P. 113 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	4p3e7 11201	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7
Theorem	4p4e8 11202	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
⊢ (4 + 4) = 8
Theorem	5p2e7 11203	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
⊢ (5 + 2) = 7
Theorem	5p3e8 11204	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
⊢ (5 + 3) = 8
Theorem	5p4e9 11205	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
⊢ (5 + 4) = 9
Theorem	5p5e10OLD 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
Theorem	6p2e8 11207	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
⊢ (6 + 2) = 8
Theorem	6p3e9 11208	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
⊢ (6 + 3) = 9
Theorem	6p4e10OLD 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
Theorem	7p2e9 11210	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
⊢ (7 + 2) = 9
Theorem	7p3e10OLD 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
Theorem	8p2e10OLD 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
Theorem	1t1e1 11213	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 · 1) = 1
Theorem	2t1e2 11214	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (2 · 1) = 2
Theorem	2t2e4 11215	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
⊢ (2 · 2) = 4
Theorem	3t1e3 11216	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (3 · 1) = 3
Theorem	3t2e6 11217	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
⊢ (3 · 2) = 6
Theorem	3t3e9 11218	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
⊢ (3 · 3) = 9
Theorem	4t2e8 11219	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
⊢ (4 · 2) = 8
Theorem	5t2e10OLD 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
Theorem	2t0e0 11221	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · 0) = 0
Theorem	4d2e2 11222	One half of four is two. (Contributed by NM, 3-Sep-1999.)
⊢ (4 / 2) = 2
Theorem	2nn 11223	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	3nn 11224	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	4nn 11225	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	5nn 11226	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	6nn 11227	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	7nn 11228	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	8nn 11229	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	9nn 11230	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	10nnOLD 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
Theorem	1lt2 11232	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
⊢ 1 < 2
Theorem	2lt3 11233	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 2 < 3
Theorem	1lt3 11234	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 1 < 3
Theorem	3lt4 11235	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 4
Theorem	2lt4 11236	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 4
Theorem	1lt4 11237	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 4
Theorem	4lt5 11238	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 5
Theorem	3lt5 11239	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 5
Theorem	2lt5 11240	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 5
Theorem	1lt5 11241	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 5
Theorem	5lt6 11242	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 11243	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 11244	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 11245	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 11246	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 11247	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 11248	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 11249	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 11250	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 11251	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 11252	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 11253	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 11254	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 11255	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 11256	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 11257	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 11258	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 11259	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 11260	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 11261	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 11262	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 11263	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 11264	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 11265	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 11266	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 11267	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9
Theorem	9lt10OLD 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
Theorem	8lt10OLD 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
Theorem	7lt10OLD 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
Theorem	6lt10OLD 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
Theorem	5lt10OLD 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
Theorem	4lt10OLD 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
Theorem	3lt10OLD 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
Theorem	2lt10OLD 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
Theorem	1lt10OLD 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
Theorem	0ne2 11277	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2
Theorem	1ne2 11278	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
⊢ 1 ≠ 2
Theorem	1le2 11279	1 is less than or equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 2
Theorem	2cnne0 11280	2 is a nonzero complex number. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
Theorem	2rene0 11281	2 is a nonzero real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 11282	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 11283	-1 · -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 11284	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 11285	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 11286	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halfge0 11287	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
⊢ 0 ≤ (1 / 2)
Theorem	halflt1 11288	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 11289	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 11290	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 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))
Theorem	it0e0 11292	i times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0
Theorem	2mulicn 11293	(2 · i) ∈ ℂ. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ∈ ℂ
Theorem	2muline0 11294	(2 · i) ≠ 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ≠ 0
5.4.5  Simple number properties
Theorem	halfcl 11295	Closure of half of a number. (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)
Theorem	rehalfcl 11296	Real closure of half. (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
Theorem	half0 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))
Theorem	2halves 11298	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	halfpos2 11299	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
Theorem	halfpos 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))