P. 124 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
5.4.3  Decimal representation of numbers The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 11113 through df-9 12316), 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 11113 and df-1 11114). With the decimal constructor df-dec 12718, it is possible to easily express larger integers in base 10. See deccl 12732 and the theorems that follow it. See also 4001prm 17211 (4001 is prime) and the proof of bpos 27468. 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 16250. 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 12301	Extend class notation to include the number 2.
class 2
Syntax	c3 12302	Extend class notation to include the number 3.
class 3
Syntax	c4 12303	Extend class notation to include the number 4.
class 4
Syntax	c5 12304	Extend class notation to include the number 5.
class 5
Syntax	c6 12305	Extend class notation to include the number 6.
class 6
Syntax	c7 12306	Extend class notation to include the number 7.
class 7
Syntax	c8 12307	Extend class notation to include the number 8.
class 8
Syntax	c9 12308	Extend class notation to include the number 9.
class 9
Definition	df-2 12309	Define the number 2. (Contributed by NM, 27-May-1999.)
⊢ 2 = (1 + 1)
Definition	df-3 12310	Define the number 3. (Contributed by NM, 27-May-1999.)
⊢ 3 = (2 + 1)
Definition	df-4 12311	Define the number 4. (Contributed by NM, 27-May-1999.)
⊢ 4 = (3 + 1)
Definition	df-5 12312	Define the number 5. (Contributed by NM, 27-May-1999.)
⊢ 5 = (4 + 1)
Definition	df-6 12313	Define the number 6. (Contributed by NM, 27-May-1999.)
⊢ 6 = (5 + 1)
Definition	df-7 12314	Define the number 7. (Contributed by NM, 27-May-1999.)
⊢ 7 = (6 + 1)
Definition	df-8 12315	Define the number 8. (Contributed by NM, 27-May-1999.)
⊢ 8 = (7 + 1)
Definition	df-9 12316	Define the number 9. (Contributed by NM, 27-May-1999.)
⊢ 9 = (8 + 1)
Theorem	1eltp012 12317	1 is an element of {0, 1, 2}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 ∈ {0, 1, 2}
Theorem	0ne1 12318	Zero is different from one (the commuted form is Axiom ax-1ne0 11175). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 1
Theorem	1m1e0 12319	One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 − 1) = 0
Theorem	2nn 12320	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	2re 12321	The number 2 is real. (Contributed by NM, 27-May-1999.)
⊢ 2 ∈ ℝ
Theorem	2cn 12322	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 12323	Alternate proof of 2cn 12322. Shorter but uses more axioms. Similar proofs are possible for 3cn 12328, ... , 9cn 12347. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 2 ∈ ℂ
Theorem	2ex 12324	The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 2 ∈ V
Theorem	2cnd 12325	The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 2 ∈ ℂ)
Theorem	3nn 12326	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	3re 12327	The number 3 is real. (Contributed by NM, 27-May-1999.)
⊢ 3 ∈ ℝ
Theorem	3cn 12328	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 12329	The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ V
Theorem	4nn 12330	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	4re 12331	The number 4 is real. (Contributed by NM, 27-May-1999.)
⊢ 4 ∈ ℝ
Theorem	4cn 12332	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 12333	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	5re 12334	The number 5 is real. (Contributed by NM, 27-May-1999.)
⊢ 5 ∈ ℝ
Theorem	5cn 12335	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 12336	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	6re 12337	The number 6 is real. (Contributed by NM, 27-May-1999.)
⊢ 6 ∈ ℝ
Theorem	6cn 12338	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 12339	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	7re 12340	The number 7 is real. (Contributed by NM, 27-May-1999.)
⊢ 7 ∈ ℝ
Theorem	7cn 12341	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 12342	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	8re 12343	The number 8 is real. (Contributed by NM, 27-May-1999.)
⊢ 8 ∈ ℝ
Theorem	8cn 12344	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 12345	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	9re 12346	The number 9 is real. (Contributed by NM, 27-May-1999.)
⊢ 9 ∈ ℝ
Theorem	9cn 12347	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 12348	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ≤ 0
Theorem	0le2 12349	The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 ≤ 2
Theorem	0le2OLD 12350	Obsolete version of 0le2 12349 as of 10-Jun-2026. (Contributed by David A. Wheeler, 7-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ≤ 2
Theorem	2pos 12351	The number 2 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 2
Theorem	2posOLD 12352	Obsolete version of 2pos 12351 as of 10-Jun-2026. (Contributed by NM, 27-May-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 < 2
Theorem	2ne0 12353	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
⊢ 2 ≠ 0
Theorem	2thalfe1 12354	2 times one half equals 1. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 · (1 / 2)) = 1
Theorem	3pos 12355	The number 3 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 3
Theorem	3ne0 12356	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 3 ≠ 0
Theorem	4pos 12357	The number 4 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 4
Theorem	4ne0 12358	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 12359	The number 5 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 5
Theorem	6pos 12360	The number 6 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 6
Theorem	7pos 12361	The number 7 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 7
Theorem	8pos 12362	The number 8 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 8
Theorem	9pos 12363	The number 9 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 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 12364	1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + -1) = 0
Theorem	0m0e0 12365	0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0 − 0) = 0
Theorem	1m0e1 12366	1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 − 0) = 1
Theorem	0p1e1 12367	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (0 + 1) = 1
Theorem	fv0p1e1 12368	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 12369	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 0) = 1
Theorem	1p1e2 12370	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
⊢ (1 + 1) = 2
Theorem	2m1e1 12371	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12401. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 − 1) = 1
Theorem	2m1e1OLD 12372	Obsolete version of 2m1e1 12371 as of 10-Jun-2026. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (2 − 1) = 1
Theorem	1e2m1 12373	1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 = (2 − 1)
Theorem	3m1e2 12374	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 12375	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (4 − 1) = 3
Theorem	5m1e4 12376	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
⊢ (5 − 1) = 4
Theorem	6m1e5 12377	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
⊢ (6 − 1) = 5
Theorem	7m1e6 12378	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
⊢ (7 − 1) = 6
Theorem	8m1e7 12379	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
⊢ (8 − 1) = 7
Theorem	9m1e8 12380	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
⊢ (9 − 1) = 8
Theorem	2p2e4 12381	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 8524 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 12382	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 12383	A number times 2. (Contributed by NM, 16-Oct-2007.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	2timesi 12384	Two times a number. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (2 · 𝐴) = (𝐴 + 𝐴)
Theorem	times2i 12385	A number times 2. (Contributed by NM, 11-May-2004.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 2) = (𝐴 + 𝐴)
Theorem	2txmxeqx 12386	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 12387	2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 / 2) = 1
Theorem	2p1e3 12388	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (2 + 1) = 3
Theorem	1p2e3 12389	1 + 2 = 3. For a shorter proof using addcomli 11408, see 1p2e3ALT 12390. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.)
⊢ (1 + 2) = 3
Theorem	1p2e3ALT 12390	Alternate proof of 1p2e3 12389, 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 12391	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (3 + 1) = 4
Theorem	4p1e5 12392	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (4 + 1) = 5
Theorem	5p1e6 12393	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (5 + 1) = 6
Theorem	6p1e7 12394	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (6 + 1) = 7
Theorem	7p1e8 12395	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (7 + 1) = 8
Theorem	8p1e9 12396	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (8 + 1) = 9
Theorem	3p2e5 12397	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
⊢ (3 + 2) = 5
Theorem	3p3e6 12398	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
⊢ (3 + 3) = 6
Theorem	4p2e6 12399	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
⊢ (4 + 2) = 6
Theorem	4p3e7 12400	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7