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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem8nn 9001 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ
 
Theorem9nn 9002 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ
 
Theorem1lt2 9003 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2
 
Theorem2lt3 9004 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3
 
Theorem1lt3 9005 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3
 
Theorem3lt4 9006 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4
 
Theorem2lt4 9007 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4
 
Theorem1lt4 9008 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4
 
Theorem4lt5 9009 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5
 
Theorem3lt5 9010 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5
 
Theorem2lt5 9011 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5
 
Theorem1lt5 9012 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5
 
Theorem5lt6 9013 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6
 
Theorem4lt6 9014 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6
 
Theorem3lt6 9015 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6
 
Theorem2lt6 9016 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6
 
Theorem1lt6 9017 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6
 
Theorem6lt7 9018 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7
 
Theorem5lt7 9019 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7
 
Theorem4lt7 9020 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7
 
Theorem3lt7 9021 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7
 
Theorem2lt7 9022 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7
 
Theorem1lt7 9023 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7
 
Theorem7lt8 9024 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8
 
Theorem6lt8 9025 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8
 
Theorem5lt8 9026 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8
 
Theorem4lt8 9027 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8
 
Theorem3lt8 9028 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8
 
Theorem2lt8 9029 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8
 
Theorem1lt8 9030 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8
 
Theorem8lt9 9031 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9
 
Theorem7lt9 9032 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9
 
Theorem6lt9 9033 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9
 
Theorem5lt9 9034 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9
 
Theorem4lt9 9035 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9
 
Theorem3lt9 9036 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9
 
Theorem2lt9 9037 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9
 
Theorem1lt9 9038 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9
 
Theorem0ne2 9039 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2
 
Theorem1ne2 9040 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2
 
Theorem1ap2 9041 1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
1 # 2
 
Theorem1le2 9042 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2
 
Theorem2cnne0 9043 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 ∈ ℂ ∧ 2 ≠ 0)
 
Theorem2rene0 9044 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 ∈ ℝ ∧ 2 ≠ 0)
 
Theorem1le3 9045 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3
 
Theoremneg1mulneg1e1 9046 -1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1
 
Theoremhalfre 9047 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℝ
 
Theoremhalfcn 9048 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℂ
 
Theoremhalfgt0 9049 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)
 
Theoremhalfge0 9050 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)
 
Theoremhalflt1 9051 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1
 
Theorem1mhlfehlf 9052 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)
 
Theorem8th4div3 9053 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)
 
Theoremhalfpm6th 9054 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 9055 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0
 
Theorem2mulicn 9056 (2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ∈ ℂ
 
Theoremiap0 9057 The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
i # 0
 
Theorem2muliap0 9058 2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
(2 · i) # 0
 
Theorem2muline0 9059 (2 · i) ≠ 0. See also 2muliap0 9058. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0
 
4.4.5  Simple number properties
 
Theoremhalfcl 9060 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)
 
Theoremrehalfcl 9061 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
 
Theoremhalf0 9062 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 9063 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
 
Theoremhalfpos2 9064 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
 
Theoremhalfpos 9065 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 9066 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))
 
Theoremhalfaddsubcl 9067 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 9068 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))
 
Theoremlt2halves 9069 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 9070 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 9071* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
¬ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ¬ ∃𝑦 ∈ ℝ (0 < 𝑦𝑦 < 𝑥))
 
Theoremavglt1 9072 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < ((𝐴 + 𝐵) / 2)))
 
Theoremavglt2 9073 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
 
Theoremavgle1 9074 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵𝐴 ≤ ((𝐴 + 𝐵) / 2)))
 
Theoremavgle2 9075 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))
 
Theorem2timesd 9076 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))
 
Theoremtimes2d 9077 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))
 
Theoremhalfcld 9078 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 2) ∈ ℂ)
 
Theorem2halvesd 9079 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
 
Theoremrehalfcld 9080 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 / 2) ∈ ℝ)
 
Theoremlt2halvesd 9081 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 9082 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       (𝐴 / 2) ∈ ℝ
 
Theoremadd1p1 9083 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
(𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
 
Theoremsub1m1 9084 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
(𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
 
Theoremcnm2m1cnm3 9085 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 9086 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 9087 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
 
Theoremarch 9088* 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 9089* 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 9090* 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)
 
4.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 9091 Extend class notation to include the class of nonnegative integers.
class 0
 
Definitiondf-n0 9092 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})
 
Theoremelnn0 9093 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
 
Theoremnnssnn0 9094 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0
 
Theoremnn0ssre 9095 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ
 
Theoremnn0sscn 9096 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
0 ⊆ ℂ
 
Theoremnn0ex 9097 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 ∈ V
 
Theoremnnnn0 9098 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
 
Theoremnnnn0i 9099 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁 ∈ ℕ       𝑁 ∈ ℕ0
 
Theoremnn0re 9100 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℝ)
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