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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem6m1e5 9001 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
(6 − 1) = 5
 
Theorem7m1e6 9002 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
(7 − 1) = 6
 
Theorem8m1e7 9003 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
(8 − 1) = 7
 
Theorem9m1e8 9004 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
(9 − 1) = 8
 
Theorem2p2e4 9005 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
 
Theorem2times 9006 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))
 
Theoremtimes2 9007 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
 
Theorem2timesi 9008 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)
 
Theoremtimes2i 9009 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)
 
Theorem2div2e1 9010 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1
 
Theorem2p1e3 9011 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3
 
Theorem1p2e3 9012 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3
 
Theorem3p1e4 9013 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4
 
Theorem4p1e5 9014 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5
 
Theorem5p1e6 9015 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6
 
Theorem6p1e7 9016 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7
 
Theorem7p1e8 9017 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8
 
Theorem8p1e9 9018 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9
 
Theorem3p2e5 9019 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5
 
Theorem3p3e6 9020 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6
 
Theorem4p2e6 9021 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6
 
Theorem4p3e7 9022 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7
 
Theorem4p4e8 9023 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8
 
Theorem5p2e7 9024 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7
 
Theorem5p3e8 9025 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8
 
Theorem5p4e9 9026 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9
 
Theorem6p2e8 9027 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8
 
Theorem6p3e9 9028 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9
 
Theorem7p2e9 9029 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9
 
Theorem1t1e1 9030 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1
 
Theorem2t1e2 9031 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2
 
Theorem2t2e4 9032 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4
 
Theorem3t1e3 9033 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3
 
Theorem3t2e6 9034 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6
 
Theorem3t3e9 9035 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9
 
Theorem4t2e8 9036 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8
 
Theorem2t0e0 9037 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0
 
Theorem4d2e2 9038 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2
 
Theorem2nn 9039 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ
 
Theorem3nn 9040 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ
 
Theorem4nn 9041 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ
 
Theorem5nn 9042 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ
 
Theorem6nn 9043 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ
 
Theorem7nn 9044 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ
 
Theorem8nn 9045 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ
 
Theorem9nn 9046 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ
 
Theorem1lt2 9047 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2
 
Theorem2lt3 9048 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3
 
Theorem1lt3 9049 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3
 
Theorem3lt4 9050 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4
 
Theorem2lt4 9051 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4
 
Theorem1lt4 9052 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4
 
Theorem4lt5 9053 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5
 
Theorem3lt5 9054 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5
 
Theorem2lt5 9055 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5
 
Theorem1lt5 9056 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5
 
Theorem5lt6 9057 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6
 
Theorem4lt6 9058 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6
 
Theorem3lt6 9059 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6
 
Theorem2lt6 9060 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6
 
Theorem1lt6 9061 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6
 
Theorem6lt7 9062 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7
 
Theorem5lt7 9063 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7
 
Theorem4lt7 9064 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7
 
Theorem3lt7 9065 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7
 
Theorem2lt7 9066 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7
 
Theorem1lt7 9067 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7
 
Theorem7lt8 9068 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8
 
Theorem6lt8 9069 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8
 
Theorem5lt8 9070 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8
 
Theorem4lt8 9071 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8
 
Theorem3lt8 9072 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8
 
Theorem2lt8 9073 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8
 
Theorem1lt8 9074 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8
 
Theorem8lt9 9075 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9
 
Theorem7lt9 9076 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9
 
Theorem6lt9 9077 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9
 
Theorem5lt9 9078 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9
 
Theorem4lt9 9079 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9
 
Theorem3lt9 9080 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9
 
Theorem2lt9 9081 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9
 
Theorem1lt9 9082 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9
 
Theorem0ne2 9083 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2
 
Theorem1ne2 9084 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2
 
Theorem1ap2 9085 1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
1 # 2
 
Theorem1le2 9086 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2
 
Theorem2cnne0 9087 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 ∈ ℂ ∧ 2 ≠ 0)
 
Theorem2rene0 9088 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 ∈ ℝ ∧ 2 ≠ 0)
 
Theorem1le3 9089 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3
 
Theoremneg1mulneg1e1 9090 -1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1
 
Theoremhalfre 9091 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℝ
 
Theoremhalfcn 9092 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℂ
 
Theoremhalfgt0 9093 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)
 
Theoremhalfge0 9094 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)
 
Theoremhalflt1 9095 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1
 
Theorem1mhlfehlf 9096 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)
 
Theorem8th4div3 9097 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)
 
Theoremhalfpm6th 9098 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 9099 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0
 
Theorem2mulicn 9100 (2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ∈ ℂ
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