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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nn0nnaddcl 11201 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | ||
Theorem | 0mnnnnn0 11202 | 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) | ||
Theorem | un0addcl 11203 | If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ 𝑇 = (𝑆 ∪ {0}) & ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 + 𝑁) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 + 𝑁) ∈ 𝑇) | ||
Theorem | un0mulcl 11204 | If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ 𝑇 = (𝑆 ∪ {0}) & ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 · 𝑁) ∈ 𝑇) | ||
Theorem | nn0addcl 11205 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) | ||
Theorem | nn0mulcl 11206 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0) | ||
Theorem | nn0addcli 11207 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 + 𝑁) ∈ ℕ0 | ||
Theorem | nn0mulcli 11208 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 · 𝑁) ∈ ℕ0 | ||
Theorem | nn0p1nn 11209 | A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 10909. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | ||
Theorem | peano2nn0 11210 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | ||
Theorem | nnm1nn0 11211 | 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) | ||
Theorem | elnn0nn 11212 | 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)) | ||
Theorem | elnnnn0 11213 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0)) | ||
Theorem | elnnnn0b 11214 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) | ||
Theorem | elnnnn0c 11215 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | ||
Theorem | nn0addge1 11216 | 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)) | ||
Theorem | nn0addge2 11217 | 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)) | ||
Theorem | nn0addge1i 11218 | 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) | ||
Theorem | nn0addge2i 11219 | 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) | ||
Theorem | nn0sub 11220 | Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0)) | ||
Theorem | ltsubnn0 11221 | Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐵 < 𝐴 → (𝐴 − 𝐵) ∈ ℕ0)) | ||
Theorem | nn0negleid 11222 | A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.) |
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴) | ||
Theorem | difgtsumgt 11223 | If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵))) | ||
Theorem | nn0le2xi 11224 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝑁 ≤ (2 · 𝑁) | ||
Theorem | nn0lele2xi 11225 | '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 ⇒ ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ (2 · 𝑀)) | ||
Theorem | frnnn0supp 11226 | Two ways to write the support of a function on ℕ0. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.) |
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ)) | ||
Theorem | frnnn0fsupp 11227 | A function on ℕ0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019.) |
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) | ||
Theorem | nnnn0d 11228 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℕ0) | ||
Theorem | nn0red 11229 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | nn0cnd 11230 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | nn0ge0d 11231 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) | ||
Theorem | nn0addcld 11232 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) | ||
Theorem | nn0mulcld 11233 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0) | ||
Theorem | nn0readdcl 11234 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | nn0n0n1ge2 11235 | A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁) | ||
Theorem | nn0n0n1ge2b 11236 | A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.) |
⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | ||
Theorem | nn0ge2m1nn 11237 | 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) | ||
Theorem | nn0ge2m1nn0 11238 | 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) | ||
Theorem | nn0nndivcl 11239 | Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ) | ||
The function values of the hash (set size) function are either nonnegative integers or positive infinity, see hashf 12987. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers ℝ*, see df-xr 9957. The definition of extended nonnegative integers can be used in Ramsey theory, because the Ramsey number is either a nonnegative integer or plus infinity, see ramcl2 15558, or for the degree of polynomials, see mdegcl 23633, or for the degree of vertices in graph theory, see vdgrf 26425. | ||
Syntax | cxnn0 11240 | The set of extended nonnegative integers. |
class ℕ0* | ||
Definition | df-xnn0 11241 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 9957. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0* = (ℕ0 ∪ {+∞}) | ||
Theorem | elxnn0 11242 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | ||
Theorem | nn0ssxnn0 11243 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0 ⊆ ℕ0* | ||
Theorem | nn0xnn0 11244 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*) | ||
Theorem | xnn0xr 11245 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | ||
Theorem | 0xnn0 11246 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ 0 ∈ ℕ0* | ||
Theorem | pnf0xnn0 11247 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ +∞ ∈ ℕ0* | ||
Theorem | nn0nepnf 11248 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) | ||
Theorem | nn0xnn0d 11249 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℕ0*) | ||
Theorem | nn0nepnfd 11250 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | xnn0nemnf 11251 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) | ||
Theorem | xnn0xrnemnf 11252 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) | ||
Theorem | xnn0nnn0pnf 11253 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) | ||
Syntax | cz 11254 | Extend class notation to include the class of integers. |
class ℤ | ||
Definition | df-z 11255 | Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.) |
⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)} | ||
Theorem | elz 11256 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | ||
Theorem | nnnegz 11257 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | ||
Theorem | zre 11258 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | ||
Theorem | zcn 11259 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | ||
Theorem | zrei 11260 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝐴 ∈ ℤ ⇒ ⊢ 𝐴 ∈ ℝ | ||
Theorem | zssre 11261 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℝ | ||
Theorem | zsscn 11262 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℂ | ||
Theorem | zex 11263 | The set of integers exists. See also zexALT 11273. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ℤ ∈ V | ||
Theorem | elnnz 11264 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | ||
Theorem | 0z 11265 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ 0 ∈ ℤ | ||
Theorem | 0zd 11266 | Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℤ) | ||
Theorem | elnn0z 11267 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) | ||
Theorem | elznn0nn 11268 | Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | ||
Theorem | elznn0 11269 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | ||
Theorem | elznn 11270 | Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ0))) | ||
Theorem | elz2 11271* | Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℤ ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 − 𝑦)) | ||
Theorem | dfz2 11272 | Alternative definition of the integers, based on elz2 11271. (Contributed by Mario Carneiro, 16-May-2014.) |
⊢ ℤ = ( − “ (ℕ × ℕ)) | ||
Theorem | zexALT 11273 | Alternate proof of zex 11263. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℤ ∈ V | ||
Theorem | nnssz 11274 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
⊢ ℕ ⊆ ℤ | ||
Theorem | nn0ssz 11275 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
⊢ ℕ0 ⊆ ℤ | ||
Theorem | nnz 11276 | A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | ||
Theorem | nn0z 11277 | A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | ||
Theorem | nnzi 11278 | A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | nn0zi 11279 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | elnnz1 11280 | Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | ||
Theorem | znnnlt1 11281 | An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.) |
⊢ (𝑁 ∈ ℤ → (¬ 𝑁 ∈ ℕ ↔ 𝑁 < 1)) | ||
Theorem | nnzrab 11282 | Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥} | ||
Theorem | nn0zrab 11283 | Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥} | ||
Theorem | 1z 11284 | One is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 1 ∈ ℤ | ||
Theorem | 1zzd 11285 | 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℤ) | ||
Theorem | 2z 11286 | 2 is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 2 ∈ ℤ | ||
Theorem | 3z 11287 | 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 3 ∈ ℤ | ||
Theorem | 4z 11288 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
⊢ 4 ∈ ℤ | ||
Theorem | znegcl 11289 | Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | ||
Theorem | neg1z 11290 | -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℤ | ||
Theorem | znegclb 11291 | A complex number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℤ ↔ -𝐴 ∈ ℤ)) | ||
Theorem | nn0negz 11292 | The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ) | ||
Theorem | nn0negzi 11293 | The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ -𝑁 ∈ ℤ | ||
Theorem | zaddcl 11294 | Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) | ||
Theorem | peano2z 11295 | Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | ||
Theorem | zsubcl 11296 | Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) | ||
Theorem | peano2zm 11297 | "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | ||
Theorem | zletr 11298 | Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.) |
⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽 ≤ 𝐾 ∧ 𝐾 ≤ 𝐿) → 𝐽 ≤ 𝐿)) | ||
Theorem | zrevaddcl 11299 | Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.) |
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ)) | ||
Theorem | znnsub 11300 | The positive difference of unequal integers is a positive integer. (Generalization of nnsub 10936.) (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) |
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