P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	8lt10 7901	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 8 < 10
Theorem	7lt10 7902	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 7 < 10
Theorem	6lt10 7903	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 6 < 10
Theorem	5lt10 7904	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 5 < 10
Theorem	4lt10 7905	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 4 < 10
Theorem	3lt10 7906	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 3 < 10
Theorem	2lt10 7907	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 2 < 10
Theorem	1lt10 7908	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 10
Theorem	0ne2 7909	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2
Theorem	1ne2 7910	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
⊢ 1 ≠ 2
Theorem	1le2 7911	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 2
Theorem	2cnne0 7912	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
Theorem	2rene0 7913	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 7914	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 7915	-1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 7916	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 7917	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 7918	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halflt1 7919	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 7920	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 7921	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 7922	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 7923	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0
Theorem	2mulicn 7924	(2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ∈ ℂ
Theorem	iap0 7925	The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ i # 0
Theorem	2muliap0 7926	2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ (2 · i) # 0
Theorem	2muline0 7927	(2 · i) ≠ 0. See also 2muliap0 7926. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ≠ 0
3.4.5  Simple number properties
Theorem	halfcl 7928	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
⊢ (A ∈ ℂ → (A / 2) ∈ ℂ)
Theorem	rehalfcl 7929	Real closure of half. (Contributed by NM, 1-Jan-2006.)
⊢ (A ∈ ℝ → (A / 2) ∈ ℝ)
Theorem	half0 7930	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 7931	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
⊢ (A ∈ ℂ → ((A / 2) + (A / 2)) = A)
Theorem	halfpos2 7932	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
⊢ (A ∈ ℝ → (0 < A ↔ 0 < (A / 2)))
Theorem	halfpos 7933	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 7934	A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
⊢ (A ∈ ℝ → (0 ≤ A ↔ 0 ≤ (A / 2)))
Theorem	halfaddsubcl 7935	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 7936	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 7937	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 7938	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 7939*	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 7940	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A < B ↔ A < ((A + B) / 2)))
Theorem	avglt2 7941	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A < B ↔ ((A + B) / 2) < B))
Theorem	avgle1 7942	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A ≤ B ↔ A ≤ ((A + B) / 2)))
Theorem	avgle2 7943	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 7944	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℂ)    ⇒   ⊢ (φ → (2 · A) = (A + A))
Theorem	times2d 7945	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℂ)    ⇒   ⊢ (φ → (A · 2) = (A + A))
Theorem	halfcld 7946	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 7947	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℂ)    ⇒   ⊢ (φ → ((A / 2) + (A / 2)) = A)
Theorem	rehalfcld 7948	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → (A / 2) ∈ ℝ)
Theorem	lt2halvesd 7949	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 7950	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
⊢ A ∈ ℝ    ⇒   ⊢ (A / 2) ∈ ℝ
Theorem	add1p1 7951	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
Theorem	sub1m1 7952	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
Theorem	cnm2m1cnm3 7953	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))
3.4.6  The Archimedean property
Theorem	arch 7954*	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 7955*	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 7956*	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 7957	Extend class notation to include the class of nonnegative integers.
class ℕ0
Definition	df-n0 7958	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 = (ℕ ∪ {0})
Theorem	elnn0 7959	Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0))
Theorem	nnssnn0 7960	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ ⊆ ℕ0
Theorem	nn0ssre 7961	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 ⊆ ℝ
Theorem	nn0sscn 7962	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℂ
Theorem	nn0ex 7963	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
⊢ ℕ0 ∈ V
Theorem	nnnn0 7964	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (A ∈ ℕ → A ∈ ℕ0)
Theorem	nnnn0i 7965	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 7966	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (A ∈ ℕ0 → A ∈ ℝ)
Theorem	nn0cn 7967	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (A ∈ ℕ0 → A ∈ ℂ)
Theorem	nn0rei 7968	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ A ∈ ℕ0    ⇒   ⊢ A ∈ ℝ
Theorem	nn0cni 7969	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
⊢ A ∈ ℕ0    ⇒   ⊢ A ∈ ℂ
Theorem	dfn2 7970	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 7971	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 7972	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 7973	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 7974	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 7975	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 7976	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 7977	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 7978	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0
Theorem	7nn0 7979	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 7 ∈ ℕ0
Theorem	8nn0 7980	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 8 ∈ ℕ0
Theorem	9nn0 7981	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 9 ∈ ℕ0
Theorem	10nn0 7982	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 10 ∈ ℕ0
Theorem	nn0ge0 7983	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 7984	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 7985	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 0 ≤ 𝑁
Theorem	nn0le0eq0 7986	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 7987	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 7988	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 7989	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 7990	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 7991	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 7992	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 7993	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 7994	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 7995	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 + 𝑁) ∈ ℕ0
Theorem	nn0mulcli 7996	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 · 𝑁) ∈ ℕ0
Theorem	nn0p1nn 7997	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 7998	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
Theorem	nnm1nn0 7999	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 8000	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))