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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem6p4e10 7801 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)
(6 + 4) = 10
 
Theorem7p2e9 7802 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9
 
Theorem7p3e10 7803 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)
(7 + 3) = 10
 
Theorem8p2e10 7804 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)
(8 + 2) = 10
 
Theorem1t1e1 7805 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1
 
Theorem2t1e2 7806 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2
 
Theorem2t2e4 7807 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4
 
Theorem3t1e3 7808 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3
 
Theorem3t2e6 7809 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6
 
Theorem3t3e9 7810 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9
 
Theorem4t2e8 7811 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8
 
Theorem5t2e10 7812 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)
(5 · 2) = 10
 
Theorem2t0e0 7813 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0
 
Theorem4d2e2 7814 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2
 
Theorem2nn 7815 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2
 
Theorem3nn 7816 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3
 
Theorem4nn 7817 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4
 
Theorem5nn 7818 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5
 
Theorem6nn 7819 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6
 
Theorem7nn 7820 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7
 
Theorem8nn 7821 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8
 
Theorem9nn 7822 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9
 
Theorem10nn 7823 10 is a positive integer. (Contributed by NM, 8-Nov-2012.)
10
 
Theorem1lt2 7824 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2
 
Theorem2lt3 7825 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3
 
Theorem1lt3 7826 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3
 
Theorem3lt4 7827 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4
 
Theorem2lt4 7828 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4
 
Theorem1lt4 7829 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4
 
Theorem4lt5 7830 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5
 
Theorem3lt5 7831 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5
 
Theorem2lt5 7832 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5
 
Theorem1lt5 7833 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5
 
Theorem5lt6 7834 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6
 
Theorem4lt6 7835 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6
 
Theorem3lt6 7836 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6
 
Theorem2lt6 7837 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6
 
Theorem1lt6 7838 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6
 
Theorem6lt7 7839 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7
 
Theorem5lt7 7840 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7
 
Theorem4lt7 7841 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7
 
Theorem3lt7 7842 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7
 
Theorem2lt7 7843 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7
 
Theorem1lt7 7844 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7
 
Theorem7lt8 7845 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8
 
Theorem6lt8 7846 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8
 
Theorem5lt8 7847 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8
 
Theorem4lt8 7848 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8
 
Theorem3lt8 7849 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8
 
Theorem2lt8 7850 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8
 
Theorem1lt8 7851 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8
 
Theorem8lt9 7852 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9
 
Theorem7lt9 7853 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9
 
Theorem6lt9 7854 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9
 
Theorem5lt9 7855 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9
 
Theorem4lt9 7856 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9
 
Theorem3lt9 7857 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9
 
Theorem2lt9 7858 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9
 
Theorem1lt9 7859 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9
 
Theorem9lt10 7860 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
9 < 10
 
Theorem8lt10 7861 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
8 < 10
 
Theorem7lt10 7862 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
7 < 10
 
Theorem6lt10 7863 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
6 < 10
 
Theorem5lt10 7864 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
5 < 10
 
Theorem4lt10 7865 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
4 < 10
 
Theorem3lt10 7866 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
3 < 10
 
Theorem2lt10 7867 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
2 < 10
 
Theorem1lt10 7868 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 10
 
Theorem0ne2 7869 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2
 
Theorem1ne2 7870 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2
 
Theorem1le2 7871 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2
 
Theorem2cnne0 7872 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 2 ≠ 0)
 
Theorem2rene0 7873 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 2 ≠ 0)
 
Theorem1le3 7874 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3
 
Theoremneg1mulneg1e1 7875 -1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1
 
Theoremhalfre 7876 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2)
 
Theoremhalfcn 7877 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2)
 
Theoremhalfgt0 7878 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)
 
Theoremhalflt1 7879 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1
 
Theorem1mhlfehlf 7880 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)
 
Theorem8th4div3 7881 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)
 
Theoremhalfpm6th 7882 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 7883 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0
 
Theorem2mulicn 7884 (2 · i) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i)
 
Theoremiap0 7885 The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
i # 0
 
Theorem2muliap0 7886 2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
(2 · i) # 0
 
Theorem2muline0 7887 (2 · i) ≠ 0. See also 2muliap0 7886. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0
 
3.4.5  Simple number properties
 
Theoremhalfcl 7888 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
(A ℂ → (A / 2) ℂ)
 
Theoremrehalfcl 7889 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(A ℝ → (A / 2) ℝ)
 
Theoremhalf0 7890 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
(A ℂ → ((A / 2) = 0 ↔ A = 0))
 
Theorem2halves 7891 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(A ℂ → ((A / 2) + (A / 2)) = A)
 
Theoremhalfpos2 7892 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(A ℝ → (0 < A ↔ 0 < (A / 2)))
 
Theoremhalfpos 7893 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(A ℝ → (0 < A ↔ (A / 2) < A))
 
Theoremhalfnneg2 7894 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
(A ℝ → (0 ≤ A ↔ 0 ≤ (A / 2)))
 
Theoremhalfaddsubcl 7895 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((A B ℂ) → (((A + B) / 2) ((AB) / 2) ℂ))
 
Theoremhalfaddsub 7896 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((A B ℂ) → ((((A + B) / 2) + ((AB) / 2)) = A (((A + B) / 2) − ((AB) / 2)) = B))
 
Theoremlt2halves 7897 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
((A B 𝐶 ℝ) → ((A < (𝐶 / 2) B < (𝐶 / 2)) → (A + B) < 𝐶))
 
Theoremaddltmul 7898 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
(((A B ℝ) (2 < A 2 < B)) → (A + B) < (A · B))
 
Theoremnominpos 7899* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
¬ x ℝ (0 < x ¬ y ℝ (0 < y y < x))
 
Theoremavglt1 7900 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((A B ℝ) → (A < BA < ((A + B) / 2)))
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