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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-6 12301 | Define the number 6. (Contributed by NM, 27-May-1999.) |
⊢ 6 = (5 + 1) | ||
Definition | df-7 12302 | Define the number 7. (Contributed by NM, 27-May-1999.) |
⊢ 7 = (6 + 1) | ||
Definition | df-8 12303 | Define the number 8. (Contributed by NM, 27-May-1999.) |
⊢ 8 = (7 + 1) | ||
Definition | df-9 12304 | Define the number 9. (Contributed by NM, 27-May-1999.) |
⊢ 9 = (8 + 1) | ||
Theorem | 0ne1 12305 | Zero is different from one (the commuted form is Axiom ax-1ne0 11199). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 ≠ 1 | ||
Theorem | 1m1e0 12306 | One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (1 − 1) = 0 | ||
Theorem | 2nn 12307 | 2 is a positive integer. (Contributed by NM, 20-Aug-2001.) |
⊢ 2 ∈ ℕ | ||
Theorem | 2re 12308 | The number 2 is real. (Contributed by NM, 27-May-1999.) |
⊢ 2 ∈ ℝ | ||
Theorem | 2cn 12309 | The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 2 ∈ ℂ | ||
Theorem | 2cnALT 12310 | Alternate proof of 2cn 12309. Shorter but uses more axioms. Similar proofs are possible for 3cn 12315, ... , 9cn 12334. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 2 ∈ ℂ | ||
Theorem | 2ex 12311 | The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 2 ∈ V | ||
Theorem | 2cnd 12312 | The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 2 ∈ ℂ) | ||
Theorem | 3nn 12313 | 3 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
⊢ 3 ∈ ℕ | ||
Theorem | 3re 12314 | The number 3 is real. (Contributed by NM, 27-May-1999.) |
⊢ 3 ∈ ℝ | ||
Theorem | 3cn 12315 | The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 3 ∈ ℂ | ||
Theorem | 3ex 12316 | The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 3 ∈ V | ||
Theorem | 4nn 12317 | 4 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
⊢ 4 ∈ ℕ | ||
Theorem | 4re 12318 | The number 4 is real. (Contributed by NM, 27-May-1999.) |
⊢ 4 ∈ ℝ | ||
Theorem | 4cn 12319 | The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 4 ∈ ℂ | ||
Theorem | 5nn 12320 | 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 ∈ ℕ | ||
Theorem | 5re 12321 | The number 5 is real. (Contributed by NM, 27-May-1999.) |
⊢ 5 ∈ ℝ | ||
Theorem | 5cn 12322 | The number 5 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 5 ∈ ℂ | ||
Theorem | 6nn 12323 | 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 ∈ ℕ | ||
Theorem | 6re 12324 | The number 6 is real. (Contributed by NM, 27-May-1999.) |
⊢ 6 ∈ ℝ | ||
Theorem | 6cn 12325 | The number 6 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 6 ∈ ℂ | ||
Theorem | 7nn 12326 | 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 7 ∈ ℕ | ||
Theorem | 7re 12327 | The number 7 is real. (Contributed by NM, 27-May-1999.) |
⊢ 7 ∈ ℝ | ||
Theorem | 7cn 12328 | The number 7 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 7 ∈ ℂ | ||
Theorem | 8nn 12329 | 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 8 ∈ ℕ | ||
Theorem | 8re 12330 | The number 8 is real. (Contributed by NM, 27-May-1999.) |
⊢ 8 ∈ ℝ | ||
Theorem | 8cn 12331 | The number 8 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 8 ∈ ℂ | ||
Theorem | 9nn 12332 | 9 is a positive integer. (Contributed by NM, 21-Oct-2012.) |
⊢ 9 ∈ ℕ | ||
Theorem | 9re 12333 | The number 9 is real. (Contributed by NM, 27-May-1999.) |
⊢ 9 ∈ ℝ | ||
Theorem | 9cn 12334 | The number 9 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 9 ∈ ℂ | ||
Theorem | 0le0 12335 | Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 0 ≤ 0 | ||
Theorem | 0le2 12336 | The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ 0 ≤ 2 | ||
Theorem | 2pos 12337 | The number 2 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 2 | ||
Theorem | 2ne0 12338 | The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.) |
⊢ 2 ≠ 0 | ||
Theorem | 3pos 12339 | The number 3 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 3 | ||
Theorem | 3ne0 12340 | The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ 3 ≠ 0 | ||
Theorem | 4pos 12341 | The number 4 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 4 | ||
Theorem | 4ne0 12342 | The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ 4 ≠ 0 | ||
Theorem | 5pos 12343 | The number 5 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 5 | ||
Theorem | 6pos 12344 | The number 6 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 6 | ||
Theorem | 7pos 12345 | The number 7 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 7 | ||
Theorem | 8pos 12346 | The number 8 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 8 | ||
Theorem | 9pos 12347 | The number 9 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 9 | ||
This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering). | ||
Theorem | neg1cn 12348 | -1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ -1 ∈ ℂ | ||
Theorem | neg1rr 12349 | -1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℝ | ||
Theorem | neg1ne0 12350 | -1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 ≠ 0 | ||
Theorem | neg1lt0 12351 | -1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 < 0 | ||
Theorem | negneg1e1 12352 | --1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ --1 = 1 | ||
Theorem | 1pneg1e0 12353 | 1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + -1) = 0 | ||
Theorem | 0m0e0 12354 | 0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (0 − 0) = 0 | ||
Theorem | 1m0e1 12355 | 1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 − 0) = 1 | ||
Theorem | 0p1e1 12356 | 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (0 + 1) = 1 | ||
Theorem | fv0p1e1 12357 | Function value at 𝑁 + 1 with 𝑁 replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1)) | ||
Theorem | 1p0e1 12358 | 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + 0) = 1 | ||
Theorem | 1p1e2 12359 | 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.) |
⊢ (1 + 1) = 2 | ||
Theorem | 2m1e1 12360 | 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12389. (Contributed by David A. Wheeler, 4-Jan-2017.) |
⊢ (2 − 1) = 1 | ||
Theorem | 1e2m1 12361 | 1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 = (2 − 1) | ||
Theorem | 3m1e2 12362 | 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (3 − 1) = 2 | ||
Theorem | 4m1e3 12363 | 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (4 − 1) = 3 | ||
Theorem | 5m1e4 12364 | 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.) |
⊢ (5 − 1) = 4 | ||
Theorem | 6m1e5 12365 | 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.) |
⊢ (6 − 1) = 5 | ||
Theorem | 7m1e6 12366 | 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.) |
⊢ (7 − 1) = 6 | ||
Theorem | 8m1e7 12367 | 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (8 − 1) = 7 | ||
Theorem | 9m1e8 12368 | 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.) |
⊢ (9 − 1) = 8 | ||
Theorem | 2p2e4 12369 | Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 8555 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.) |
⊢ (2 + 2) = 4 | ||
Theorem | 2times 12370 | Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.) |
⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | ||
Theorem | times2 12371 | A number times 2. (Contributed by NM, 16-Oct-2007.) |
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | ||
Theorem | 2timesi 12372 | Two times a number. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (2 · 𝐴) = (𝐴 + 𝐴) | ||
Theorem | times2i 12373 | A number times 2. (Contributed by NM, 11-May-2004.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 2) = (𝐴 + 𝐴) | ||
Theorem | 2txmxeqx 12374 | Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ (𝑋 ∈ ℂ → ((2 · 𝑋) − 𝑋) = 𝑋) | ||
Theorem | 2div2e1 12375 | 2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 / 2) = 1 | ||
Theorem | 2p1e3 12376 | 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (2 + 1) = 3 | ||
Theorem | 1p2e3 12377 | 1 + 2 = 3. For a shorter proof using addcomli 11428, see 1p2e3ALT 12378. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.) |
⊢ (1 + 2) = 3 | ||
Theorem | 1p2e3ALT 12378 | Alternate proof of 1p2e3 12377, shorter but using more axioms. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (1 + 2) = 3 | ||
Theorem | 3p1e4 12379 | 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (3 + 1) = 4 | ||
Theorem | 4p1e5 12380 | 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (4 + 1) = 5 | ||
Theorem | 5p1e6 12381 | 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (5 + 1) = 6 | ||
Theorem | 6p1e7 12382 | 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (6 + 1) = 7 | ||
Theorem | 7p1e8 12383 | 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (7 + 1) = 8 | ||
Theorem | 8p1e9 12384 | 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (8 + 1) = 9 | ||
Theorem | 3p2e5 12385 | 3 + 2 = 5. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 2) = 5 | ||
Theorem | 3p3e6 12386 | 3 + 3 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 3) = 6 | ||
Theorem | 4p2e6 12387 | 4 + 2 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 2) = 6 | ||
Theorem | 4p3e7 12388 | 4 + 3 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 3) = 7 | ||
Theorem | 4p4e8 12389 | 4 + 4 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 4) = 8 | ||
Theorem | 5p2e7 12390 | 5 + 2 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 2) = 7 | ||
Theorem | 5p3e8 12391 | 5 + 3 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 3) = 8 | ||
Theorem | 5p4e9 12392 | 5 + 4 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 4) = 9 | ||
Theorem | 6p2e8 12393 | 6 + 2 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 2) = 8 | ||
Theorem | 6p3e9 12394 | 6 + 3 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 3) = 9 | ||
Theorem | 7p2e9 12395 | 7 + 2 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (7 + 2) = 9 | ||
Theorem | 1t1e1 12396 | 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (1 · 1) = 1 | ||
Theorem | 2t1e2 12397 | 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (2 · 1) = 2 | ||
Theorem | 2t2e4 12398 | 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.) |
⊢ (2 · 2) = 4 | ||
Theorem | 3t1e3 12399 | 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (3 · 1) = 3 | ||
Theorem | 3t2e6 12400 | 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.) |
⊢ (3 · 2) = 6 |
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