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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | div4p1lem1div2 9101 | 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)) | ||
Theorem | arch 9102* | 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) | ||
Theorem | nnrecl 9103* | 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) | ||
Theorem | bndndx 9104* | 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 ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘) | ||
Syntax | cn0 9105 | Extend class notation to include the class of nonnegative integers. |
class ℕ0 | ||
Definition | df-n0 9106 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 = (ℕ ∪ {0}) | ||
Theorem | elnn0 9107 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | ||
Theorem | nnssnn0 9108 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ ⊆ ℕ0 | ||
Theorem | nn0ssre 9109 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 ⊆ ℝ | ||
Theorem | nn0sscn 9110 | Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.) |
⊢ ℕ0 ⊆ ℂ | ||
Theorem | nn0ex 9111 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
⊢ ℕ0 ∈ V | ||
Theorem | nnnn0 9112 | A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0) | ||
Theorem | nnnn0i 9113 | A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝑁 ∈ ℕ0 | ||
Theorem | nn0re 9114 | A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | ||
Theorem | nn0cn 9115 | A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) | ||
Theorem | nn0rei 9116 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 ∈ ℝ | ||
Theorem | nn0cni 9117 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 ∈ ℂ | ||
Theorem | dfn2 9118 | 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}) | ||
Theorem | elnnne0 9119 | The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | ||
Theorem | 0nn0 9120 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 0 ∈ ℕ0 | ||
Theorem | 1nn0 9121 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 1 ∈ ℕ0 | ||
Theorem | 2nn0 9122 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 2 ∈ ℕ0 | ||
Theorem | 3nn0 9123 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 3 ∈ ℕ0 | ||
Theorem | 4nn0 9124 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 4 ∈ ℕ0 | ||
Theorem | 5nn0 9125 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 5 ∈ ℕ0 | ||
Theorem | 6nn0 9126 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 6 ∈ ℕ0 | ||
Theorem | 7nn0 9127 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 7 ∈ ℕ0 | ||
Theorem | 8nn0 9128 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 8 ∈ ℕ0 | ||
Theorem | 9nn0 9129 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 9 ∈ ℕ0 | ||
Theorem | nn0ge0 9130 | 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) | ||
Theorem | nn0nlt0 9131 | 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) | ||
Theorem | nn0ge0i 9132 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 0 ≤ 𝑁 | ||
Theorem | nn0le0eq0 9133 | 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)) | ||
Theorem | nn0p1gt0 9134 | A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) | ||
Theorem | nnnn0addcl 9135 | 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) | ||
Theorem | nn0nnaddcl 9136 | 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 9137 | 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 9138 | 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 9139 | 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 9140 | 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 9141 | 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 9142 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 + 𝑁) ∈ ℕ0 | ||
Theorem | nn0mulcli 9143 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 · 𝑁) ∈ ℕ0 | ||
Theorem | nn0p1nn 9144 | 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) | ||
Theorem | peano2nn0 9145 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | ||
Theorem | nnm1nn0 9146 | 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 9147 | 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 9148 | 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 9149 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) | ||
Theorem | elnnnn0c 9150 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | ||
Theorem | nn0addge1 9151 | 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 9152 | 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 9153 | 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 9154 | 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 | nn0le2xi 9155 | 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 9156 | '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 | nn0supp 9157 | Two ways to write the support of a function on ℕ0. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) = (◡𝐹 “ ℕ)) | ||
Theorem | nnnn0d 9158 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℕ0) | ||
Theorem | nn0red 9159 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | nn0cnd 9160 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | nn0ge0d 9161 | 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 9162 | 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 9163 | 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 9164 | 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 | nn0ge2m1nn 9165 | 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 9166 | 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 9167 | 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. 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 7928. | ||
Syntax | cxnn0 9168 | The set of extended nonnegative integers. |
class ℕ0* | ||
Definition | df-xnn0 9169 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 7928. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7076 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0* = (ℕ0 ∪ {+∞}) | ||
Theorem | elxnn0 9170 | 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 9171 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0 ⊆ ℕ0* | ||
Theorem | nn0xnn0 9172 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*) | ||
Theorem | xnn0xr 9173 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | ||
Theorem | 0xnn0 9174 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ 0 ∈ ℕ0* | ||
Theorem | pnf0xnn0 9175 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ +∞ ∈ ℕ0* | ||
Theorem | nn0nepnf 9176 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) | ||
Theorem | nn0xnn0d 9177 | 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 9178 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | xnn0nemnf 9179 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) | ||
Theorem | xnn0xrnemnf 9180 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) | ||
Theorem | xnn0nnn0pnf 9181 | 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 9182 | Extend class notation to include the class of integers. |
class ℤ | ||
Definition | df-z 9183 | 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 9184 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | ||
Theorem | nnnegz 9185 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | ||
Theorem | zre 9186 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | ||
Theorem | zcn 9187 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | ||
Theorem | zrei 9188 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝐴 ∈ ℤ ⇒ ⊢ 𝐴 ∈ ℝ | ||
Theorem | zssre 9189 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℝ | ||
Theorem | zsscn 9190 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℂ | ||
Theorem | zex 9191 | The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ℤ ∈ V | ||
Theorem | elnnz 9192 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | ||
Theorem | 0z 9193 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ 0 ∈ ℤ | ||
Theorem | 0zd 9194 | Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℤ) | ||
Theorem | elnn0z 9195 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) | ||
Theorem | elznn0nn 9196 | 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 9197 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | ||
Theorem | elznn 9198 | 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 | nnssz 9199 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
⊢ ℕ ⊆ ℤ | ||
Theorem | nn0ssz 9200 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
⊢ ℕ0 ⊆ ℤ |
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