorem	nn0mulcli 9287	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 · 𝑁) ∈ ℕ0
Theorem	nn0p1nn 9288	A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
Theorem	peano2nn0 9289	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
Theorem	nnm1nn0 9290	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 9291	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 9292	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 9293	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
Theorem	elnnnn0c 9294	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
Theorem	nn0addge1 9295	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 9296	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 9297	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 9298	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	nn0le2xi 9299	A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 𝑁 ≤ (2 · 𝑁)
Theorem	nn0lele2xi 9300	'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))