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Theorem List for Metamath Proof Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhalfre 11001 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℝ
 
Theoremhalfcn 11002 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℂ
 
Theoremhalfgt0 11003 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)
 
Theoremhalfge0 11004 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)
 
Theoremhalflt1 11005 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1
 
Theorem1mhlfehlf 11006 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)
 
Theorem8th4div3 11007 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)
 
Theoremhalfpm6th 11008 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))
 
Theoremit0e0 11009 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0
 
Theorem2mulicn 11010 (2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ∈ ℂ
 
Theorem2muline0 11011 (2 · i) ≠ 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0
 
5.4.5  Simple number properties
 
Theoremhalfcl 11012 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)
 
Theoremrehalfcl 11013 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
 
Theoremhalf0 11014 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
(𝐴 ∈ ℂ → ((𝐴 / 2) = 0 ↔ 𝐴 = 0))
 
Theorem2halves 11015 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
 
Theoremhalfpos2 11016 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
 
Theoremhalfpos 11017 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))
 
Theoremhalfnneg2 11018 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))
 
Theoremhalfaddsubcl 11019 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) / 2) ∈ ℂ ∧ ((𝐴𝐵) / 2) ∈ ℂ))
 
Theoremhalfaddsub 11020 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐵) / 2) + ((𝐴𝐵) / 2)) = 𝐴 ∧ (((𝐴 + 𝐵) / 2) − ((𝐴𝐵) / 2)) = 𝐵))
 
Theoremsubhalfhalf 11021 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))
 
Theoremlt2halves 11022 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))
 
Theoremaddltmul 11023 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))
 
Theoremnominpos 11024* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
¬ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ¬ ∃𝑦 ∈ ℝ (0 < 𝑦𝑦 < 𝑥))
 
Theoremavglt1 11025 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < ((𝐴 + 𝐵) / 2)))
 
Theoremavglt2 11026 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
 
Theoremavgle1 11027 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵𝐴 ≤ ((𝐴 + 𝐵) / 2)))
 
Theoremavgle2 11028 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵))
 
Theoremavgle 11029 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))
 
Theorem2timesd 11030 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))
 
Theoremtimes2d 11031 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))
 
Theoremhalfcld 11032 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 2) ∈ ℂ)
 
Theorem2halvesd 11033 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
 
Theoremrehalfcld 11034 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 / 2) ∈ ℝ)
 
Theoremlt2halvesd 11035 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐶 / 2))    &   (𝜑𝐵 < (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) < 𝐶)
 
Theoremrehalfcli 11036 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       (𝐴 / 2) ∈ ℝ
 
Theoremlt2addmuld 11037 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵 < 𝐶)       (𝜑 → (𝐴 + 𝐵) < (2 · 𝐶))
 
Theoremadd1p1 11038 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
(𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
 
Theoremsub1m1 11039 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
(𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
 
Theoremcnm2m1cnm3 11040 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))
 
Theoremxp1d2m1eqxm1d2 11041 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))
 
Theoremdiv4p1lem1div2 11042 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
 
Theoremnnunb 11043* 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 < 𝑥𝑦 = 𝑥)
 
Theoremarch 11044* 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)
 
Theoremnnrecl 11045* 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)
 
Theorembndndx 11046* 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 ∧ 𝐴𝑥) → ∃𝑘 ∈ ℕ 𝐴𝑘)
 
5.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 11047 Extend class notation to include the class of nonnegative integers.
class 0
 
Definitiondf-n0 11048 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})
 
Theoremelnn0 11049 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
 
Theoremnnssnn0 11050 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0
 
Theoremnn0ssre 11051 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ
 
Theoremnn0sscn 11052 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
0 ⊆ ℂ
 
Theoremnn0ex 11053 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 ∈ V
 
Theoremnnnn0 11054 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
 
Theoremnnnn0i 11055 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁 ∈ ℕ       𝑁 ∈ ℕ0
 
Theoremnn0re 11056 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℝ)
 
Theoremnn0cn 11057 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℂ)
 
Theoremnn0rei 11058 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℝ
 
Theoremnn0cni 11059 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℂ
 
Theoremdfn2 11060 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})
 
Theoremelnnne0 11061 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
 
Theorem0nn0 11062 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
0 ∈ ℕ0
 
Theorem1nn0 11063 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
1 ∈ ℕ0
 
Theorem2nn0 11064 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
2 ∈ ℕ0
 
Theorem3nn0 11065 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
3 ∈ ℕ0
 
Theorem4nn0 11066 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
4 ∈ ℕ0
 
Theorem5nn0 11067 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
5 ∈ ℕ0
 
Theorem6nn0 11068 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
6 ∈ ℕ0
 
Theorem7nn0 11069 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
7 ∈ ℕ0
 
Theorem8nn0 11070 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
8 ∈ ℕ0
 
Theorem9nn0 11071 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
9 ∈ ℕ0
 
Theorem10nn0OLD 11072 Obsolete version of 10nn0 11256 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
10 ∈ ℕ0
 
Theoremnn0ge0 11073 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)
 
Theoremnn0nlt0 11074 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)
 
Theoremnn0ge0i 11075 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 ∈ ℕ0       0 ≤ 𝑁
 
Theoremnn0le0eq0 11076 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))
 
Theoremnn0p1gt0 11077 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
 
Theoremnnnn0addcl 11078 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)
 
Theoremnn0nnaddcl 11079 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
 
Theorem0mnnnnn0 11080 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)
 
Theoremun0addcl 11081 If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((𝜑 ∧ (𝑀𝑆𝑁𝑆)) → (𝑀 + 𝑁) ∈ 𝑆)       ((𝜑 ∧ (𝑀𝑇𝑁𝑇)) → (𝑀 + 𝑁) ∈ 𝑇)
 
Theoremun0mulcl 11082 If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((𝜑 ∧ (𝑀𝑆𝑁𝑆)) → (𝑀 · 𝑁) ∈ 𝑆)       ((𝜑 ∧ (𝑀𝑇𝑁𝑇)) → (𝑀 · 𝑁) ∈ 𝑇)
 
Theoremnn0addcl 11083 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
 
Theoremnn0mulcl 11084 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0)
 
Theoremnn0addcli 11085 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 + 𝑁) ∈ ℕ0
 
Theoremnn0mulcli 11086 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 · 𝑁) ∈ ℕ0
 
Theoremnn0p1nn 11087 A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 10787. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
 
Theorempeano2nn0 11088 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
 
Theoremnnm1nn0 11089 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)
 
Theoremelnn0nn 11090 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))
 
Theoremelnnnn0 11091 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0))
 
Theoremelnnnn0b 11092 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
 
Theoremelnnnn0c 11093 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
 
Theoremnn0addge1 11094 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))
 
Theoremnn0addge2 11095 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))
 
Theoremnn0addge1i 11096 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)
 
Theoremnn0addge2i 11097 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)
 
Theoremnn0sub 11098 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))
 
Theoremltsubnn0 11099 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 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐵 < 𝐴 → (𝐴𝐵) ∈ ℕ0))
 
Theoremnn0negleid 11100 A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
(𝐴 ∈ ℕ0 → -𝐴𝐴)
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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