P. 123 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12201-12300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnaddcld 12201	Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ)
Theorem	nnmulcld 12202	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nndivred 12203	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
5.4.3  Decimal representation of numbers The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 11037 through df-9 12219), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 11037 and df-1 11038). With the decimal constructor df-dec 12612, it is possible to easily express larger integers in base 10. See deccl 12626 and the theorems that follow it. See also 4001prm 17076 (4001 is prime) and the proof of bpos 27264. Note that the decimal constructor builds on the definitions in this section. Note: The number 10 will be represented by its digits using the decimal constructor only, i.e., by ;10. Therefore, only decimal digits are needed (as symbols) for the decimal representation of a number. Integers can also be exhibited as sums of powers of 10 (e.g., the number 103 can be expressed as ((;10↑2) + 3)) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as (7↑7) − 2. Decimals can be expressed as ratios of integers, as in cos2bnd 16117. Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.
Syntax	c2 12204	Extend class notation to include the number 2.
class 2
Syntax	c3 12205	Extend class notation to include the number 3.
class 3
Syntax	c4 12206	Extend class notation to include the number 4.
class 4
Syntax	c5 12207	Extend class notation to include the number 5.
class 5
Syntax	c6 12208	Extend class notation to include the number 6.
class 6
Syntax	c7 12209	Extend class notation to include the number 7.
class 7
Syntax	c8 12210	Extend class notation to include the number 8.
class 8
Syntax	c9 12211	Extend class notation to include the number 9.
class 9
Definition	df-2 12212	Define the number 2. (Contributed by NM, 27-May-1999.)
⊢ 2 = (1 + 1)
Definition	df-3 12213	Define the number 3. (Contributed by NM, 27-May-1999.)
⊢ 3 = (2 + 1)
Definition	df-4 12214	Define the number 4. (Contributed by NM, 27-May-1999.)
⊢ 4 = (3 + 1)
Definition	df-5 12215	Define the number 5. (Contributed by NM, 27-May-1999.)
⊢ 5 = (4 + 1)
Definition	df-6 12216	Define the number 6. (Contributed by NM, 27-May-1999.)
⊢ 6 = (5 + 1)
Definition	df-7 12217	Define the number 7. (Contributed by NM, 27-May-1999.)
⊢ 7 = (6 + 1)
Definition	df-8 12218	Define the number 8. (Contributed by NM, 27-May-1999.)
⊢ 8 = (7 + 1)
Definition	df-9 12219	Define the number 9. (Contributed by NM, 27-May-1999.)
⊢ 9 = (8 + 1)
Theorem	0ne1 12220	Zero is different from one (the commuted form is Axiom ax-1ne0 11099). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 1
Theorem	1m1e0 12221	One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 − 1) = 0
Theorem	2nn 12222	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	2re 12223	The number 2 is real. (Contributed by NM, 27-May-1999.)
⊢ 2 ∈ ℝ
Theorem	2cn 12224	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 2 ∈ ℂ
Theorem	2cnALT 12225	Alternate proof of 2cn 12224. Shorter but uses more axioms. Similar proofs are possible for 3cn 12230, ... , 9cn 12249. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 2 ∈ ℂ
Theorem	2ex 12226	The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 2 ∈ V
Theorem	2cnd 12227	The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 2 ∈ ℂ)
Theorem	3nn 12228	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	3re 12229	The number 3 is real. (Contributed by NM, 27-May-1999.)
⊢ 3 ∈ ℝ
Theorem	3cn 12230	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 3 ∈ ℂ
Theorem	3ex 12231	The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ V
Theorem	4nn 12232	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	4re 12233	The number 4 is real. (Contributed by NM, 27-May-1999.)
⊢ 4 ∈ ℝ
Theorem	4cn 12234	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 4 ∈ ℂ
Theorem	5nn 12235	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	5re 12236	The number 5 is real. (Contributed by NM, 27-May-1999.)
⊢ 5 ∈ ℝ
Theorem	5cn 12237	The number 5 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 5 ∈ ℂ
Theorem	6nn 12238	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	6re 12239	The number 6 is real. (Contributed by NM, 27-May-1999.)
⊢ 6 ∈ ℝ
Theorem	6cn 12240	The number 6 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 6 ∈ ℂ
Theorem	7nn 12241	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	7re 12242	The number 7 is real. (Contributed by NM, 27-May-1999.)
⊢ 7 ∈ ℝ
Theorem	7cn 12243	The number 7 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 7 ∈ ℂ
Theorem	8nn 12244	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	8re 12245	The number 8 is real. (Contributed by NM, 27-May-1999.)
⊢ 8 ∈ ℝ
Theorem	8cn 12246	The number 8 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 8 ∈ ℂ
Theorem	9nn 12247	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	9re 12248	The number 9 is real. (Contributed by NM, 27-May-1999.)
⊢ 9 ∈ ℝ
Theorem	9cn 12249	The number 9 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 9 ∈ ℂ
Theorem	0le0 12250	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ≤ 0
Theorem	0le2 12251	The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ 0 ≤ 2
Theorem	2pos 12252	The number 2 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 2
Theorem	2ne0 12253	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
⊢ 2 ≠ 0
Theorem	3pos 12254	The number 3 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 3
Theorem	3ne0 12255	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 3 ≠ 0
Theorem	4pos 12256	The number 4 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 4
Theorem	4ne0 12257	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 12258	The number 5 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 5
Theorem	6pos 12259	The number 6 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 6
Theorem	7pos 12260	The number 7 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 7
Theorem	8pos 12261	The number 8 is positive. (Contributed by NM, 27-May-1999.)
⊢ 0 < 8
Theorem	9pos 12262	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	1pneg1e0 12263	1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + -1) = 0
Theorem	0m0e0 12264	0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0 − 0) = 0
Theorem	1m0e1 12265	1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 − 0) = 1
Theorem	0p1e1 12266	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (0 + 1) = 1
Theorem	fv0p1e1 12267	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 12268	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 0) = 1
Theorem	1p1e2 12269	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
⊢ (1 + 1) = 2
Theorem	2m1e1 12270	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12299. (Contributed by David A. Wheeler, 4-Jan-2017.)
⊢ (2 − 1) = 1
Theorem	1e2m1 12271	1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 = (2 − 1)
Theorem	3m1e2 12272	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 12273	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (4 − 1) = 3
Theorem	5m1e4 12274	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
⊢ (5 − 1) = 4
Theorem	6m1e5 12275	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
⊢ (6 − 1) = 5
Theorem	7m1e6 12276	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
⊢ (7 − 1) = 6
Theorem	8m1e7 12277	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
⊢ (8 − 1) = 7
Theorem	9m1e8 12278	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
⊢ (9 − 1) = 8
Theorem	2p2e4 12279	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 8470 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 12280	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 12281	A number times 2. (Contributed by NM, 16-Oct-2007.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	2timesi 12282	Two times a number. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (2 · 𝐴) = (𝐴 + 𝐴)
Theorem	times2i 12283	A number times 2. (Contributed by NM, 11-May-2004.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 2) = (𝐴 + 𝐴)
Theorem	2txmxeqx 12284	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 12285	2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 / 2) = 1
Theorem	2p1e3 12286	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (2 + 1) = 3
Theorem	1p2e3 12287	1 + 2 = 3. For a shorter proof using addcomli 11329, see 1p2e3ALT 12288. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.)
⊢ (1 + 2) = 3
Theorem	1p2e3ALT 12288	Alternate proof of 1p2e3 12287, 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 12289	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (3 + 1) = 4
Theorem	4p1e5 12290	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (4 + 1) = 5
Theorem	5p1e6 12291	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (5 + 1) = 6
Theorem	6p1e7 12292	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (6 + 1) = 7
Theorem	7p1e8 12293	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (7 + 1) = 8
Theorem	8p1e9 12294	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (8 + 1) = 9
Theorem	3p2e5 12295	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
⊢ (3 + 2) = 5
Theorem	3p3e6 12296	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
⊢ (3 + 3) = 6
Theorem	4p2e6 12297	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
⊢ (4 + 2) = 6
Theorem	4p3e7 12298	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7
Theorem	4p4e8 12299	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
⊢ (4 + 4) = 8
Theorem	5p2e7 12300	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
⊢ (5 + 2) = 7