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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 1lt4 11801 | 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 4 | ||
Theorem | 4lt5 11802 | 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 5 | ||
Theorem | 3lt5 11803 | 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 5 | ||
Theorem | 2lt5 11804 | 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 5 | ||
Theorem | 1lt5 11805 | 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 5 | ||
Theorem | 5lt6 11806 | 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 6 | ||
Theorem | 4lt6 11807 | 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 6 | ||
Theorem | 3lt6 11808 | 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 6 | ||
Theorem | 2lt6 11809 | 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 6 | ||
Theorem | 1lt6 11810 | 1 is less than 6. (Contributed by NM, 19-Oct-2012.) |
⊢ 1 < 6 | ||
Theorem | 6lt7 11811 | 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 < 7 | ||
Theorem | 5lt7 11812 | 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 7 | ||
Theorem | 4lt7 11813 | 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 7 | ||
Theorem | 3lt7 11814 | 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 7 | ||
Theorem | 2lt7 11815 | 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 7 | ||
Theorem | 1lt7 11816 | 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 7 | ||
Theorem | 7lt8 11817 | 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 7 < 8 | ||
Theorem | 6lt8 11818 | 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 < 8 | ||
Theorem | 5lt8 11819 | 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 8 | ||
Theorem | 4lt8 11820 | 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 8 | ||
Theorem | 3lt8 11821 | 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 8 | ||
Theorem | 2lt8 11822 | 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 8 | ||
Theorem | 1lt8 11823 | 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 8 | ||
Theorem | 8lt9 11824 | 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.) |
⊢ 8 < 9 | ||
Theorem | 7lt9 11825 | 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 7 < 9 | ||
Theorem | 6lt9 11826 | 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 6 < 9 | ||
Theorem | 5lt9 11827 | 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 5 < 9 | ||
Theorem | 4lt9 11828 | 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 4 < 9 | ||
Theorem | 3lt9 11829 | 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 3 < 9 | ||
Theorem | 2lt9 11830 | 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 2 < 9 | ||
Theorem | 1lt9 11831 | 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) |
⊢ 1 < 9 | ||
Theorem | 0ne2 11832 | 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 ≠ 2 | ||
Theorem | 1ne2 11833 | 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.) |
⊢ 1 ≠ 2 | ||
Theorem | 1le2 11834 | 1 is less than or equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 ≤ 2 | ||
Theorem | 2cnne0 11835 | 2 is a nonzero complex number. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | ||
Theorem | 2rene0 11836 | 2 is a nonzero real number. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 ∈ ℝ ∧ 2 ≠ 0) | ||
Theorem | 1le3 11837 | 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 ≤ 3 | ||
Theorem | neg1mulneg1e1 11838 | -1 · -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (-1 · -1) = 1 | ||
Theorem | halfre 11839 | One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 / 2) ∈ ℝ | ||
Theorem | halfcn 11840 | One-half is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 / 2) ∈ ℂ | ||
Theorem | halfgt0 11841 | One-half is greater than zero. (Contributed by NM, 24-Feb-2005.) |
⊢ 0 < (1 / 2) | ||
Theorem | halfge0 11842 | One-half is not negative. (Contributed by AV, 7-Jun-2020.) |
⊢ 0 ≤ (1 / 2) | ||
Theorem | halflt1 11843 | One-half is less than one. (Contributed by NM, 24-Feb-2005.) |
⊢ (1 / 2) < 1 | ||
Theorem | 1mhlfehlf 11844 | Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.) |
⊢ (1 − (1 / 2)) = (1 / 2) | ||
Theorem | 8th4div3 11845 | An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.) |
⊢ ((1 / 8) · (4 / 3)) = (1 / 6) | ||
Theorem | halfpm6th 11846 | 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 11847 | i times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (i · 0) = 0 | ||
Theorem | 2mulicn 11848 | (2 · i) ∈ ℂ. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · i) ∈ ℂ | ||
Theorem | 2muline0 11849 | (2 · i) ≠ 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · i) ≠ 0 | ||
Theorem | halfcl 11850 | Closure of half of a number. (Contributed by NM, 1-Jan-2006.) |
⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) | ||
Theorem | rehalfcl 11851 | Real closure of half. (Contributed by NM, 1-Jan-2006.) |
⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) | ||
Theorem | half0 11852 | 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 11853 | Two halves make a whole. (Contributed by NM, 11-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) | ||
Theorem | halfpos2 11854 | A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2))) | ||
Theorem | halfpos 11855 | 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 11856 | A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.) |
⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2))) | ||
Theorem | halfaddsubcl 11857 | 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 11858 | 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 | subhalfhalf 11859 | 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)) | ||
Theorem | lt2halves 11860 | 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 11861 | 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 11862* | 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 11863 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) | ||
Theorem | avglt2 11864 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵)) | ||
Theorem | avgle1 11865 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ((𝐴 + 𝐵) / 2))) | ||
Theorem | avgle2 11866 | 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 | avgle 11867 | 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)) | ||
Theorem | 2timesd 11868 | Two times a number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴)) | ||
Theorem | times2d 11869 | A number times 2. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴)) | ||
Theorem | halfcld 11870 | 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 11871 | Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) | ||
Theorem | rehalfcld 11872 | Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ) | ||
Theorem | lt2halvesd 11873 | 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 11874 | Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 / 2) ∈ ℝ | ||
Theorem | lt2addmuld 11875 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐶) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) < (2 · 𝐶)) | ||
Theorem | add1p1 11876 | Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.) |
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) | ||
Theorem | sub1m1 11877 | Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.) |
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2)) | ||
Theorem | cnm2m1cnm3 11878 | 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 11879 | 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 11880 | 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)) | ||
Theorem | nnunb 11881* | 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 < 𝑥 ∨ 𝑦 = 𝑥) | ||
Theorem | arch 11882* | 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 11883* | 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 11884* | 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) | ||
Syntax | cn0 11885 | Extend class notation to include the class of nonnegative integers. |
class ℕ0 | ||
Definition | df-n0 11886 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 = (ℕ ∪ {0}) | ||
Theorem | elnn0 11887 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | ||
Theorem | nnssnn0 11888 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ ⊆ ℕ0 | ||
Theorem | nn0ssre 11889 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 ⊆ ℝ | ||
Theorem | nn0sscn 11890 | Nonnegative integers are a subset of the complex numbers. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
⊢ ℕ0 ⊆ ℂ | ||
Theorem | nn0ex 11891 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
⊢ ℕ0 ∈ V | ||
Theorem | nnnn0 11892 | A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0) | ||
Theorem | nnnn0i 11893 | A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝑁 ∈ ℕ0 | ||
Theorem | nn0re 11894 | A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | ||
Theorem | nn0cn 11895 | A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) | ||
Theorem | nn0rei 11896 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 ∈ ℝ | ||
Theorem | nn0cni 11897 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 ∈ ℂ | ||
Theorem | dfn2 11898 | 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 11899 | 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 11900 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 0 ∈ ℕ0 |
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