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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 1z 12001 | One is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 1 ∈ ℤ | ||
Theorem | 1zzd 12002 | One is an integer, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℤ) | ||
Theorem | 2z 12003 | 2 is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 2 ∈ ℤ | ||
Theorem | 3z 12004 | 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 3 ∈ ℤ | ||
Theorem | 4z 12005 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
⊢ 4 ∈ ℤ | ||
Theorem | znegcl 12006 | Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | ||
Theorem | neg1z 12007 | -1 is an integer. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℤ | ||
Theorem | znegclb 12008 | 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 12009 | The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ) | ||
Theorem | nn0negzi 12010 | The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ -𝑁 ∈ ℤ | ||
Theorem | zaddcl 12011 | 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 12012 | Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | ||
Theorem | zsubcl 12013 | Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) | ||
Theorem | peano2zm 12014 | "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | ||
Theorem | zletr 12015 | 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 12016 | Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.) |
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ)) | ||
Theorem | znnsub 12017 | The positive difference of unequal integers is a positive integer. (Generalization of nnsub 11670.) (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) | ||
Theorem | znn0sub 12018 | The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 11936.) (Contributed by NM, 14-Jul-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0)) | ||
Theorem | nzadd 12019 | 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 12020 | Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | ||
Theorem | zltp1le 12021 | 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 12022 | Integer ordering relation. (Contributed by NM, 10-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) | ||
Theorem | zlem1lt 12023 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | zltlem1 12024 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | zgt0ge1 12025 | 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 12026 | 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 12027 | Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | ||
Theorem | nnaddm1cl 12028 | 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 12029 | 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 12030 | Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) | ||
Theorem | nn0ltlem1 12031 | 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 12032 | Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0) | ||
Theorem | nn0lt10b 12033 | 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 12034 | 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 12035 | 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 12036 | Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | nnlem1lt 12037 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | nnltlem1 12038 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | nnm1ge0 12039 | 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 12040 | 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 12041* | 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 12042 | 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 12043 | 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 12044* | 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 12045* | 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 12046 | 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 12047 | 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 12048 | 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 12049 | One-half is not an integer. (Contributed by NM, 31-Jul-2004.) |
⊢ ¬ (1 / 2) ∈ ℤ | ||
Theorem | 3halfnz 12050 | Three halves is not an integer. (Contributed by AV, 2-Jun-2020.) |
⊢ ¬ (3 / 2) ∈ ℤ | ||
Theorem | suprzcl 12051* | 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 12052* | Two ways to express "𝐴 is a prime number (or 1)." See also isprm 16007. (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 12053 | 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 12054 | 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 12055 | 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 12056 | 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 12057 | An integer is even or odd. (Contributed by NM, 1-Jan-2006.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | ||
Theorem | zeo2 12058 | 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 12059* | 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 12060* | 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 12061* | 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 12062* | An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 11640 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 12063* | 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 12064* | 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 12065* | 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 12066* | 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 12067* | 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 12066 or nn0indALT 12067 may be used; see comment for nnind 11645. (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 12068* | 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 12069* | 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 12070* | 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 12071* | 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 12072* | 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 12073* | 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 12074 | A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) | ||
Theorem | nnzd 12075 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) | ||
Theorem | zred 12076 | An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | zcnd 12077 | An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | znegcld 12078 | Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℤ) | ||
Theorem | peano2zd 12079 | Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) | ||
Theorem | zaddcld 12080 | Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) | ||
Theorem | zsubcld 12081 | Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) | ||
Theorem | zmulcld 12082 | Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) | ||
Theorem | znnn0nn 12083 | 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 12084 | 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 12085* | 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 12086 | 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 12087 | Constant used for decimal constructor. |
class ;𝐴𝐵 | ||
Definition | df-dec 12088 | Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 28153. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.) |
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | ||
Theorem | 9p1e10 12089 | 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.) |
⊢ (9 + 1) = ;10 | ||
Theorem | dfdec10 12090 | Version of the definition of the "decimal constructor" using ;10 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.) |
⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | ||
Theorem | decex 12091 | A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ ;𝐴𝐵 ∈ V | ||
Theorem | deceq1 12092 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) | ||
Theorem | deceq2 12093 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) | ||
Theorem | deceq1i 12094 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐶 | ||
Theorem | deceq2i 12095 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐶𝐴 = ;𝐶𝐵 | ||
Theorem | deceq12i 12096 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐷 | ||
Theorem | numnncl 12097 | Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ | ||
Theorem | num0u 12098 | Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0) | ||
Theorem | num0h 12099 | Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 = ((𝑇 · 0) + 𝐴) | ||
Theorem | numcl 12100 | Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
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