P. 79 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	6p4e10 7801	6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)
⊢ (6 + 4) = 10
Theorem	7p2e9 7802	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
⊢ (7 + 2) = 9
Theorem	7p3e10 7803	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)
⊢ (7 + 3) = 10
Theorem	8p2e10 7804	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)
⊢ (8 + 2) = 10
Theorem	1t1e1 7805	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 · 1) = 1
Theorem	2t1e2 7806	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (2 · 1) = 2
Theorem	2t2e4 7807	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
⊢ (2 · 2) = 4
Theorem	3t1e3 7808	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (3 · 1) = 3
Theorem	3t2e6 7809	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
⊢ (3 · 2) = 6
Theorem	3t3e9 7810	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
⊢ (3 · 3) = 9
Theorem	4t2e8 7811	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
⊢ (4 · 2) = 8
Theorem	5t2e10 7812	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)
⊢ (5 · 2) = 10
Theorem	2t0e0 7813	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · 0) = 0
Theorem	4d2e2 7814	One half of four is two. (Contributed by NM, 3-Sep-1999.)
⊢ (4 / 2) = 2
Theorem	2nn 7815	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	3nn 7816	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	4nn 7817	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	5nn 7818	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	6nn 7819	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	7nn 7820	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	8nn 7821	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	9nn 7822	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	10nn 7823	10 is a positive integer. (Contributed by NM, 8-Nov-2012.)
⊢ 10 ∈ ℕ
Theorem	1lt2 7824	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
⊢ 1 < 2
Theorem	2lt3 7825	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 2 < 3
Theorem	1lt3 7826	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 1 < 3
Theorem	3lt4 7827	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 4
Theorem	2lt4 7828	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 4
Theorem	1lt4 7829	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 4
Theorem	4lt5 7830	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 5
Theorem	3lt5 7831	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 5
Theorem	2lt5 7832	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 5
Theorem	1lt5 7833	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 5
Theorem	5lt6 7834	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 7835	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 7836	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 7837	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 7838	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 7839	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 7840	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 7841	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 7842	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 7843	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 7844	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 7845	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 7846	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 7847	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 7848	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 7849	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 7850	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 7851	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 7852	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 7853	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 7854	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 7855	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 7856	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 7857	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 7858	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 7859	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9
Theorem	9lt10 7860	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 9 < 10
Theorem	8lt10 7861	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 8 < 10
Theorem	7lt10 7862	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 7 < 10
Theorem	6lt10 7863	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 6 < 10
Theorem	5lt10 7864	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 5 < 10
Theorem	4lt10 7865	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 4 < 10
Theorem	3lt10 7866	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 3 < 10
Theorem	2lt10 7867	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ 2 < 10
Theorem	1lt10 7868	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 10
Theorem	0ne2 7869	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2
Theorem	1ne2 7870	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
⊢ 1 ≠ 2
Theorem	1le2 7871	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 2
Theorem	2cnne0 7872	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
Theorem	2rene0 7873	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 7874	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 7875	-1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 7876	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 7877	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 7878	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halflt1 7879	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 7880	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 7881	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 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))
Theorem	it0e0 7883	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0
Theorem	2mulicn 7884	(2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ∈ ℂ
Theorem	iap0 7885	The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ i # 0
Theorem	2muliap0 7886	2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ (2 · i) # 0
Theorem	2muline0 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
Theorem	halfcl 7888	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
⊢ (A ∈ ℂ → (A / 2) ∈ ℂ)
Theorem	rehalfcl 7889	Real closure of half. (Contributed by NM, 1-Jan-2006.)
⊢ (A ∈ ℝ → (A / 2) ∈ ℝ)
Theorem	half0 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))
Theorem	2halves 7891	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
⊢ (A ∈ ℂ → ((A / 2) + (A / 2)) = A)
Theorem	halfpos2 7892	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
⊢ (A ∈ ℝ → (0 < A ↔ 0 < (A / 2)))
Theorem	halfpos 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))
Theorem	halfnneg2 7894	A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
⊢ (A ∈ ℝ → (0 ≤ A ↔ 0 ≤ (A / 2)))
Theorem	halfaddsubcl 7895	Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (((A + B) / 2) ∈ ℂ ∧ ((A − B) / 2) ∈ ℂ))
Theorem	halfaddsub 7896	Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → ((((A + B) / 2) + ((A − B) / 2)) = A ∧ (((A + B) / 2) − ((A − B) / 2)) = B))
Theorem	lt2halves 7897	A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((A < (𝐶 / 2) ∧ B < (𝐶 / 2)) → (A + B) < 𝐶))
Theorem	addltmul 7898	Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (2 < A ∧ 2 < B)) → (A + B) < (A · B))
Theorem	nominpos 7899*	There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
⊢ ¬ ∃x ∈ ℝ (0 < x ∧ ¬ ∃y ∈ ℝ (0 < y ∧ y < x))
Theorem	avglt1 7900	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A < B ↔ A < ((A + B) / 2)))