P. 92 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neg1rr 9101	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℝ
Theorem	neg1ne0 9102	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 ≠ 0
Theorem	neg1lt0 9103	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 < 0
Theorem	neg1ap0 9104	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ -1 # 0
Theorem	negneg1e1 9105	--1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ --1 = 1
Theorem	1pneg1e0 9106	1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + -1) = 0
Theorem	0m0e0 9107	0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0 − 0) = 0
Theorem	1m0e1 9108	1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 − 0) = 1
Theorem	0p1e1 9109	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (0 + 1) = 1
Theorem	fv0p1e1 9110	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 9111	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 0) = 1
Theorem	1p1e2 9112	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
⊢ (1 + 1) = 2
Theorem	2m1e1 9113	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9141. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (2 − 1) = 1
Theorem	1e2m1 9114	1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 = (2 − 1)
Theorem	3m1e2 9115	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
⊢ (3 − 1) = 2
Theorem	4m1e3 9116	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (4 − 1) = 3
Theorem	5m1e4 9117	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
⊢ (5 − 1) = 4
Theorem	6m1e5 9118	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
⊢ (6 − 1) = 5
Theorem	7m1e6 9119	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
⊢ (7 − 1) = 6
Theorem	8m1e7 9120	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
⊢ (8 − 1) = 7
Theorem	9m1e8 9121	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
⊢ (9 − 1) = 8
Theorem	2p2e4 9122	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 9123	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 9124	A number times 2. (Contributed by NM, 16-Oct-2007.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	2timesi 9125	Two times a number. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (2 · 𝐴) = (𝐴 + 𝐴)
Theorem	times2i 9126	A number times 2. (Contributed by NM, 11-May-2004.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 2) = (𝐴 + 𝐴)
Theorem	2txmxeqx 9127	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 9128	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 / 2) = 1
Theorem	2p1e3 9129	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (2 + 1) = 3
Theorem	1p2e3 9130	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 2) = 3
Theorem	3p1e4 9131	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (3 + 1) = 4
Theorem	4p1e5 9132	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (4 + 1) = 5
Theorem	5p1e6 9133	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (5 + 1) = 6
Theorem	6p1e7 9134	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (6 + 1) = 7
Theorem	7p1e8 9135	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (7 + 1) = 8
Theorem	8p1e9 9136	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (8 + 1) = 9
Theorem	3p2e5 9137	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
⊢ (3 + 2) = 5
Theorem	3p3e6 9138	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
⊢ (3 + 3) = 6
Theorem	4p2e6 9139	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
⊢ (4 + 2) = 6
Theorem	4p3e7 9140	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7
Theorem	4p4e8 9141	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
⊢ (4 + 4) = 8
Theorem	5p2e7 9142	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
⊢ (5 + 2) = 7
Theorem	5p3e8 9143	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
⊢ (5 + 3) = 8
Theorem	5p4e9 9144	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
⊢ (5 + 4) = 9
Theorem	6p2e8 9145	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
⊢ (6 + 2) = 8
Theorem	6p3e9 9146	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
⊢ (6 + 3) = 9
Theorem	7p2e9 9147	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
⊢ (7 + 2) = 9
Theorem	1t1e1 9148	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 · 1) = 1
Theorem	2t1e2 9149	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (2 · 1) = 2
Theorem	2t2e4 9150	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
⊢ (2 · 2) = 4
Theorem	3t1e3 9151	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (3 · 1) = 3
Theorem	3t2e6 9152	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
⊢ (3 · 2) = 6
Theorem	3t3e9 9153	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
⊢ (3 · 3) = 9
Theorem	4t2e8 9154	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
⊢ (4 · 2) = 8
Theorem	2t0e0 9155	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · 0) = 0
Theorem	4d2e2 9156	One half of four is two. (Contributed by NM, 3-Sep-1999.)
⊢ (4 / 2) = 2
Theorem	2nn 9157	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	3nn 9158	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	4nn 9159	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	5nn 9160	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	6nn 9161	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	7nn 9162	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	8nn 9163	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	9nn 9164	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	1lt2 9165	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
⊢ 1 < 2
Theorem	2lt3 9166	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 2 < 3
Theorem	1lt3 9167	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 1 < 3
Theorem	3lt4 9168	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 4
Theorem	2lt4 9169	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 4
Theorem	1lt4 9170	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 4
Theorem	4lt5 9171	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 5
Theorem	3lt5 9172	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 5
Theorem	2lt5 9173	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 5
Theorem	1lt5 9174	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 5
Theorem	5lt6 9175	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 9176	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 9177	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 9178	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 9179	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 9180	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 9181	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 9182	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 9183	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 9184	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 9185	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 9186	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 9187	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 9188	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 9189	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 9190	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 9191	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 9192	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 9193	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 9194	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 9195	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 9196	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 9197	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 9198	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 9199	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 9200	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9