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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | add1p1 12501 | Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.) |
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) | ||
Theorem | sub1m1 12502 | Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.) |
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2)) | ||
Theorem | cnm2m1cnm3 12503 | 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)) | ||
Theorem | xp1d2m1eqxm1d2 12504 | 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)) | ||
Theorem | div4p1lem1div2 12505 | 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 | nnunb 12506* | The set of positive integers is unbounded above. Theorem I.28 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.) |
⊢ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥) | ||
Theorem | arch 12507* | 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 12508* | 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 12509* | 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 12510 | Extend class notation to include the class of nonnegative integers. |
class ℕ0 | ||
Definition | df-n0 12511 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 = (ℕ ∪ {0}) | ||
Theorem | elnn0 12512 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | ||
Theorem | nnssnn0 12513 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ ⊆ ℕ0 | ||
Theorem | nn0ssre 12514 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ ℕ0 ⊆ ℝ | ||
Theorem | nn0sscn 12515 | Nonnegative integers are a subset of the complex numbers. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
⊢ ℕ0 ⊆ ℂ | ||
Theorem | nn0ex 12516 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
⊢ ℕ0 ∈ V | ||
Theorem | nnnn0 12517 | A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0) | ||
Theorem | nnnn0i 12518 | A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝑁 ∈ ℕ0 | ||
Theorem | nn0re 12519 | A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | ||
Theorem | nn0cn 12520 | A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) | ||
Theorem | nn0rei 12521 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 ∈ ℝ | ||
Theorem | nn0cni 12522 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 ∈ ℂ | ||
Theorem | dfn2 12523 | 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 12524 | 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 12525 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 0 ∈ ℕ0 | ||
Theorem | 1nn0 12526 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 1 ∈ ℕ0 | ||
Theorem | 2nn0 12527 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 2 ∈ ℕ0 | ||
Theorem | 3nn0 12528 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 3 ∈ ℕ0 | ||
Theorem | 4nn0 12529 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 4 ∈ ℕ0 | ||
Theorem | 5nn0 12530 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 5 ∈ ℕ0 | ||
Theorem | 6nn0 12531 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 6 ∈ ℕ0 | ||
Theorem | 7nn0 12532 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 7 ∈ ℕ0 | ||
Theorem | 8nn0 12533 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 8 ∈ ℕ0 | ||
Theorem | 9nn0 12534 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 9 ∈ ℕ0 | ||
Theorem | nn0ge0 12535 | 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 12536 | 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 12537 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 0 ≤ 𝑁 | ||
Theorem | nn0le0eq0 12538 | 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 12539 | 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 12540 | 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 12541 | 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 12542 | 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 12543 | 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 12544 | 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 12545 | 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 12546 | 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 12547 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 + 𝑁) ∈ ℕ0 | ||
Theorem | nn0mulcli 12548 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 · 𝑁) ∈ ℕ0 | ||
Theorem | nn0p1nn 12549 | A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 12262. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | ||
Theorem | peano2nn0 12550 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | ||
Theorem | nnm1nn0 12551 | 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 12552 | 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 12553 | 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 12554 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) | ||
Theorem | elnnnn0c 12555 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | ||
Theorem | nn0addge1 12556 | 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 12557 | 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 12558 | 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 12559 | 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 12560 | 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 12561 | 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 12562 | A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.) |
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴) | ||
Theorem | difgtsumgt 12563 | 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 12564 | 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 12565 | '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 | fcdmnn0supp 12566 | Two ways to write the support of a function into 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 | fcdmnn0fsupp 12567 | A function into 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 | fcdmnn0suppg 12568 | Version of fcdmnn0supp 12566 avoiding ax-rep 5286 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ)) | ||
Theorem | fcdmnn0fsuppg 12569 | Version of fcdmnn0fsupp 12567 avoiding ax-rep 5286 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) | ||
Theorem | nnnn0d 12570 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℕ0) | ||
Theorem | nn0red 12571 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | nn0cnd 12572 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | nn0ge0d 12573 | 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 12574 | 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 12575 | 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 12576 | 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 12577 | 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 12578 | 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 12579 | 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 12580 | 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 12581 | 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 14341. 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 11289. 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 17004, or for the degree of polynomials, see mdegcl 26066, or for the degree of vertices in graph theory, see vtxdgf 29377. | ||
Syntax | cxnn0 12582 | The set of extended nonnegative integers. |
class ℕ0* | ||
Definition | df-xnn0 12583 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 11289. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0* = (ℕ0 ∪ {+∞}) | ||
Theorem | elxnn0 12584 | 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 12585 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0 ⊆ ℕ0* | ||
Theorem | nn0xnn0 12586 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*) | ||
Theorem | xnn0xr 12587 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | ||
Theorem | 0xnn0 12588 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ 0 ∈ ℕ0* | ||
Theorem | pnf0xnn0 12589 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ +∞ ∈ ℕ0* | ||
Theorem | nn0nepnf 12590 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) | ||
Theorem | nn0xnn0d 12591 | 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 12592 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | xnn0nemnf 12593 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) | ||
Theorem | xnn0xrnemnf 12594 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) | ||
Theorem | xnn0nnn0pnf 12595 | 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 12596 | Extend class notation to include the class of integers. |
class ℤ | ||
Definition | df-z 12597 | 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 12598 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | ||
Theorem | nnnegz 12599 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | ||
Theorem | zre 12600 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) |
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