P. 83 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1lt2 8201	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
⊢ 1 < 2
Theorem	2lt3 8202	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 2 < 3
Theorem	1lt3 8203	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 1 < 3
Theorem	3lt4 8204	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 4
Theorem	2lt4 8205	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 4
Theorem	1lt4 8206	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 4
Theorem	4lt5 8207	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 5
Theorem	3lt5 8208	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 5
Theorem	2lt5 8209	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 5
Theorem	1lt5 8210	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 5
Theorem	5lt6 8211	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 8212	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 8213	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 8214	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 8215	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 8216	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 8217	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 8218	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 8219	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 8220	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 8221	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 8222	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 8223	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 8224	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 8225	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 8226	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 8227	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 8228	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 8229	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 8230	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 8231	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 8232	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 8233	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 8234	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 8235	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 8236	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9
Theorem	0ne2 8237	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2
Theorem	1ne2 8238	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
⊢ 1 ≠ 2
Theorem	1le2 8239	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 2
Theorem	2cnne0 8240	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
Theorem	2rene0 8241	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 8242	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 8243	-1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 8244	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 8245	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 8246	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halfge0 8247	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
⊢ 0 ≤ (1 / 2)
Theorem	halflt1 8248	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 8249	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 8250	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 8251	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 8252	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0
Theorem	2mulicn 8253	(2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ∈ ℂ
Theorem	iap0 8254	The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ i # 0
Theorem	2muliap0 8255	2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ (2 · i) # 0
Theorem	2muline0 8256	(2 · i) ≠ 0. See also 2muliap0 8255. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ≠ 0
3.4.5  Simple number properties
Theorem	halfcl 8257	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)
Theorem	rehalfcl 8258	Real closure of half. (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
Theorem	half0 8259	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 8260	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	halfpos2 8261	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
Theorem	halfpos 8262	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 8263	A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))
Theorem	halfaddsubcl 8264	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 8265	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	lt2halves 8266	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 8267	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 8268*	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 8269	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
Theorem	avglt2 8270	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
Theorem	avgle1 8271	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ((𝐴 + 𝐵) / 2)))
Theorem	avgle2 8272	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 8273	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))
Theorem	times2d 8274	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	halfcld 8275	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 8276	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	rehalfcld 8277	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ)
Theorem	lt2halvesd 8278	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 8279	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 / 2) ∈ ℝ
Theorem	add1p1 8280	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
Theorem	sub1m1 8281	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
Theorem	cnm2m1cnm3 8282	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 8283	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 8284	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))
3.4.6  The Archimedean property
Theorem	arch 8285*	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 8286*	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 8287*	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)
3.4.7  Nonnegative integers (as a subset of complex numbers)
Syntax	cn0 8288	Extend class notation to include the class of nonnegative integers.
class ℕ0
Definition	df-n0 8289	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 = (ℕ ∪ {0})
Theorem	elnn0 8290	Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
Theorem	nnssnn0 8291	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ ⊆ ℕ0
Theorem	nn0ssre 8292	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 ⊆ ℝ
Theorem	nn0sscn 8293	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℂ
Theorem	nn0ex 8294	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
⊢ ℕ0 ∈ V
Theorem	nnnn0 8295	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Theorem	nnnn0i 8296	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 8297	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
Theorem	nn0cn 8298	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
Theorem	nn0rei 8299	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nn0cni 8300	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℂ