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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnngt0 8201 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(𝐴 ∈ ℕ → 0 < 𝐴)

Theoremnnnlt1 8202 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℕ → ¬ 𝐴 < 1)

Theorem0nnn 8203 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
¬ 0 ∈ ℕ

Theoremnnne0 8204 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)

Theoremnnap0 8205 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝐴 ∈ ℕ → 𝐴 # 0)

Theoremnngt0i 8206 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℕ       0 < 𝐴

Theoremnnne0i 8207 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ≠ 0

Theoremnn2ge 8208* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ (𝐴𝑥𝐵𝑥))

Theoremnn1gt1 8209 A positive integer is either one or greater than one. This is for ; 0elnn 4386 is a similar theorem for ω (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
(𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))

Theoremnngt1ne1 8210 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
(𝐴 ∈ ℕ → (1 < 𝐴𝐴 ≠ 1))

Theoremnndivre 8211 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ)

Theoremnnrecre 8212 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)

Theoremnnrecgt0 8213 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 0 < (1 / 𝐴))

Theoremnnsub 8214 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ))

Theoremnnsubi 8215 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ)

Theoremnndiv 8216* Two ways to express "𝐴 divides 𝐵 " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ))

Theoremnndivtr 8217 Transitive property of divisibility: if 𝐴 divides 𝐵 and 𝐵 divides 𝐶, then 𝐴 divides 𝐶. Typically, 𝐶 would be an integer, although the theorem holds for complex 𝐶. (Contributed by NM, 3-May-2005.)
(((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ)) → (𝐶 / 𝐴) ∈ ℕ)

Theoremnnge1d 8218 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 1 ≤ 𝐴)

Theoremnngt0d 8219 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 0 < 𝐴)

Theoremnnne0d 8220 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ≠ 0)

Theoremnnap0d 8221 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 # 0)

Theoremnnrecred 8222 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremnnaddcld 8223 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ)

Theoremnnmulcld 8224 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ)

Theoremnndivred 8225 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)

3.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7120 through df-9 8242), 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 7120 and df-1 7121).

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.

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 8226 Extend class notation to include the number 2.
class 2

Syntaxc3 8227 Extend class notation to include the number 3.
class 3

Syntaxc4 8228 Extend class notation to include the number 4.
class 4

Syntaxc5 8229 Extend class notation to include the number 5.
class 5

Syntaxc6 8230 Extend class notation to include the number 6.
class 6

Syntaxc7 8231 Extend class notation to include the number 7.
class 7

Syntaxc8 8232 Extend class notation to include the number 8.
class 8

Syntaxc9 8233 Extend class notation to include the number 9.
class 9

Syntaxc10 8234 Extend class notation to include the number 10.
class 10

Definitiondf-2 8235 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)

Definitiondf-3 8236 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)

Definitiondf-4 8237 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)

Definitiondf-5 8238 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)

Definitiondf-6 8239 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)

Definitiondf-7 8240 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)

Definitiondf-8 8241 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)

Definitiondf-9 8242 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)

Theorem0ne1 8243 0 ≠ 1 (common case). See aso 1ap0 7827. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1

Theorem1ne0 8244 1 ≠ 0. See aso 1ap0 7827. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0

Theorem1m1e0 8245 (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0

Theorem2re 8246 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ

Theorem2cn 8247 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2 ∈ ℂ

Theorem2ex 8248 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V

Theorem2cnd 8249 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)

Theorem3re 8250 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ

Theorem3cn 8251 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3 ∈ ℂ

Theorem3ex 8252 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V

Theorem4re 8253 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ

Theorem4cn 8254 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4 ∈ ℂ

Theorem5re 8255 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 ∈ ℝ

Theorem5cn 8256 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5 ∈ ℂ

Theorem6re 8257 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 ∈ ℝ

Theorem6cn 8258 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6 ∈ ℂ

Theorem7re 8259 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 ∈ ℝ

Theorem7cn 8260 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7 ∈ ℂ

Theorem8re 8261 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 ∈ ℝ

Theorem8cn 8262 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8 ∈ ℂ

Theorem9re 8263 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 ∈ ℝ

Theorem9cn 8264 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9 ∈ ℂ

Theorem0le0 8265 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0

Theorem0le2 8266 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2

Theorem2pos 8267 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2

Theorem2ne0 8268 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0

Theorem2ap0 8269 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0

Theorem3pos 8270 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3

Theorem3ne0 8271 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0

Theorem3ap0 8272 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
3 # 0

Theorem4pos 8273 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4

Theorem4ne0 8274 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0

Theorem4ap0 8275 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
4 # 0

Theorem5pos 8276 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5

Theorem6pos 8277 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6

Theorem7pos 8278 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7

Theorem8pos 8279 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8

Theorem9pos 8280 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9

3.4.4  Some properties of specific numbers

This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.

Theoremneg1cn 8281 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ

Theoremneg1rr 8282 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ

Theoremneg1ne0 8283 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 8284 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremneg1ap0 8285 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0

Theoremnegneg1e1 8286 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 8287 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 8288 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 8289 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 8290 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theorem1p0e1 8291 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 8292 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 8293 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8314. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 8294 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 8295 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2

Theorem2p2e4 8296 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
(2 + 2) = 4

Theorem2times 8297 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 8298 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))

Theorem2timesi 8299 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)

Theoremtimes2i 8300 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)

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