Home | Metamath
Proof Explorer Theorem List (p. 121 of 450) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28695) |
Hilbert Space Explorer
(28696-30218) |
Users' Mathboxes
(30219-44926) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfz2 12001 | Alternative definition of the integers, based on elz2 12000. (Contributed by Mario Carneiro, 16-May-2014.) |
⊢ ℤ = ( − “ (ℕ × ℕ)) | ||
Theorem | zexALT 12002 | Alternate proof of zex 11991. (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 12003 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022.) |
⊢ ℕ ⊆ ℤ | ||
Theorem | nn0ssz 12004 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
⊢ ℕ0 ⊆ ℤ | ||
Theorem | nnz 12005 | A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | ||
Theorem | nn0z 12006 | A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | ||
Theorem | nnzi 12007 | A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | nn0zi 12008 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | elnnz1 12009 | 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 12010 | 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 12011 | Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥} | ||
Theorem | nn0zrab 12012 | Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥} | ||
Theorem | 1z 12013 | One is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 1 ∈ ℤ | ||
Theorem | 1zzd 12014 | One is an integer, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℤ) | ||
Theorem | 2z 12015 | 2 is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 2 ∈ ℤ | ||
Theorem | 3z 12016 | 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 3 ∈ ℤ | ||
Theorem | 4z 12017 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
⊢ 4 ∈ ℤ | ||
Theorem | znegcl 12018 | Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | ||
Theorem | neg1z 12019 | -1 is an integer. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℤ | ||
Theorem | znegclb 12020 | 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 12021 | The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ) | ||
Theorem | nn0negzi 12022 | The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ -𝑁 ∈ ℤ | ||
Theorem | zaddcl 12023 | 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 12024 | Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | ||
Theorem | zsubcl 12025 | Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) | ||
Theorem | peano2zm 12026 | "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | ||
Theorem | zletr 12027 | 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 12028 | Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.) |
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ)) | ||
Theorem | znnsub 12029 | The positive difference of unequal integers is a positive integer. (Generalization of nnsub 11682.) (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) | ||
Theorem | znn0sub 12030 | The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 11948.) (Contributed by NM, 14-Jul-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0)) | ||
Theorem | nzadd 12031 | The sum of a real number not being an integer and an integer is not an integer. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ (ℝ ∖ ℤ) ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℤ)) | ||
Theorem | zmulcl 12032 | Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | ||
Theorem | zltp1le 12033 | Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | ||
Theorem | zleltp1 12034 | Integer ordering relation. (Contributed by NM, 10-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) | ||
Theorem | zlem1lt 12035 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | zltlem1 12036 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | zgt0ge1 12037 | An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.) |
⊢ (𝑍 ∈ ℤ → (0 < 𝑍 ↔ 1 ≤ 𝑍)) | ||
Theorem | nnleltp1 12038 | Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ≤ 𝐵 ↔ 𝐴 < (𝐵 + 1))) | ||
Theorem | nnltp1le 12039 | Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | ||
Theorem | nnaddm1cl 12040 | Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) − 1) ∈ ℕ) | ||
Theorem | nn0ltp1le 12041 | Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | ||
Theorem | nn0leltp1 12042 | Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) | ||
Theorem | nn0ltlem1 12043 | Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | nn0sub2 12044 | Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0) | ||
Theorem | nn0lt10b 12045 | A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0)) | ||
Theorem | nn0lt2 12046 | A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 = 0 ∨ 𝑁 = 1)) | ||
Theorem | nn0le2is012 12047 | A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) | ||
Theorem | nn0lem1lt 12048 | Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | nnlem1lt 12049 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | nnltlem1 12050 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | nnm1ge0 12051 | A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.) |
⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) | ||
Theorem | nn0ge0div 12052 | Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → 0 ≤ (𝐾 / 𝐿)) | ||
Theorem | zdiv 12053* | Two ways to express "𝑀 divides 𝑁. (Contributed by NM, 3-Oct-2008.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ)) | ||
Theorem | zdivadd 12054 | Property of divisibility: if 𝐷 divides 𝐴 and 𝐵 then it divides 𝐴 + 𝐵. (Contributed by NM, 3-Oct-2008.) |
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ)) → ((𝐴 + 𝐵) / 𝐷) ∈ ℤ) | ||
Theorem | zdivmul 12055 | Property of divisibility: if 𝐷 divides 𝐴 then it divides 𝐵 · 𝐴. (Contributed by NM, 3-Oct-2008.) |
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 / 𝐷) ∈ ℤ) → ((𝐵 · 𝐴) / 𝐷) ∈ ℤ) | ||
Theorem | zextle 12056* | An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 = 𝑁) | ||
Theorem | zextlt 12057* | An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 < 𝑀 ↔ 𝑘 < 𝑁)) → 𝑀 = 𝑁) | ||
Theorem | recnz 12058 | The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → ¬ (1 / 𝐴) ∈ ℤ) | ||
Theorem | btwnnz 12059 | A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 𝐵 ∧ 𝐵 < (𝐴 + 1)) → ¬ 𝐵 ∈ ℤ) | ||
Theorem | gtndiv 12060 | A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ ∧ 𝐵 < 𝐴) → ¬ (𝐵 / 𝐴) ∈ ℤ) | ||
Theorem | halfnz 12061 | One-half is not an integer. (Contributed by NM, 31-Jul-2004.) |
⊢ ¬ (1 / 2) ∈ ℤ | ||
Theorem | 3halfnz 12062 | Three halves is not an integer. (Contributed by AV, 2-Jun-2020.) |
⊢ ¬ (3 / 2) ∈ ℤ | ||
Theorem | suprzcl 12063* | The supremum of a bounded-above set of integers is a member of the set. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴) | ||
Theorem | prime 12064* | Two ways to express "𝐴 is a prime number (or 1)." See also isprm 16017. (Contributed by NM, 4-May-2005.) |
⊢ (𝐴 ∈ ℕ → (∀𝑥 ∈ ℕ ((𝐴 / 𝑥) ∈ ℕ → (𝑥 = 1 ∨ 𝑥 = 𝐴)) ↔ ∀𝑥 ∈ ℕ ((1 < 𝑥 ∧ 𝑥 ≤ 𝐴 ∧ (𝐴 / 𝑥) ∈ ℕ) → 𝑥 = 𝐴))) | ||
Theorem | msqznn 12065 | The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) → (𝐴 · 𝐴) ∈ ℕ) | ||
Theorem | zneo 12066 | No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ≠ ((2 · 𝐵) + 1)) | ||
Theorem | nneo 12067 | A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.) |
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ)) | ||
Theorem | nneoi 12068 | A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ) | ||
Theorem | zeo 12069 | An integer is even or odd. (Contributed by NM, 1-Jan-2006.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | ||
Theorem | zeo2 12070 | An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℤ)) | ||
Theorem | peano2uz2 12071* | Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ {𝑥 ∈ ℤ ∣ 𝐴 ≤ 𝑥}) → (𝐵 + 1) ∈ {𝑥 ∈ ℤ ∣ 𝐴 ≤ 𝑥}) | ||
Theorem | peano5uzi 12072* | Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.) |
⊢ 𝑁 ∈ ℤ ⇒ ⊢ ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴) | ||
Theorem | peano5uzti 12073* | Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴)) | ||
Theorem | dfuzi 12074* | An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 11651 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.) |
⊢ 𝑁 ∈ ℤ ⇒ ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} = ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | ||
Theorem | uzind 12075* | Induction on the upper integers that start at 𝑀. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.) |
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜏) | ||
Theorem | uzind2 12076* | Induction on the upper integers that start after an integer 𝑀. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.) |
⊢ (𝑗 = (𝑀 + 1) → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 < 𝑘) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → 𝜏) | ||
Theorem | uzind3 12077* | Induction on the upper integers that start at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.) |
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑚 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑚 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → 𝜏) | ||
Theorem | nn0ind 12078* | Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.) |
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ ℕ0 → 𝜏) | ||
Theorem | nn0indALT 12079* | Principle of Mathematical Induction (inference schema) on nonnegative integers. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either nn0ind 12078 or nn0indALT 12079 may be used; see comment for nnind 11656. (Contributed by NM, 28-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) & ⊢ 𝜓 & ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) ⇒ ⊢ (𝐴 ∈ ℕ0 → 𝜏) | ||
Theorem | nn0indd 12080* | Principle of Mathematical Induction (inference schema) on nonnegative integers, a deduction version. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
⊢ (𝑥 = 0 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → 𝜂) | ||
Theorem | fzind 12081* | Induction on the integers from 𝑀 to 𝑁 inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏)) & ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓) & ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃)) ⇒ ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏) | ||
Theorem | fnn0ind 12082* | Induction on the integers from 0 to 𝑁 inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏)) & ⊢ (𝑁 ∈ ℕ0 → 𝜓) & ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ∧ 𝑦 < 𝑁) → (𝜒 → 𝜃)) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → 𝜏) | ||
Theorem | nn0ind-raph 12083* | Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ ℕ0 → 𝜏) | ||
Theorem | zindd 12084* | Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.) |
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) & ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) & ⊢ (𝜁 → 𝜓) & ⊢ (𝜁 → (𝑦 ∈ ℕ0 → (𝜒 → 𝜏))) & ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) ⇒ ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) | ||
Theorem | btwnz 12085* | Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.) |
⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℤ 𝑥 < 𝐴 ∧ ∃𝑦 ∈ ℤ 𝐴 < 𝑦)) | ||
Theorem | nn0zd 12086 | A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) | ||
Theorem | nnzd 12087 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) | ||
Theorem | zred 12088 | An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | zcnd 12089 | An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | znegcld 12090 | Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℤ) | ||
Theorem | peano2zd 12091 | Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) | ||
Theorem | zaddcld 12092 | Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) | ||
Theorem | zsubcld 12093 | Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) | ||
Theorem | zmulcld 12094 | Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) | ||
Theorem | znnn0nn 12095 | The negative of a negative integer, is a natural number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) | ||
Theorem | zadd2cl 12096 | Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
⊢ (𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ) | ||
Theorem | zriotaneg 12097* | The negative of the unique integer such that 𝜑. (Contributed by AV, 1-Dec-2018.) |
⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ ℤ 𝜑 → (℩𝑥 ∈ ℤ 𝜑) = -(℩𝑦 ∈ ℤ 𝜓)) | ||
Theorem | suprfinzcl 12098 | The supremum of a nonempty finite set of integers is a member of the set. (Contributed by AV, 1-Oct-2019.) |
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴) | ||
Syntax | cdc 12099 | Constant used for decimal constructor. |
class ;𝐴𝐵 | ||
Definition | df-dec 12100 | Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 28225. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.) |
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |