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Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1m0e1 8801 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1
 
Theorem0p1e1 8802 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1
 
Theoremfv0p1e1 8803 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))
 
Theorem1p0e1 8804 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1
 
Theorem1p1e2 8805 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2
 
Theorem2m1e1 8806 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8833. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1
 
Theorem1e2m1 8807 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)
 
Theorem3m1e2 8808 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2
 
Theorem4m1e3 8809 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
(4 − 1) = 3
 
Theorem5m1e4 8810 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
(5 − 1) = 4
 
Theorem6m1e5 8811 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
(6 − 1) = 5
 
Theorem7m1e6 8812 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
(7 − 1) = 6
 
Theorem8m1e7 8813 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
(8 − 1) = 7
 
Theorem9m1e8 8814 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
(9 − 1) = 8
 
Theorem2p2e4 8815 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 8816 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 8817 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
 
Theorem2timesi 8818 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)
 
Theoremtimes2i 8819 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)
 
Theorem2div2e1 8820 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1
 
Theorem2p1e3 8821 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3
 
Theorem1p2e3 8822 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3
 
Theorem3p1e4 8823 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4
 
Theorem4p1e5 8824 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5
 
Theorem5p1e6 8825 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6
 
Theorem6p1e7 8826 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7
 
Theorem7p1e8 8827 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8
 
Theorem8p1e9 8828 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9
 
Theorem3p2e5 8829 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5
 
Theorem3p3e6 8830 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6
 
Theorem4p2e6 8831 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6
 
Theorem4p3e7 8832 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7
 
Theorem4p4e8 8833 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8
 
Theorem5p2e7 8834 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7
 
Theorem5p3e8 8835 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8
 
Theorem5p4e9 8836 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9
 
Theorem6p2e8 8837 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8
 
Theorem6p3e9 8838 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9
 
Theorem7p2e9 8839 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9
 
Theorem1t1e1 8840 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1
 
Theorem2t1e2 8841 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2
 
Theorem2t2e4 8842 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4
 
Theorem3t1e3 8843 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3
 
Theorem3t2e6 8844 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6
 
Theorem3t3e9 8845 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9
 
Theorem4t2e8 8846 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8
 
Theorem2t0e0 8847 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0
 
Theorem4d2e2 8848 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2
 
Theorem2nn 8849 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ
 
Theorem3nn 8850 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ
 
Theorem4nn 8851 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ
 
Theorem5nn 8852 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ
 
Theorem6nn 8853 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ
 
Theorem7nn 8854 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ
 
Theorem8nn 8855 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ
 
Theorem9nn 8856 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ
 
Theorem1lt2 8857 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2
 
Theorem2lt3 8858 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3
 
Theorem1lt3 8859 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3
 
Theorem3lt4 8860 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4
 
Theorem2lt4 8861 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4
 
Theorem1lt4 8862 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4
 
Theorem4lt5 8863 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5
 
Theorem3lt5 8864 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5
 
Theorem2lt5 8865 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5
 
Theorem1lt5 8866 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5
 
Theorem5lt6 8867 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6
 
Theorem4lt6 8868 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6
 
Theorem3lt6 8869 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6
 
Theorem2lt6 8870 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6
 
Theorem1lt6 8871 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6
 
Theorem6lt7 8872 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7
 
Theorem5lt7 8873 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7
 
Theorem4lt7 8874 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7
 
Theorem3lt7 8875 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7
 
Theorem2lt7 8876 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7
 
Theorem1lt7 8877 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7
 
Theorem7lt8 8878 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8
 
Theorem6lt8 8879 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8
 
Theorem5lt8 8880 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8
 
Theorem4lt8 8881 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8
 
Theorem3lt8 8882 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8
 
Theorem2lt8 8883 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8
 
Theorem1lt8 8884 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8
 
Theorem8lt9 8885 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9
 
Theorem7lt9 8886 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9
 
Theorem6lt9 8887 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9
 
Theorem5lt9 8888 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9
 
Theorem4lt9 8889 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9
 
Theorem3lt9 8890 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9
 
Theorem2lt9 8891 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9
 
Theorem1lt9 8892 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9
 
Theorem0ne2 8893 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2
 
Theorem1ne2 8894 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2
 
Theorem1ap2 8895 1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
1 # 2
 
Theorem1le2 8896 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2
 
Theorem2cnne0 8897 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 ∈ ℂ ∧ 2 ≠ 0)
 
Theorem2rene0 8898 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 ∈ ℝ ∧ 2 ≠ 0)
 
Theorem1le3 8899 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3
 
Theoremneg1mulneg1e1 8900 -1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1
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