P. 93 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	6m1e5 9201	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
⊢ (6 − 1) = 5
Theorem	7m1e6 9202	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
⊢ (7 − 1) = 6
Theorem	8m1e7 9203	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
⊢ (8 − 1) = 7
Theorem	9m1e8 9204	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
⊢ (9 − 1) = 8
Theorem	2p2e4 9205	Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: https://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
⊢ (2 + 2) = 4
Theorem	2times 9206	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 9207	A number times 2. (Contributed by NM, 16-Oct-2007.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	2timesi 9208	Two times a number. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (2 · 𝐴) = (𝐴 + 𝐴)
Theorem	times2i 9209	A number times 2. (Contributed by NM, 11-May-2004.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 2) = (𝐴 + 𝐴)
Theorem	2txmxeqx 9210	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 9211	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 / 2) = 1
Theorem	2p1e3 9212	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (2 + 1) = 3
Theorem	1p2e3 9213	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 2) = 3
Theorem	3p1e4 9214	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (3 + 1) = 4
Theorem	4p1e5 9215	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (4 + 1) = 5
Theorem	5p1e6 9216	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (5 + 1) = 6
Theorem	6p1e7 9217	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (6 + 1) = 7
Theorem	7p1e8 9218	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (7 + 1) = 8
Theorem	8p1e9 9219	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (8 + 1) = 9
Theorem	3p2e5 9220	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
⊢ (3 + 2) = 5
Theorem	3p3e6 9221	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
⊢ (3 + 3) = 6
Theorem	4p2e6 9222	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
⊢ (4 + 2) = 6
Theorem	4p3e7 9223	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7
Theorem	4p4e8 9224	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
⊢ (4 + 4) = 8
Theorem	5p2e7 9225	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
⊢ (5 + 2) = 7
Theorem	5p3e8 9226	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
⊢ (5 + 3) = 8
Theorem	5p4e9 9227	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
⊢ (5 + 4) = 9
Theorem	6p2e8 9228	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
⊢ (6 + 2) = 8
Theorem	6p3e9 9229	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
⊢ (6 + 3) = 9
Theorem	7p2e9 9230	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
⊢ (7 + 2) = 9
Theorem	1t1e1 9231	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 · 1) = 1
Theorem	2t1e2 9232	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (2 · 1) = 2
Theorem	2t2e4 9233	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
⊢ (2 · 2) = 4
Theorem	3t1e3 9234	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (3 · 1) = 3
Theorem	3t2e6 9235	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
⊢ (3 · 2) = 6
Theorem	3t3e9 9236	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
⊢ (3 · 3) = 9
Theorem	4t2e8 9237	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
⊢ (4 · 2) = 8
Theorem	2t0e0 9238	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · 0) = 0
Theorem	4d2e2 9239	One half of four is two. (Contributed by NM, 3-Sep-1999.)
⊢ (4 / 2) = 2
Theorem	2nn 9240	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	3nn 9241	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	4nn 9242	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	5nn 9243	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	6nn 9244	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	7nn 9245	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	8nn 9246	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	9nn 9247	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	1lt2 9248	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
⊢ 1 < 2
Theorem	2lt3 9249	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 2 < 3
Theorem	1lt3 9250	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 1 < 3
Theorem	3lt4 9251	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 4
Theorem	2lt4 9252	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 4
Theorem	1lt4 9253	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 4
Theorem	4lt5 9254	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 5
Theorem	3lt5 9255	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 5
Theorem	2lt5 9256	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 5
Theorem	1lt5 9257	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 5
Theorem	5lt6 9258	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 9259	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 9260	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 9261	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 9262	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 9263	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 9264	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 9265	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 9266	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 9267	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 9268	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 9269	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 9270	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 9271	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 9272	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 9273	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 9274	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 9275	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 9276	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 9277	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 9278	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 9279	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 9280	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 9281	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 9282	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 9283	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9
Theorem	0ne2 9284	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2
Theorem	1ne2 9285	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
⊢ 1 ≠ 2
Theorem	1ap2 9286	1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
⊢ 1 # 2
Theorem	1le2 9287	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 2
Theorem	2cnne0 9288	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
Theorem	2rene0 9289	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 9290	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 9291	-1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 9292	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 9293	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 9294	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halfge0 9295	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
⊢ 0 ≤ (1 / 2)
Theorem	halflt1 9296	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 9297	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 9298	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 9299	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 9300	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0