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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnge1d 11101 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 1 ≤ 𝐴)

Theoremnngt0d 11102 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 0 < 𝐴)

Theoremnnne0d 11103 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ≠ 0)

Theoremnnrecred 11104 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremnnaddcld 11105 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ)

Theoremnnmulcld 11106 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ)

Theoremnndivred 11107 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ 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 9981 through df-9 11124), 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 9981 and df-1 9982).

With the decimal constructor df-dec 11532, it is possible to easily express larger integers in base 10. See deccl 11550 and the theorems that follow it. See also 4001prm 15899 (4001 is prime) and the proof of bpos 25063. Note that the decimal constructor builds on the definitions in this section.

Note: The symbol 10 representing the number 10 is deprecated (and will be removed in the near future). The number 10 should 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 14962.

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.

Syntaxc2 11108 Extend class notation to include the number 2.
class 2

Syntaxc3 11109 Extend class notation to include the number 3.
class 3

Syntaxc4 11110 Extend class notation to include the number 4.
class 4

Syntaxc5 11111 Extend class notation to include the number 5.
class 5

Syntaxc6 11112 Extend class notation to include the number 6.
class 6

Syntaxc7 11113 Extend class notation to include the number 7.
class 7

Syntaxc8 11114 Extend class notation to include the number 8.
class 8

Syntaxc9 11115 Extend class notation to include the number 9.
class 9

Syntaxc10 11116 Extend class notation to include the number 10.
class 10

Definitiondf-2 11117 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)

Definitiondf-3 11118 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)

Definitiondf-4 11119 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)

Definitiondf-5 11120 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)

Definitiondf-6 11121 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)

Definitiondf-7 11122 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)

Definitiondf-8 11123 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)

Definitiondf-9 11124 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)

Definitiondf-10OLD 11125 Define the number 10. See remarks under df-2 11117. (Contributed by NM, 5-Feb-2007.) Obsolete as of 9-Sep-2021. (New usage is discouraged.)
10 = (9 + 1)

Theorem0ne1 11126 0 ≠ 1; the reverse order is already proved. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1

Theorem1m1e0 11127 (1 − 1) = 0. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0

Theorem2re 11128 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ

Theorem2cn 11129 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2 ∈ ℂ

Theorem2ex 11130 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V

Theorem2cnd 11131 2 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)

Theorem3re 11132 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ

Theorem3cn 11133 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3 ∈ ℂ

Theorem3ex 11134 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V

Theorem4re 11135 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ

Theorem4cn 11136 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4 ∈ ℂ

Theorem5re 11137 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 ∈ ℝ

Theorem5cn 11138 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5 ∈ ℂ

Theorem6re 11139 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 ∈ ℝ

Theorem6cn 11140 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6 ∈ ℂ

Theorem7re 11141 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 ∈ ℝ

Theorem7cn 11142 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7 ∈ ℂ

Theorem8re 11143 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 ∈ ℝ

Theorem8cn 11144 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8 ∈ ℂ

Theorem9re 11145 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 ∈ ℝ

Theorem9cn 11146 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9 ∈ ℂ

Theorem10reOLD 11147 Obsolete version of 10re 11555 as of 8-Sep-2021. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
10 ∈ ℝ

Theorem0le0 11148 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0

Theorem0le2 11149 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2

Theorem2pos 11150 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2

Theorem2ne0 11151 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0

Theorem3pos 11152 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3

Theorem3ne0 11153 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0

Theorem4pos 11154 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4

Theorem4ne0 11155 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0

Theorem5pos 11156 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5

Theorem6pos 11157 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6

Theorem7pos 11158 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7

Theorem8pos 11159 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8

Theorem9pos 11160 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9

Theorem10posOLD 11161 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) Obsolete version of 10pos 11553 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
0 < 10

5.4.4  Some properties of specific numbers

This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering).

Theoremneg1cn 11162 -1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ

Theoremneg1rr 11163 -1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ

Theoremneg1ne0 11164 -1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 11165 -1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremnegneg1e1 11166 --1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 11167 1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 11168 0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 11169 1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 11170 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theorem1p0e1 11171 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 11172 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 11173 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 11202. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 11174 1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 11175 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

Theorem4m1e3 11176 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
(4 − 1) = 3

Theorem5m1e4 11177 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
(5 − 1) = 4

Theorem6m1e5 11178 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
(6 − 1) = 5

Theorem7m1e6 11179 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
(7 − 1) = 6

Theorem8m1e7 11180 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
(8 − 1) = 7

Theorem9m1e8 11181 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
(9 − 1) = 8

Theorem2p2e4 11182 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 7666 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

Theorem2times 11183 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))

Theoremtimes2 11184 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))

Theorem2timesi 11185 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)

Theoremtimes2i 11186 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)

Theorem2txmxeqx 11187 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)

Theorem2div2e1 11188 2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1

Theorem2p1e3 11189 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3

Theorem1p2e3 11190 1 + 2 = 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3

Theorem3p1e4 11191 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4

Theorem4p1e5 11192 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5

Theorem5p1e6 11193 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6

Theorem6p1e7 11194 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7

Theorem7p1e8 11195 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8

Theorem8p1e9 11196 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9

Theorem9p1e10OLD 11197 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) Obsolete version of 9p1e10 11534 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(9 + 1) = 10

Theorem3p2e5 11198 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5

Theorem3p3e6 11199 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6

Theorem4p2e6 11200 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6

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