P. 92 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	5lt6 9101	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 9102	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 9103	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 9104	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 9105	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 9106	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 9107	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 9108	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 9109	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 9110	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 9111	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 9112	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 9113	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 9114	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 9115	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 9116	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 9117	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 9118	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 9119	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 9120	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 9121	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 9122	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 9123	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 9124	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 9125	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 9126	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9
Theorem	0ne2 9127	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2
Theorem	1ne2 9128	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
⊢ 1 ≠ 2
Theorem	1ap2 9129	1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
⊢ 1 # 2
Theorem	1le2 9130	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 2
Theorem	2cnne0 9131	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
Theorem	2rene0 9132	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 9133	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 9134	-1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 9135	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 9136	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 9137	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halfge0 9138	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
⊢ 0 ≤ (1 / 2)
Theorem	halflt1 9139	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 9140	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 9141	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 9142	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 9143	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0
Theorem	2mulicn 9144	(2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ∈ ℂ
Theorem	iap0 9145	The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ i # 0
Theorem	2muliap0 9146	2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ (2 · i) # 0
Theorem	2muline0 9147	(2 · i) ≠ 0. See also 2muliap0 9146. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ≠ 0
4.4.5  Simple number properties
Theorem	halfcl 9148	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)
Theorem	rehalfcl 9149	Real closure of half. (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
Theorem	half0 9150	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 9151	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	halfpos2 9152	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
Theorem	halfpos 9153	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 9154	A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))
Theorem	halfaddsubcl 9155	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 9156	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 9157	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 9158	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 9159*	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 9160	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
Theorem	avglt2 9161	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
Theorem	avgle1 9162	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ((𝐴 + 𝐵) / 2)))
Theorem	avgle2 9163	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 9164	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))
Theorem	times2d 9165	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	halfcld 9166	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 9167	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	rehalfcld 9168	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ)
Theorem	lt2halvesd 9169	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 9170	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 / 2) ∈ ℝ
Theorem	add1p1 9171	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
Theorem	sub1m1 9172	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
Theorem	cnm2m1cnm3 9173	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 9174	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 9175	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 9176*	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 9177*	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 9178*	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 9179	Extend class notation to include the class of nonnegative integers.
class ℕ0
Definition	df-n0 9180	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 = (ℕ ∪ {0})
Theorem	elnn0 9181	Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
Theorem	nnssnn0 9182	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ ⊆ ℕ0
Theorem	nn0ssre 9183	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 ⊆ ℝ
Theorem	nn0sscn 9184	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℂ
Theorem	nn0ex 9185	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
⊢ ℕ0 ∈ V
Theorem	nnnn0 9186	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Theorem	nnnn0i 9187	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 9188	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
Theorem	nn0cn 9189	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
Theorem	nn0rei 9190	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nn0cni 9191	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	dfn2 9192	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 9193	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 9194	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 9195	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 9196	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 9197	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 9198	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 9199	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 9200	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0