Theorem List for Intuitionistic Logic Explorer - 7801-7900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | neg1rr 7801 |
-1 is a real number (common case). (Contributed by David A. Wheeler,
5-Dec-2018.)
|
⊢ -1 ∈
ℝ |
|
Theorem | neg1ne0 7802 |
-1 is nonzero (common case). (Contributed by David A. Wheeler,
8-Dec-2018.)
|
⊢ -1 ≠ 0 |
|
Theorem | neg1lt0 7803 |
-1 is less than 0 (common case). (Contributed by David A. Wheeler,
8-Dec-2018.)
|
⊢ -1 < 0 |
|
Theorem | neg1ap0 7804 |
-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|
⊢ -1 # 0 |
|
Theorem | negneg1e1 7805 |
--1 is 1 (common case). (Contributed by David A.
Wheeler,
8-Dec-2018.)
|
⊢ --1 = 1 |
|
Theorem | 1pneg1e0 7806 |
1 + -1 is 0 (common case). (Contributed by David A.
Wheeler,
8-Dec-2018.)
|
⊢ (1 + -1) = 0 |
|
Theorem | 0m0e0 7807 |
0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler,
8-Dec-2018.)
|
⊢ (0 − 0) = 0 |
|
Theorem | 1m0e1 7808 |
1 - 0 = 1 (common case). (Contributed by David A. Wheeler,
8-Dec-2018.)
|
⊢ (1 − 0) = 1 |
|
Theorem | 0p1e1 7809 |
0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|
⊢ (0 + 1) = 1 |
|
Theorem | 1p0e1 7810 |
1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
|
⊢ (1 + 0) = 1 |
|
Theorem | 1p1e2 7811 |
1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
|
⊢ (1 + 1) = 2 |
|
Theorem | 2m1e1 7812 |
2 - 1 = 1. The result is on the right-hand-side to be consistent with
similar proofs like 4p4e8 7834. (Contributed by David A. Wheeler,
4-Jan-2017.)
|
⊢ (2 − 1) = 1 |
|
Theorem | 1e2m1 7813 |
1 = 2 - 1 (common case). (Contributed by David A. Wheeler,
8-Dec-2018.)
|
⊢ 1 = (2 − 1) |
|
Theorem | 3m1e2 7814 |
3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM,
10-Dec-2017.)
|
⊢ (3 − 1) = 2 |
|
Theorem | 2p2e4 7815 |
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 |
|
Theorem | 2times 7816 |
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 7817 |
A number times 2. (Contributed by NM, 16-Oct-2007.)
|
⊢ (A ∈ ℂ → (A · 2) = (A + A)) |
|
Theorem | 2timesi 7818 |
Two times a number. (Contributed by NM, 1-Aug-1999.)
|
⊢ A ∈ ℂ ⇒ ⊢ (2 · A) = (A +
A) |
|
Theorem | times2i 7819 |
A number times 2. (Contributed by NM, 11-May-2004.)
|
⊢ A ∈ ℂ ⇒ ⊢ (A · 2) = (A + A) |
|
Theorem | 2div2e1 7820 |
2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler,
8-Dec-2018.)
|
⊢ (2 / 2) = 1 |
|
Theorem | 2p1e3 7821 |
2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (2 + 1) = 3 |
|
Theorem | 1p2e3 7822 |
1 + 2 = 3 (common case). (Contributed by David A. Wheeler,
8-Dec-2018.)
|
⊢ (1 + 2) = 3 |
|
Theorem | 3p1e4 7823 |
3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (3 + 1) = 4 |
|
Theorem | 4p1e5 7824 |
4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (4 + 1) = 5 |
|
Theorem | 5p1e6 7825 |
5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (5 + 1) = 6 |
|
Theorem | 6p1e7 7826 |
6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (6 + 1) = 7 |
|
Theorem | 7p1e8 7827 |
7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (7 + 1) = 8 |
|
Theorem | 8p1e9 7828 |
8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (8 + 1) = 9 |
|
Theorem | 9p1e10 7829 |
9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)
|
⊢ (9 + 1) = 10 |
|
Theorem | 3p2e5 7830 |
3 + 2 = 5. (Contributed by NM, 11-May-2004.)
|
⊢ (3 + 2) = 5 |
|
Theorem | 3p3e6 7831 |
3 + 3 = 6. (Contributed by NM, 11-May-2004.)
|
⊢ (3 + 3) = 6 |
|
Theorem | 4p2e6 7832 |
4 + 2 = 6. (Contributed by NM, 11-May-2004.)
|
⊢ (4 + 2) = 6 |
|
Theorem | 4p3e7 7833 |
4 + 3 = 7. (Contributed by NM, 11-May-2004.)
|
⊢ (4 + 3) = 7 |
|
Theorem | 4p4e8 7834 |
4 + 4 = 8. (Contributed by NM, 11-May-2004.)
|
⊢ (4 + 4) = 8 |
|
Theorem | 5p2e7 7835 |
5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|
⊢ (5 + 2) = 7 |
|
Theorem | 5p3e8 7836 |
5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|
⊢ (5 + 3) = 8 |
|
Theorem | 5p4e9 7837 |
5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|
⊢ (5 + 4) = 9 |
|
Theorem | 5p5e10 7838 |
5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)
|
⊢ (5 + 5) = 10 |
|
Theorem | 6p2e8 7839 |
6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|
⊢ (6 + 2) = 8 |
|
Theorem | 6p3e9 7840 |
6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|
⊢ (6 + 3) = 9 |
|
Theorem | 6p4e10 7841 |
6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)
|
⊢ (6 + 4) = 10 |
|
Theorem | 7p2e9 7842 |
7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|
⊢ (7 + 2) = 9 |
|
Theorem | 7p3e10 7843 |
7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)
|
⊢ (7 + 3) = 10 |
|
Theorem | 8p2e10 7844 |
8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)
|
⊢ (8 + 2) = 10 |
|
Theorem | 1t1e1 7845 |
1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|
⊢ (1 · 1) = 1 |
|
Theorem | 2t1e2 7846 |
2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|
⊢ (2 · 1) = 2 |
|
Theorem | 2t2e4 7847 |
2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|
⊢ (2 · 2) = 4 |
|
Theorem | 3t1e3 7848 |
3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|
⊢ (3 · 1) = 3 |
|
Theorem | 3t2e6 7849 |
3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|
⊢ (3 · 2) = 6 |
|
Theorem | 3t3e9 7850 |
3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|
⊢ (3 · 3) = 9 |
|
Theorem | 4t2e8 7851 |
4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|
⊢ (4 · 2) = 8 |
|
Theorem | 5t2e10 7852 |
5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)
|
⊢ (5 · 2) = 10 |
|
Theorem | 2t0e0 7853 |
2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|
⊢ (2 · 0) = 0 |
|
Theorem | 4d2e2 7854 |
One half of four is two. (Contributed by NM, 3-Sep-1999.)
|
⊢ (4 / 2) = 2 |
|
Theorem | 2nn 7855 |
2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|
⊢ 2 ∈
ℕ |
|
Theorem | 3nn 7856 |
3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|
⊢ 3 ∈
ℕ |
|
Theorem | 4nn 7857 |
4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|
⊢ 4 ∈
ℕ |
|
Theorem | 5nn 7858 |
5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 5 ∈
ℕ |
|
Theorem | 6nn 7859 |
6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 6 ∈
ℕ |
|
Theorem | 7nn 7860 |
7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 7 ∈
ℕ |
|
Theorem | 8nn 7861 |
8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 8 ∈
ℕ |
|
Theorem | 9nn 7862 |
9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|
⊢ 9 ∈
ℕ |
|
Theorem | 10nn 7863 |
10 is a positive integer. (Contributed by NM, 8-Nov-2012.)
|
⊢ 10 ∈
ℕ |
|
Theorem | 1lt2 7864 |
1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|
⊢ 1 < 2 |
|
Theorem | 2lt3 7865 |
2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|
⊢ 2 < 3 |
|
Theorem | 1lt3 7866 |
1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|
⊢ 1 < 3 |
|
Theorem | 3lt4 7867 |
3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 3 < 4 |
|
Theorem | 2lt4 7868 |
2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 2 < 4 |
|
Theorem | 1lt4 7869 |
1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 1 < 4 |
|
Theorem | 4lt5 7870 |
4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 4 < 5 |
|
Theorem | 3lt5 7871 |
3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 3 < 5 |
|
Theorem | 2lt5 7872 |
2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 2 < 5 |
|
Theorem | 1lt5 7873 |
1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 1 < 5 |
|
Theorem | 5lt6 7874 |
5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 5 < 6 |
|
Theorem | 4lt6 7875 |
4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 4 < 6 |
|
Theorem | 3lt6 7876 |
3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 3 < 6 |
|
Theorem | 2lt6 7877 |
2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 2 < 6 |
|
Theorem | 1lt6 7878 |
1 is less than 6. (Contributed by NM, 19-Oct-2012.)
|
⊢ 1 < 6 |
|
Theorem | 6lt7 7879 |
6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 6 < 7 |
|
Theorem | 5lt7 7880 |
5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 5 < 7 |
|
Theorem | 4lt7 7881 |
4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 4 < 7 |
|
Theorem | 3lt7 7882 |
3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 3 < 7 |
|
Theorem | 2lt7 7883 |
2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 2 < 7 |
|
Theorem | 1lt7 7884 |
1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 1 < 7 |
|
Theorem | 7lt8 7885 |
7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 7 < 8 |
|
Theorem | 6lt8 7886 |
6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 6 < 8 |
|
Theorem | 5lt8 7887 |
5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 5 < 8 |
|
Theorem | 4lt8 7888 |
4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 4 < 8 |
|
Theorem | 3lt8 7889 |
3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 3 < 8 |
|
Theorem | 2lt8 7890 |
2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 2 < 8 |
|
Theorem | 1lt8 7891 |
1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|
⊢ 1 < 8 |
|
Theorem | 8lt9 7892 |
8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
|
⊢ 8 < 9 |
|
Theorem | 7lt9 7893 |
7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|
⊢ 7 < 9 |
|
Theorem | 6lt9 7894 |
6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|
⊢ 6 < 9 |
|
Theorem | 5lt9 7895 |
5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|
⊢ 5 < 9 |
|
Theorem | 4lt9 7896 |
4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|
⊢ 4 < 9 |
|
Theorem | 3lt9 7897 |
3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|
⊢ 3 < 9 |
|
Theorem | 2lt9 7898 |
2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|
⊢ 2 < 9 |
|
Theorem | 1lt9 7899 |
1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario
Carneiro, 9-Mar-2015.)
|
⊢ 1 < 9 |
|
Theorem | 9lt10 7900 |
9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
|
⊢ 9 < 10 |