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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0m0e0 8101 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0
 
Theorem1m0e1 8102 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1
 
Theorem0p1e1 8103 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1
 
Theorem1p0e1 8104 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1
 
Theorem1p1e2 8105 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2
 
Theorem2m1e1 8106 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8127. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1
 
Theorem1e2m1 8107 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)
 
Theorem3m1e2 8108 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2
 
Theorem2p2e4 8109 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
(2 + 2) = 4
 
Theorem2times 8110 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 8111 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
 
Theorem2timesi 8112 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)
 
Theoremtimes2i 8113 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)
 
Theorem2div2e1 8114 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1
 
Theorem2p1e3 8115 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3
 
Theorem1p2e3 8116 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3
 
Theorem3p1e4 8117 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4
 
Theorem4p1e5 8118 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5
 
Theorem5p1e6 8119 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6
 
Theorem6p1e7 8120 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7
 
Theorem7p1e8 8121 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8
 
Theorem8p1e9 8122 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9
 
Theorem3p2e5 8123 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5
 
Theorem3p3e6 8124 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6
 
Theorem4p2e6 8125 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6
 
Theorem4p3e7 8126 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7
 
Theorem4p4e8 8127 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8
 
Theorem5p2e7 8128 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7
 
Theorem5p3e8 8129 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8
 
Theorem5p4e9 8130 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9
 
Theorem6p2e8 8131 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8
 
Theorem6p3e9 8132 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9
 
Theorem7p2e9 8133 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9
 
Theorem1t1e1 8134 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1
 
Theorem2t1e2 8135 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2
 
Theorem2t2e4 8136 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4
 
Theorem3t1e3 8137 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3
 
Theorem3t2e6 8138 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6
 
Theorem3t3e9 8139 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9
 
Theorem4t2e8 8140 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8
 
Theorem2t0e0 8141 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0
 
Theorem4d2e2 8142 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2
 
Theorem2nn 8143 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ
 
Theorem3nn 8144 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ
 
Theorem4nn 8145 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ
 
Theorem5nn 8146 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ
 
Theorem6nn 8147 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ
 
Theorem7nn 8148 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ
 
Theorem8nn 8149 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ
 
Theorem9nn 8150 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ
 
Theorem1lt2 8151 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2
 
Theorem2lt3 8152 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3
 
Theorem1lt3 8153 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3
 
Theorem3lt4 8154 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4
 
Theorem2lt4 8155 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4
 
Theorem1lt4 8156 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4
 
Theorem4lt5 8157 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5
 
Theorem3lt5 8158 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5
 
Theorem2lt5 8159 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5
 
Theorem1lt5 8160 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5
 
Theorem5lt6 8161 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6
 
Theorem4lt6 8162 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6
 
Theorem3lt6 8163 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6
 
Theorem2lt6 8164 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6
 
Theorem1lt6 8165 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6
 
Theorem6lt7 8166 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7
 
Theorem5lt7 8167 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7
 
Theorem4lt7 8168 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7
 
Theorem3lt7 8169 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7
 
Theorem2lt7 8170 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7
 
Theorem1lt7 8171 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7
 
Theorem7lt8 8172 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8
 
Theorem6lt8 8173 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8
 
Theorem5lt8 8174 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8
 
Theorem4lt8 8175 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8
 
Theorem3lt8 8176 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8
 
Theorem2lt8 8177 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8
 
Theorem1lt8 8178 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8
 
Theorem8lt9 8179 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9
 
Theorem7lt9 8180 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9
 
Theorem6lt9 8181 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9
 
Theorem5lt9 8182 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9
 
Theorem4lt9 8183 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9
 
Theorem3lt9 8184 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9
 
Theorem2lt9 8185 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9
 
Theorem1lt9 8186 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9
 
Theorem0ne2 8187 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2
 
Theorem1ne2 8188 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2
 
Theorem1le2 8189 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2
 
Theorem2cnne0 8190 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 ∈ ℂ ∧ 2 ≠ 0)
 
Theorem2rene0 8191 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 ∈ ℝ ∧ 2 ≠ 0)
 
Theorem1le3 8192 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3
 
Theoremneg1mulneg1e1 8193 -1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1
 
Theoremhalfre 8194 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℝ
 
Theoremhalfcn 8195 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℂ
 
Theoremhalfgt0 8196 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)
 
Theoremhalfge0 8197 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)
 
Theoremhalflt1 8198 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1
 
Theorem1mhlfehlf 8199 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)
 
Theorem8th4div3 8200 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)
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