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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | xnn0xr 12601 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | ||
Theorem | 0xnn0 12602 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ 0 ∈ ℕ0* | ||
Theorem | pnf0xnn0 12603 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ +∞ ∈ ℕ0* | ||
Theorem | nn0nepnf 12604 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) | ||
Theorem | nn0xnn0d 12605 | 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 12606 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | xnn0nemnf 12607 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) | ||
Theorem | xnn0xrnemnf 12608 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) | ||
Theorem | xnn0nnn0pnf 12609 | 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 12610 | Extend class notation to include the class of integers. |
class ℤ | ||
Definition | df-z 12611 | 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 12612 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | ||
Theorem | nnnegz 12613 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | ||
Theorem | zre 12614 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | ||
Theorem | zcn 12615 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | ||
Theorem | zrei 12616 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝐴 ∈ ℤ ⇒ ⊢ 𝐴 ∈ ℝ | ||
Theorem | zssre 12617 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℝ | ||
Theorem | zsscn 12618 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℂ | ||
Theorem | zex 12619 | The set of integers exists. See also zexALT 12630. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ℤ ∈ V | ||
Theorem | elnnz 12620 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | ||
Theorem | 0z 12621 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ 0 ∈ ℤ | ||
Theorem | 0zd 12622 | Zero is an integer, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℤ) | ||
Theorem | elnn0z 12623 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) | ||
Theorem | elznn0nn 12624 | 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 12625 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | ||
Theorem | elznn 12626 | 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 | zle0orge1 12627 | There is no integer in the open unit interval, i.e., an integer is either less than or equal to 0 or greater than or equal to 1. (Contributed by AV, 4-Jun-2023.) |
⊢ (𝑍 ∈ ℤ → (𝑍 ≤ 0 ∨ 1 ≤ 𝑍)) | ||
Theorem | elz2 12628* | 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 12629 | Alternative definition of the integers, based on elz2 12628. (Contributed by Mario Carneiro, 16-May-2014.) |
⊢ ℤ = ( − “ (ℕ × ℕ)) | ||
Theorem | zexALT 12630 | Alternate proof of zex 12619. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℤ ∈ V | ||
Theorem | nnz 12631 | A positive integer is an integer. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022.) |
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | ||
Theorem | nnssz 12632 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
⊢ ℕ ⊆ ℤ | ||
Theorem | nn0ssz 12633 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
⊢ ℕ0 ⊆ ℤ | ||
Theorem | nnzOLD 12634 | Obsolete version of nnz 12631 as of 1-Feb-2025. (Contributed by NM, 9-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | ||
Theorem | nn0z 12635 | A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | ||
Theorem | nn0zd 12636 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) | ||
Theorem | nnzd 12637 | A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) | ||
Theorem | nnzi 12638 | A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | nn0zi 12639 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | elnnz1 12640 | 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 12641 | 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 12642 | Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥} | ||
Theorem | nn0zrab 12643 | Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥} | ||
Theorem | 1z 12644 | One is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 1 ∈ ℤ | ||
Theorem | 1zzd 12645 | One is an integer, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℤ) | ||
Theorem | 2z 12646 | 2 is an integer. (Contributed by NM, 10-May-2004.) |
⊢ 2 ∈ ℤ | ||
Theorem | 3z 12647 | 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 3 ∈ ℤ | ||
Theorem | 4z 12648 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
⊢ 4 ∈ ℤ | ||
Theorem | znegcl 12649 | Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | ||
Theorem | neg1z 12650 | -1 is an integer. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℤ | ||
Theorem | znegclb 12651 | 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 12652 | The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ) | ||
Theorem | nn0negzi 12653 | The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ -𝑁 ∈ ℤ | ||
Theorem | zaddcl 12654 | 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 12655 | Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | ||
Theorem | zsubcl 12656 | Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) | ||
Theorem | peano2zm 12657 | "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | ||
Theorem | zletr 12658 | 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 12659 | Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.) |
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ)) | ||
Theorem | znnsub 12660 | The positive difference of unequal integers is a positive integer. (Generalization of nnsub 12307.) (Contributed by NM, 11-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) | ||
Theorem | znn0sub 12661 | The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 12573.) (Contributed by NM, 14-Jul-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0)) | ||
Theorem | nzadd 12662 | 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 12663 | Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | ||
Theorem | zltp1le 12664 | 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 12665 | Integer ordering relation. (Contributed by NM, 10-May-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) | ||
Theorem | zlem1lt 12666 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | zltlem1 12667 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | zltlem1d 12668 | Integer ordering relation, a deduction version. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | zgt0ge1 12669 | 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 12670 | 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 12671 | Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | ||
Theorem | nnaddm1cl 12672 | 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 12673 | 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 12674 | Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) | ||
Theorem | nn0ltlem1 12675 | 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 12676 | Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0) | ||
Theorem | nn0lt10b 12677 | 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 12678 | 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 12679 | 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 12680 | Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | nnlem1lt 12681 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
Theorem | nnltlem1 12682 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | ||
Theorem | nnm1ge0 12683 | 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 12684 | 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 12685* | 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 12686 | 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 12687 | 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 12688* | 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 12689* | 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 12690 | 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 12691 | 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 12692 | 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 12693 | One-half is not an integer. (Contributed by NM, 31-Jul-2004.) |
⊢ ¬ (1 / 2) ∈ ℤ | ||
Theorem | 3halfnz 12694 | Three halves is not an integer. (Contributed by AV, 2-Jun-2020.) |
⊢ ¬ (3 / 2) ∈ ℤ | ||
Theorem | suprzcl 12695* | 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 12696* | Two ways to express "𝐴 is a prime number (or 1)". See also isprm 16706. (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 12697 | 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 12698 | 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 12699 | 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 12700 | 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) |
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