P. 122 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	times2i 12101	A number times 2. (Contributed by NM, 11-May-2004.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 2) = (𝐴 + 𝐴)
Theorem	2txmxeqx 12102	Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ (𝑋 ∈ ℂ → ((2 · 𝑋) − 𝑋) = 𝑋)
Theorem	2div2e1 12103	2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 / 2) = 1
Theorem	2p1e3 12104	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (2 + 1) = 3
Theorem	1p2e3 12105	1 + 2 = 3. For a shorter proof using addcomli 11156, see 1p2e3ALT 12106. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.)
⊢ (1 + 2) = 3
Theorem	1p2e3ALT 12106	Alternate proof of 1p2e3 12105, shorter but using more axioms. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (1 + 2) = 3
Theorem	3p1e4 12107	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (3 + 1) = 4
Theorem	4p1e5 12108	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (4 + 1) = 5
Theorem	5p1e6 12109	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (5 + 1) = 6
Theorem	6p1e7 12110	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (6 + 1) = 7
Theorem	7p1e8 12111	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (7 + 1) = 8
Theorem	8p1e9 12112	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (8 + 1) = 9
Theorem	3p2e5 12113	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
⊢ (3 + 2) = 5
Theorem	3p3e6 12114	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
⊢ (3 + 3) = 6
Theorem	4p2e6 12115	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
⊢ (4 + 2) = 6
Theorem	4p3e7 12116	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7
Theorem	4p4e8 12117	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
⊢ (4 + 4) = 8
Theorem	5p2e7 12118	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
⊢ (5 + 2) = 7
Theorem	5p3e8 12119	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
⊢ (5 + 3) = 8
Theorem	5p4e9 12120	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
⊢ (5 + 4) = 9
Theorem	6p2e8 12121	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
⊢ (6 + 2) = 8
Theorem	6p3e9 12122	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
⊢ (6 + 3) = 9
Theorem	7p2e9 12123	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
⊢ (7 + 2) = 9
Theorem	1t1e1 12124	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 · 1) = 1
Theorem	2t1e2 12125	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (2 · 1) = 2
Theorem	2t2e4 12126	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
⊢ (2 · 2) = 4
Theorem	3t1e3 12127	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (3 · 1) = 3
Theorem	3t2e6 12128	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
⊢ (3 · 2) = 6
Theorem	3t3e9 12129	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
⊢ (3 · 3) = 9
Theorem	4t2e8 12130	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
⊢ (4 · 2) = 8
Theorem	2t0e0 12131	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · 0) = 0
Theorem	4d2e2 12132	One half of four is two. (Contributed by NM, 3-Sep-1999.)
⊢ (4 / 2) = 2
Theorem	1lt2 12133	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
⊢ 1 < 2
Theorem	2lt3 12134	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 2 < 3
Theorem	1lt3 12135	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 1 < 3
Theorem	3lt4 12136	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 4
Theorem	2lt4 12137	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 4
Theorem	1lt4 12138	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 4
Theorem	4lt5 12139	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 5
Theorem	3lt5 12140	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 5
Theorem	2lt5 12141	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 5
Theorem	1lt5 12142	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 5
Theorem	5lt6 12143	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 12144	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 12145	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 12146	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 12147	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 12148	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 12149	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 12150	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 12151	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 12152	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 12153	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 12154	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 12155	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 12156	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 12157	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 12158	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 12159	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 12160	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 12161	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 12162	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 12163	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 12164	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 12165	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 12166	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 12167	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 12168	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9
Theorem	0ne2 12169	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2
Theorem	1ne2 12170	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
⊢ 1 ≠ 2
Theorem	1le2 12171	1 is less than or equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 2
Theorem	2cnne0 12172	2 is a nonzero complex number. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
Theorem	2rene0 12173	2 is a nonzero real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 ∈ ℝ ∧ 2 ≠ 0)
Theorem	1le3 12174	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 ≤ 3
Theorem	neg1mulneg1e1 12175	-1 · -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1 · -1) = 1
Theorem	halfre 12176	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℝ
Theorem	halfcn 12177	One-half is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 / 2) ∈ ℂ
Theorem	halfgt0 12178	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
⊢ 0 < (1 / 2)
Theorem	halfge0 12179	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
⊢ 0 ≤ (1 / 2)
Theorem	halflt1 12180	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
⊢ (1 / 2) < 1
Theorem	1mhlfehlf 12181	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (1 − (1 / 2)) = (1 / 2)
Theorem	8th4div3 12182	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
⊢ ((1 / 8) · (4 / 3)) = (1 / 6)
Theorem	halfpm6th 12183	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 12184	i times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (i · 0) = 0
Theorem	2mulicn 12185	(2 · i) ∈ ℂ. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ∈ ℂ
Theorem	2muline0 12186	(2 · i) ≠ 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · i) ≠ 0
5.4.5  Simple number properties
Theorem	halfcl 12187	Closure of half of a number. (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)
Theorem	rehalfcl 12188	Real closure of half. (Contributed by NM, 1-Jan-2006.)
⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)
Theorem	half0 12189	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 12190	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
⊢ (𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)
Theorem	halfpos2 12191	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))
Theorem	halfpos 12192	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 12193	A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))
Theorem	halfaddsubcl 12194	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 12195	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	subhalfhalf 12196	Subtracting the half of a number from the number yields the half of the number. (Contributed by AV, 28-Jun-2021.)
⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2))
Theorem	lt2halves 12197	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 12198	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 12199*	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 12200	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))