P. 125 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12401-12500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	4ne0 12401	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 12402	The number 5 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 5
Theorem	6pos 12403	The number 6 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 6
Theorem	7pos 12404	The number 7 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 7
Theorem	8pos 12405	The number 8 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 8
Theorem	9pos 12406	The number 9 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 9
5.4.4  Some properties of specific numbers This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering).
Theorem	neg1cn 12407	-1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ -1 ∈ ℂ
Theorem	neg1rr 12408	-1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℝ
Theorem	neg1ne0 12409	-1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 ≠ 0
Theorem	neg1lt0 12410	-1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 < 0
Theorem	negneg1e1 12411	--1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ --1 = 1
Theorem	1pneg1e0 12412	1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + -1) = 0
Theorem	0m0e0 12413	0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0 − 0) = 0
Theorem	1m0e1 12414	1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 − 0) = 1
Theorem	0p1e1 12415	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (0 + 1) = 1
Theorem	fv0p1e1 12416	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))
Theorem	1p0e1 12417	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 0) = 1
Theorem	1p1e2 12418	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
⊢ (1 + 1) = 2
Theorem	2m1e1 12419	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12448. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (2 − 1) = 1
Theorem	1e2m1 12420	1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 = (2 − 1)
Theorem	3m1e2 12421	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (3 − 1) = 2
Theorem	4m1e3 12422	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (4 − 1) = 3
Theorem	5m1e4 12423	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
⊢ (5 − 1) = 4
Theorem	6m1e5 12424	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
⊢ (6 − 1) = 5
Theorem	7m1e6 12425	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
⊢ (7 − 1) = 6
Theorem	8m1e7 12426	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
⊢ (8 − 1) = 7
Theorem	9m1e8 12427	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
⊢ (9 − 1) = 8
Theorem	2p2e4 12428	Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 8597 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.)
⊢ (2 + 2) = 4
Theorem	2times 12429	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 12430	A number times 2. (Contributed by NM, 16-Oct-2007.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	2timesi 12431	Two times a number. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (2 · 𝐴) = (𝐴 + 𝐴)
Theorem	times2i 12432	A number times 2. (Contributed by NM, 11-May-2004.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 2) = (𝐴 + 𝐴)
Theorem	2txmxeqx 12433	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 12434	2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 / 2) = 1
Theorem	2p1e3 12435	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (2 + 1) = 3
Theorem	1p2e3 12436	1 + 2 = 3. For a shorter proof using addcomli 11482, see 1p2e3ALT 12437. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.)
⊢ (1 + 2) = 3
Theorem	1p2e3ALT 12437	Alternate proof of 1p2e3 12436, 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 12438	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (3 + 1) = 4
Theorem	4p1e5 12439	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (4 + 1) = 5
Theorem	5p1e6 12440	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (5 + 1) = 6
Theorem	6p1e7 12441	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (6 + 1) = 7
Theorem	7p1e8 12442	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (7 + 1) = 8
Theorem	8p1e9 12443	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (8 + 1) = 9
Theorem	3p2e5 12444	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
⊢ (3 + 2) = 5
Theorem	3p3e6 12445	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
⊢ (3 + 3) = 6
Theorem	4p2e6 12446	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
⊢ (4 + 2) = 6
Theorem	4p3e7 12447	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7
Theorem	4p4e8 12448	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
⊢ (4 + 4) = 8
Theorem	5p2e7 12449	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
⊢ (5 + 2) = 7
Theorem	5p3e8 12450	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
⊢ (5 + 3) = 8
Theorem	5p4e9 12451	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
⊢ (5 + 4) = 9
Theorem	6p2e8 12452	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
⊢ (6 + 2) = 8
Theorem	6p3e9 12453	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
⊢ (6 + 3) = 9
Theorem	7p2e9 12454	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
⊢ (7 + 2) = 9
Theorem	1t1e1 12455	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 · 1) = 1
Theorem	2t1e2 12456	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (2 · 1) = 2
Theorem	2t2e4 12457	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
⊢ (2 · 2) = 4
Theorem	3t1e3 12458	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (3 · 1) = 3
Theorem	3t2e6 12459	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
⊢ (3 · 2) = 6
Theorem	3t3e9 12460	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
⊢ (3 · 3) = 9
Theorem	4t2e8 12461	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
⊢ (4 · 2) = 8
Theorem	2t0e0 12462	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 · 0) = 0
Theorem	4d2e2 12463	One half of four is two. (Contributed by NM, 3-Sep-1999.)
⊢ (4 / 2) = 2
Theorem	1lt2 12464	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
⊢ 1 < 2
Theorem	2lt3 12465	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 2 < 3
Theorem	1lt3 12466	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
⊢ 1 < 3
Theorem	3lt4 12467	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 4
Theorem	2lt4 12468	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 4
Theorem	1lt4 12469	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 4
Theorem	4lt5 12470	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 5
Theorem	3lt5 12471	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 5
Theorem	2lt5 12472	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 5
Theorem	1lt5 12473	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 5
Theorem	5lt6 12474	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 6
Theorem	4lt6 12475	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 6
Theorem	3lt6 12476	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 6
Theorem	2lt6 12477	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 6
Theorem	1lt6 12478	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
⊢ 1 < 6
Theorem	6lt7 12479	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 7
Theorem	5lt7 12480	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 7
Theorem	4lt7 12481	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 7
Theorem	3lt7 12482	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 7
Theorem	2lt7 12483	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 7
Theorem	1lt7 12484	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 7
Theorem	7lt8 12485	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 < 8
Theorem	6lt8 12486	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 < 8
Theorem	5lt8 12487	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 < 8
Theorem	4lt8 12488	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 4 < 8
Theorem	3lt8 12489	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 3 < 8
Theorem	2lt8 12490	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 2 < 8
Theorem	1lt8 12491	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 1 < 8
Theorem	8lt9 12492	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
⊢ 8 < 9
Theorem	7lt9 12493	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 7 < 9
Theorem	6lt9 12494	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 6 < 9
Theorem	5lt9 12495	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 5 < 9
Theorem	4lt9 12496	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 4 < 9
Theorem	3lt9 12497	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 3 < 9
Theorem	2lt9 12498	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 2 < 9
Theorem	1lt9 12499	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
⊢ 1 < 9
Theorem	0ne2 12500	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 2