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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 3t3e9 12401 | 3 times 3 equals 9. (Contributed by NM, 11-May-2004.) |
⊢ (3 · 3) = 9 | ||
Theorem | 4t2e8 12402 | 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.) |
⊢ (4 · 2) = 8 | ||
Theorem | 2t0e0 12403 | 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · 0) = 0 | ||
Theorem | 4d2e2 12404 | One half of four is two. (Contributed by NM, 3-Sep-1999.) |
⊢ (4 / 2) = 2 | ||
Theorem | 1lt2 12405 | 1 is less than 2. (Contributed by NM, 24-Feb-2005.) |
⊢ 1 < 2 | ||
Theorem | 2lt3 12406 | 2 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 2 < 3 | ||
Theorem | 1lt3 12407 | 1 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 1 < 3 | ||
Theorem | 3lt4 12408 | 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 4 | ||
Theorem | 2lt4 12409 | 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 4 | ||
Theorem | 1lt4 12410 | 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 4 | ||
Theorem | 4lt5 12411 | 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 5 | ||
Theorem | 3lt5 12412 | 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 5 | ||
Theorem | 2lt5 12413 | 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 5 | ||
Theorem | 1lt5 12414 | 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 5 | ||
Theorem | 5lt6 12415 | 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 6 | ||
Theorem | 4lt6 12416 | 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 6 | ||
Theorem | 3lt6 12417 | 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 6 | ||
Theorem | 2lt6 12418 | 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 6 | ||
Theorem | 1lt6 12419 | 1 is less than 6. (Contributed by NM, 19-Oct-2012.) |
⊢ 1 < 6 | ||
Theorem | 6lt7 12420 | 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 < 7 | ||
Theorem | 5lt7 12421 | 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 7 | ||
Theorem | 4lt7 12422 | 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 7 | ||
Theorem | 3lt7 12423 | 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 7 | ||
Theorem | 2lt7 12424 | 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 7 | ||
Theorem | 1lt7 12425 | 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 7 | ||
Theorem | 7lt8 12426 | 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 7 < 8 | ||
Theorem | 6lt8 12427 | 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 < 8 | ||
Theorem | 5lt8 12428 | 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 8 | ||
Theorem | 4lt8 12429 | 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 8 | ||
Theorem | 3lt8 12430 | 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 8 | ||
Theorem | 2lt8 12431 | 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 8 | ||
Theorem | 1lt8 12432 | 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 8 | ||
Theorem | 8lt9 12433 | 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.) |
⊢ 8 < 9 | ||
Theorem | 7lt9 12434 | 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 7 < 9 | ||
Theorem | 6lt9 12435 | 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 6 < 9 | ||
Theorem | 5lt9 12436 | 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 5 < 9 | ||
Theorem | 4lt9 12437 | 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 4 < 9 | ||
Theorem | 3lt9 12438 | 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 3 < 9 | ||
Theorem | 2lt9 12439 | 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 2 < 9 | ||
Theorem | 1lt9 12440 | 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) |
⊢ 1 < 9 | ||
Theorem | 0ne2 12441 | 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 ≠ 2 | ||
Theorem | 1ne2 12442 | 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.) |
⊢ 1 ≠ 2 | ||
Theorem | 1le2 12443 | 1 is less than or equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 ≤ 2 | ||
Theorem | 2cnne0 12444 | 2 is a nonzero complex number. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | ||
Theorem | 2rene0 12445 | 2 is a nonzero real number. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 ∈ ℝ ∧ 2 ≠ 0) | ||
Theorem | 1le3 12446 | 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 ≤ 3 | ||
Theorem | neg1mulneg1e1 12447 | -1 · -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (-1 · -1) = 1 | ||
Theorem | halfre 12448 | One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 / 2) ∈ ℝ | ||
Theorem | halfcn 12449 | One-half is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 / 2) ∈ ℂ | ||
Theorem | halfgt0 12450 | One-half is greater than zero. (Contributed by NM, 24-Feb-2005.) |
⊢ 0 < (1 / 2) | ||
Theorem | halfge0 12451 | One-half is not negative. (Contributed by AV, 7-Jun-2020.) |
⊢ 0 ≤ (1 / 2) | ||
Theorem | halflt1 12452 | One-half is less than one. (Contributed by NM, 24-Feb-2005.) |
⊢ (1 / 2) < 1 | ||
Theorem | 1mhlfehlf 12453 | Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.) |
⊢ (1 − (1 / 2)) = (1 / 2) | ||
Theorem | 8th4div3 12454 | An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.) |
⊢ ((1 / 8) · (4 / 3)) = (1 / 6) | ||
Theorem | halfpm6th 12455 | 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 12456 | i times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (i · 0) = 0 | ||
Theorem | 2mulicn 12457 | (2 · i) ∈ ℂ. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · i) ∈ ℂ | ||
Theorem | 2muline0 12458 | (2 · i) ≠ 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · i) ≠ 0 | ||
Theorem | halfcl 12459 | Closure of half of a number. (Contributed by NM, 1-Jan-2006.) |
⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) | ||
Theorem | rehalfcl 12460 | Real closure of half. (Contributed by NM, 1-Jan-2006.) |
⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) | ||
Theorem | half0 12461 | 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 12462 | Two halves make a whole. (Contributed by NM, 11-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) | ||
Theorem | halfpos2 12463 | A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2))) | ||
Theorem | halfpos 12464 | 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 12465 | A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.) |
⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2))) | ||
Theorem | halfaddsubcl 12466 | 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 12467 | 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 12468 | 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 12469 | 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 12470 | 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 12471* | 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 12472 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) | ||
Theorem | avglt2 12473 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵)) | ||
Theorem | avgle1 12474 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ((𝐴 + 𝐵) / 2))) | ||
Theorem | avgle2 12475 | 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 12476 | 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 12477 | Two times a number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴)) | ||
Theorem | times2d 12478 | A number times 2. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴)) | ||
Theorem | halfcld 12479 | 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 12480 | Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴) | ||
Theorem | rehalfcld 12481 | Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ) | ||
Theorem | lt2halvesd 12482 | 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 12483 | Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 / 2) ∈ ℝ | ||
Theorem | lt2addmuld 12484 | 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 12485 | Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.) |
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) | ||
Theorem | sub1m1 12486 | Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.) |
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2)) | ||
Theorem | cnm2m1cnm3 12487 | 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 12488 | 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 12489 | 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 12490* | 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 12491* | 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 12492* | 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 12493* | 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 12494 | Extend class notation to include the class of nonnegative integers. |
class ℕ0 | ||
Definition | df-n0 12495 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 = (ℕ ∪ {0}) | ||
Theorem | elnn0 12496 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | ||
Theorem | nnssnn0 12497 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ ⊆ ℕ0 | ||
Theorem | nn0ssre 12498 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 ⊆ ℝ | ||
Theorem | nn0sscn 12499 | 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 12500 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
⊢ ℕ0 ∈ V |
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