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Type | Label | Description |
---|---|---|
Statement | ||
This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering). | ||
Theorem | neg1cn 12201 | -1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ -1 ∈ ℂ | ||
Theorem | neg1rr 12202 | -1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℝ | ||
Theorem | neg1ne0 12203 | -1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 ≠ 0 | ||
Theorem | neg1lt0 12204 | -1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 < 0 | ||
Theorem | negneg1e1 12205 | --1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ --1 = 1 | ||
Theorem | 1pneg1e0 12206 | 1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + -1) = 0 | ||
Theorem | 0m0e0 12207 | 0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (0 − 0) = 0 | ||
Theorem | 1m0e1 12208 | 1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 − 0) = 1 | ||
Theorem | 0p1e1 12209 | 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (0 + 1) = 1 | ||
Theorem | fv0p1e1 12210 | 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 12211 | 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + 0) = 1 | ||
Theorem | 1p1e2 12212 | 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.) |
⊢ (1 + 1) = 2 | ||
Theorem | 2m1e1 12213 | 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12242. (Contributed by David A. Wheeler, 4-Jan-2017.) |
⊢ (2 − 1) = 1 | ||
Theorem | 1e2m1 12214 | 1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 = (2 − 1) | ||
Theorem | 3m1e2 12215 | 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 12216 | 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (4 − 1) = 3 | ||
Theorem | 5m1e4 12217 | 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.) |
⊢ (5 − 1) = 4 | ||
Theorem | 6m1e5 12218 | 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.) |
⊢ (6 − 1) = 5 | ||
Theorem | 7m1e6 12219 | 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.) |
⊢ (7 − 1) = 6 | ||
Theorem | 8m1e7 12220 | 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (8 − 1) = 7 | ||
Theorem | 9m1e8 12221 | 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.) |
⊢ (9 − 1) = 8 | ||
Theorem | 2p2e4 12222 | 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 8455 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 12223 | 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 12224 | A number times 2. (Contributed by NM, 16-Oct-2007.) |
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | ||
Theorem | 2timesi 12225 | Two times a number. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (2 · 𝐴) = (𝐴 + 𝐴) | ||
Theorem | times2i 12226 | A number times 2. (Contributed by NM, 11-May-2004.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 2) = (𝐴 + 𝐴) | ||
Theorem | 2txmxeqx 12227 | 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 12228 | 2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 / 2) = 1 | ||
Theorem | 2p1e3 12229 | 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (2 + 1) = 3 | ||
Theorem | 1p2e3 12230 | 1 + 2 = 3. For a shorter proof using addcomli 11281, see 1p2e3ALT 12231. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.) |
⊢ (1 + 2) = 3 | ||
Theorem | 1p2e3ALT 12231 | Alternate proof of 1p2e3 12230, 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 12232 | 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (3 + 1) = 4 | ||
Theorem | 4p1e5 12233 | 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (4 + 1) = 5 | ||
Theorem | 5p1e6 12234 | 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (5 + 1) = 6 | ||
Theorem | 6p1e7 12235 | 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (6 + 1) = 7 | ||
Theorem | 7p1e8 12236 | 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (7 + 1) = 8 | ||
Theorem | 8p1e9 12237 | 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (8 + 1) = 9 | ||
Theorem | 3p2e5 12238 | 3 + 2 = 5. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 2) = 5 | ||
Theorem | 3p3e6 12239 | 3 + 3 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 3) = 6 | ||
Theorem | 4p2e6 12240 | 4 + 2 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 2) = 6 | ||
Theorem | 4p3e7 12241 | 4 + 3 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 3) = 7 | ||
Theorem | 4p4e8 12242 | 4 + 4 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 4) = 8 | ||
Theorem | 5p2e7 12243 | 5 + 2 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 2) = 7 | ||
Theorem | 5p3e8 12244 | 5 + 3 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 3) = 8 | ||
Theorem | 5p4e9 12245 | 5 + 4 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 4) = 9 | ||
Theorem | 6p2e8 12246 | 6 + 2 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 2) = 8 | ||
Theorem | 6p3e9 12247 | 6 + 3 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 3) = 9 | ||
Theorem | 7p2e9 12248 | 7 + 2 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (7 + 2) = 9 | ||
Theorem | 1t1e1 12249 | 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (1 · 1) = 1 | ||
Theorem | 2t1e2 12250 | 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (2 · 1) = 2 | ||
Theorem | 2t2e4 12251 | 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.) |
⊢ (2 · 2) = 4 | ||
Theorem | 3t1e3 12252 | 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (3 · 1) = 3 | ||
Theorem | 3t2e6 12253 | 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.) |
⊢ (3 · 2) = 6 | ||
Theorem | 3t3e9 12254 | 3 times 3 equals 9. (Contributed by NM, 11-May-2004.) |
⊢ (3 · 3) = 9 | ||
Theorem | 4t2e8 12255 | 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.) |
⊢ (4 · 2) = 8 | ||
Theorem | 2t0e0 12256 | 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · 0) = 0 | ||
Theorem | 4d2e2 12257 | One half of four is two. (Contributed by NM, 3-Sep-1999.) |
⊢ (4 / 2) = 2 | ||
Theorem | 1lt2 12258 | 1 is less than 2. (Contributed by NM, 24-Feb-2005.) |
⊢ 1 < 2 | ||
Theorem | 2lt3 12259 | 2 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 2 < 3 | ||
Theorem | 1lt3 12260 | 1 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 1 < 3 | ||
Theorem | 3lt4 12261 | 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 4 | ||
Theorem | 2lt4 12262 | 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 4 | ||
Theorem | 1lt4 12263 | 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 4 | ||
Theorem | 4lt5 12264 | 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 5 | ||
Theorem | 3lt5 12265 | 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 5 | ||
Theorem | 2lt5 12266 | 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 5 | ||
Theorem | 1lt5 12267 | 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 5 | ||
Theorem | 5lt6 12268 | 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 6 | ||
Theorem | 4lt6 12269 | 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 6 | ||
Theorem | 3lt6 12270 | 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 6 | ||
Theorem | 2lt6 12271 | 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 6 | ||
Theorem | 1lt6 12272 | 1 is less than 6. (Contributed by NM, 19-Oct-2012.) |
⊢ 1 < 6 | ||
Theorem | 6lt7 12273 | 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 < 7 | ||
Theorem | 5lt7 12274 | 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 7 | ||
Theorem | 4lt7 12275 | 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 7 | ||
Theorem | 3lt7 12276 | 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 7 | ||
Theorem | 2lt7 12277 | 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 7 | ||
Theorem | 1lt7 12278 | 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 7 | ||
Theorem | 7lt8 12279 | 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 7 < 8 | ||
Theorem | 6lt8 12280 | 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 < 8 | ||
Theorem | 5lt8 12281 | 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 < 8 | ||
Theorem | 4lt8 12282 | 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 8 | ||
Theorem | 3lt8 12283 | 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 8 | ||
Theorem | 2lt8 12284 | 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 8 | ||
Theorem | 1lt8 12285 | 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 8 | ||
Theorem | 8lt9 12286 | 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.) |
⊢ 8 < 9 | ||
Theorem | 7lt9 12287 | 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 7 < 9 | ||
Theorem | 6lt9 12288 | 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 6 < 9 | ||
Theorem | 5lt9 12289 | 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 5 < 9 | ||
Theorem | 4lt9 12290 | 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 4 < 9 | ||
Theorem | 3lt9 12291 | 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 3 < 9 | ||
Theorem | 2lt9 12292 | 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 2 < 9 | ||
Theorem | 1lt9 12293 | 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) |
⊢ 1 < 9 | ||
Theorem | 0ne2 12294 | 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 ≠ 2 | ||
Theorem | 1ne2 12295 | 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.) |
⊢ 1 ≠ 2 | ||
Theorem | 1le2 12296 | 1 is less than or equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 ≤ 2 | ||
Theorem | 2cnne0 12297 | 2 is a nonzero complex number. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | ||
Theorem | 2rene0 12298 | 2 is a nonzero real number. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 ∈ ℝ ∧ 2 ≠ 0) | ||
Theorem | 1le3 12299 | 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 ≤ 3 | ||
Theorem | neg1mulneg1e1 12300 | -1 · -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (-1 · -1) = 1 |
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