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

Theoremhalfpm6th 8201 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
(((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))

Theoremit0e0 8202 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0

Theorem2mulicn 8203 (2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ∈ ℂ

Theoremiap0 8204 The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
i # 0

Theorem2muliap0 8205 2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
(2 · i) # 0

Theorem2muline0 8206 (2 · i) ≠ 0. See also 2muliap0 8205. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0

3.4.5  Simple number properties

Theoremhalfcl 8207 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)

Theoremrehalfcl 8208 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)

Theoremhalf0 8209 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
(𝐴 ∈ ℂ → ((𝐴 / 2) = 0 ↔ 𝐴 = 0))

Theorem2halves 8210 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)

Theoremhalfpos2 8211 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))

Theoremhalfpos 8212 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (𝐴 / 2) < 𝐴))

Theoremhalfnneg2 8213 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))

Theoremhalfaddsubcl 8214 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) / 2) ∈ ℂ ∧ ((𝐴𝐵) / 2) ∈ ℂ))

Theoremhalfaddsub 8215 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐵) / 2) + ((𝐴𝐵) / 2)) = 𝐴 ∧ (((𝐴 + 𝐵) / 2) − ((𝐴𝐵) / 2)) = 𝐵))

Theoremlt2halves 8216 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶))

Theoremaddltmul 8217 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))

Theoremnominpos 8218* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
¬ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ¬ ∃𝑦 ∈ ℝ (0 < 𝑦𝑦 < 𝑥))

Theoremavglt1 8219 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < ((𝐴 + 𝐵) / 2)))

Theoremavglt2 8220 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))

Theoremavgle1 8221 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵𝐴 ≤ ((𝐴 + 𝐵) / 2)))

Theoremavgle2 8222 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵))

Theorem2timesd 8223 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))

Theoremtimes2d 8224 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))

Theoremhalfcld 8225 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 2) ∈ ℂ)

Theorem2halvesd 8226 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)

Theoremrehalfcld 8227 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 / 2) ∈ ℝ)

Theoremlt2halvesd 8228 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐶 / 2))    &   (𝜑𝐵 < (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) < 𝐶)

Theoremrehalfcli 8229 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       (𝐴 / 2) ∈ ℝ

Theoremadd1p1 8230 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
(𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))

Theoremsub1m1 8231 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
(𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))

Theoremcnm2m1cnm3 8232 Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝐴 ∈ ℂ → ((𝐴 − 2) − 1) = (𝐴 − 3))

Theoremxp1d2m1eqxm1d2 8233 A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
(𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2))

Theoremdiv4p1lem1div2 8234 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))

3.4.6  The Archimedean property

Theoremarch 8235* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
(𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)

Theoremnnrecl 8236* There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)

Theorembndndx 8237* A bounded real sequence 𝐴(𝑘) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
(∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴𝑥) → ∃𝑘 ∈ ℕ 𝐴𝑘)

3.4.7  Nonnegative integers (as a subset of complex numbers)

Syntaxcn0 8238 Extend class notation to include the class of nonnegative integers.
class 0

Definitiondf-n0 8239 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})

Theoremelnn0 8240 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))

Theoremnnssnn0 8241 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0

Theoremnn0ssre 8242 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ

Theoremnn0sscn 8243 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
0 ⊆ ℂ

Theoremnn0ex 8244 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 ∈ V

Theoremnnnn0 8245 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)

Theoremnnnn0i 8246 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁 ∈ ℕ       𝑁 ∈ ℕ0

Theoremnn0re 8247 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℝ)

Theoremnn0cn 8248 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℂ)

Theoremnn0rei 8249 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℝ

Theoremnn0cni 8250 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℂ

Theoremdfn2 8251 The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
ℕ = (ℕ0 ∖ {0})

Theoremelnnne0 8252 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))

Theorem0nn0 8253 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
0 ∈ ℕ0

Theorem1nn0 8254 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
1 ∈ ℕ0

Theorem2nn0 8255 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
2 ∈ ℕ0

Theorem3nn0 8256 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
3 ∈ ℕ0

Theorem4nn0 8257 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
4 ∈ ℕ0

Theorem5nn0 8258 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
5 ∈ ℕ0

Theorem6nn0 8259 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
6 ∈ ℕ0

Theorem7nn0 8260 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
7 ∈ ℕ0

Theorem8nn0 8261 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
8 ∈ ℕ0

Theorem9nn0 8262 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
9 ∈ ℕ0

Theoremnn0ge0 8263 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ0 → 0 ≤ 𝑁)

Theoremnn0nlt0 8264 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℕ0 → ¬ 𝐴 < 0)

Theoremnn0ge0i 8265 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 ∈ ℕ0       0 ≤ 𝑁

Theoremnn0le0eq0 8266 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
(𝑁 ∈ ℕ0 → (𝑁 ≤ 0 ↔ 𝑁 = 0))

Theoremnn0p1gt0 8267 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))

Theoremnnnn0addcl 8268 A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ)

Theoremnn0nnaddcl 8269 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)

Theorem0mnnnnn0 8270 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
(𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0)

Theoremun0addcl 8271 If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((𝜑 ∧ (𝑀𝑆𝑁𝑆)) → (𝑀 + 𝑁) ∈ 𝑆)       ((𝜑 ∧ (𝑀𝑇𝑁𝑇)) → (𝑀 + 𝑁) ∈ 𝑇)

Theoremun0mulcl 8272 If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((𝜑 ∧ (𝑀𝑆𝑁𝑆)) → (𝑀 · 𝑁) ∈ 𝑆)       ((𝜑 ∧ (𝑀𝑇𝑁𝑇)) → (𝑀 · 𝑁) ∈ 𝑇)

Theoremnn0addcl 8273 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)

Theoremnn0mulcl 8274 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0)

Theoremnn0addcli 8275 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 + 𝑁) ∈ ℕ0

Theoremnn0mulcli 8276 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 · 𝑁) ∈ ℕ0

Theoremnn0p1nn 8277 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)

Theorempeano2nn0 8278 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)

Theoremnnm1nn0 8279 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)

Theoremelnn0nn 8280 The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ))

Theoremelnnnn0 8281 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0))

Theoremelnnnn0b 8282 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))

Theoremelnnnn0c 8283 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))

Theoremnn0addge1 8284 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝑁))

Theoremnn0addge2 8285 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝑁 + 𝐴))

Theoremnn0addge1i 8286 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
𝐴 ∈ ℝ    &   𝑁 ∈ ℕ0       𝐴 ≤ (𝐴 + 𝑁)

Theoremnn0addge2i 8287 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
𝐴 ∈ ℝ    &   𝑁 ∈ ℕ0       𝐴 ≤ (𝑁 + 𝐴)

Theoremnn0le2xi 8288 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 ∈ ℕ0       𝑁 ≤ (2 · 𝑁)

Theoremnn0lele2xi 8289 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑁𝑀𝑁 ≤ (2 · 𝑀))

Theoremnn0supp 8290 Two ways to write the support of a function on 0. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐹:𝐼⟶ℕ0 → (𝐹 “ (V ∖ {0})) = (𝐹 “ ℕ))

Theoremnnnn0d 8291 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℕ0)

Theoremnn0red 8292 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℝ)

Theoremnn0cnd 8293 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℂ)

Theoremnn0ge0d 8294 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑 → 0 ≤ 𝐴)

Theoremnn0addcld 8295 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ0)

Theoremnn0mulcld 8296 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ0)

Theoremnn0readdcl 8297 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℝ)

Theoremnn0ge2m1nn 8298 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)

Theoremnn0ge2m1nn0 8299 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ0)

Theoremnn0nndivcl 8300 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 ∈ ℕ0𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ)

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