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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem4p3e7 11201 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7

Theorem4p4e8 11202 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8

Theorem5p2e7 11203 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7

Theorem5p3e8 11204 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8

Theorem5p4e9 11205 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9

Theorem5p5e10OLD 11206 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 5p5e10 11634 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(5 + 5) = 10

Theorem6p2e8 11207 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8

Theorem6p3e9 11208 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9

Theorem6p4e10OLD 11209 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 6p4e10 11636 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(6 + 4) = 10

Theorem7p2e9 11210 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9

Theorem7p3e10OLD 11211 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 7p3e10 11641 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(7 + 3) = 10

Theorem8p2e10OLD 11212 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 8p2e10 11648 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(8 + 2) = 10

Theorem1t1e1 11213 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1

Theorem2t1e2 11214 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2

Theorem2t2e4 11215 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4

Theorem3t1e3 11216 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3

Theorem3t2e6 11217 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6

Theorem3t3e9 11218 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9

Theorem4t2e8 11219 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8

Theorem5t2e10OLD 11220 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 5t2e10 11672 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 2) = 10

Theorem2t0e0 11221 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0

Theorem4d2e2 11222 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2

Theorem2nn 11223 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ

Theorem3nn 11224 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ

Theorem4nn 11225 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ

Theorem5nn 11226 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ

Theorem6nn 11227 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ

Theorem7nn 11228 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ

Theorem8nn 11229 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ

Theorem9nn 11230 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ

Theorem10nnOLD 11231 Obsolete version of 10nn 11552 as of 6-Sep-2021. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
10 ∈ ℕ

Theorem1lt2 11232 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2

Theorem2lt3 11233 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3

Theorem1lt3 11234 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3

Theorem3lt4 11235 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4

Theorem2lt4 11236 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4

Theorem1lt4 11237 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4

Theorem4lt5 11238 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5

Theorem3lt5 11239 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5

Theorem2lt5 11240 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5

Theorem1lt5 11241 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5

Theorem5lt6 11242 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6

Theorem4lt6 11243 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6

Theorem3lt6 11244 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6

Theorem2lt6 11245 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6

Theorem1lt6 11246 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6

Theorem6lt7 11247 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7

Theorem5lt7 11248 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7

Theorem4lt7 11249 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7

Theorem3lt7 11250 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7

Theorem2lt7 11251 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7

Theorem1lt7 11252 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7

Theorem7lt8 11253 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8

Theorem6lt8 11254 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8

Theorem5lt8 11255 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8

Theorem4lt8 11256 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8

Theorem3lt8 11257 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8

Theorem2lt8 11258 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8

Theorem1lt8 11259 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8

Theorem8lt9 11260 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9

Theorem7lt9 11261 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9

Theorem6lt9 11262 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9

Theorem5lt9 11263 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9

Theorem4lt9 11264 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9

Theorem3lt9 11265 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9

Theorem2lt9 11266 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9

Theorem1lt9 11267 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9

Theorem9lt10OLD 11268 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) Obsolete version of 9lt10 11711 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
9 < 10

Theorem8lt10OLD 11269 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) Obsolete version of 8lt10 11712 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
8 < 10

Theorem7lt10OLD 11270 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) Obsolete version of 7lt10 11713 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
7 < 10

Theorem6lt10OLD 11271 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) Obsolete version of 6lt10 11714 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
6 < 10

Theorem5lt10OLD 11272 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) Obsolete version of 5lt10 11715 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
5 < 10

Theorem4lt10OLD 11273 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) Obsolete version of 4lt10 11716 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
4 < 10

Theorem3lt10OLD 11274 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) Obsolete version of 3lt10 11717 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
3 < 10

Theorem2lt10OLD 11275 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) Obsolete version of 2lt10 11718 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
2 < 10

Theorem1lt10OLD 11276 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) Obsolete version of 1lt10 11719 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
1 < 10

Theorem0ne2 11277 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2

Theorem1ne2 11278 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2

Theorem1le2 11279 1 is less than or equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2

Theorem2cnne0 11280 2 is a nonzero complex number. (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 ∈ ℂ ∧ 2 ≠ 0)

Theorem2rene0 11281 2 is a nonzero real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 ∈ ℝ ∧ 2 ≠ 0)

Theorem1le3 11282 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3

Theoremneg1mulneg1e1 11283 -1 · -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1

Theoremhalfre 11284 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℝ

Theoremhalfcn 11285 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℂ

Theoremhalfgt0 11286 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)

Theoremhalfge0 11287 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)

Theoremhalflt1 11288 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1

Theorem1mhlfehlf 11289 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)

Theorem8th4div3 11290 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)

Theoremhalfpm6th 11291 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 11292 i times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0

Theorem2mulicn 11293 (2 · i) ∈ ℂ. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ∈ ℂ

Theorem2muline0 11294 (2 · i) ≠ 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0

5.4.5  Simple number properties

Theoremhalfcl 11295 Closure of half of a number. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)

Theoremrehalfcl 11296 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)

Theoremhalf0 11297 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 11298 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)

Theoremhalfpos2 11299 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))

Theoremhalfpos 11300 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))

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