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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | subfzo0 13801 | The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.) |
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → (-𝑁 < (𝐼 − 𝐽) ∧ (𝐼 − 𝐽) < 𝑁)) | ||
Syntax | cfl 13802 | Extend class notation with floor (greatest integer) function. |
class ⌊ | ||
Syntax | cceil 13803 | Extend class notation to include the ceiling function. |
class ⌈ | ||
Definition | df-fl 13804* |
Define the floor (greatest integer less than or equal to) function. See
flval 13806 for its value, fllelt 13809 for its basic property, and flcl 13807
for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 30375).
The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.) |
⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) | ||
Definition | df-ceil 13805 |
The ceiling (least integer greater than or equal to) function. Defined in
ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of
Mathematical Functions" , front introduction, "Common Notations
and
Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4.
See ceilval 13850 for its value, ceilge 13857 and ceilm1lt 13860 for its basic
properties, and ceilcl 13854 for its closure. For example,
(⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1
(ex-ceil 30376).
The symbol ⌈ is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥)) | ||
Theorem | flval 13806* | Value of the floor (greatest integer) function. The floor of 𝐴 is the (unique) integer less than or equal to 𝐴 whose successor is strictly greater than 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) | ||
Theorem | flcl 13807 | The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | reflcl 13808 | The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | ||
Theorem | fllelt 13809 | A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.) |
⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | ||
Theorem | flcld 13810 | The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | flle 13811 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | ||
Theorem | flltp1 13812 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) | ||
Theorem | fllep1 13813 | A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1)) | ||
Theorem | fraclt1 13814 | The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1) | ||
Theorem | fracle1 13815 | The fractional part of a real number is less than or equal to one. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) ≤ 1) | ||
Theorem | fracge0 13816 | The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴))) | ||
Theorem | flge 13817 | The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Fan Zheng, 14-Jul-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴))) | ||
Theorem | fllt 13818 | The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵)) | ||
Theorem | flflp1 13819 | Move floor function between strict and non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((⌊‘𝐴) ≤ 𝐵 ↔ 𝐴 < ((⌊‘𝐵) + 1))) | ||
Theorem | flid 13820 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) | ||
Theorem | flidm 13821 | The floor function is idempotent. (Contributed by NM, 17-Aug-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴)) | ||
Theorem | flidz 13822 | A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.) |
⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ)) | ||
Theorem | flltnz 13823 | The floor of a non-integer real is less than it. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴) | ||
Theorem | flwordi 13824 | Ordering relation for the floor function. (Contributed by NM, 31-Dec-2005.) (Proof shortened by Fan Zheng, 14-Jul-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵)) | ||
Theorem | flword2 13825 | Ordering relation for the floor function. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴))) | ||
Theorem | flval2 13826* | An alternate way to define the floor function. (Contributed by NM, 16-Nov-2004.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦 ≤ 𝐴 → 𝑦 ≤ 𝑥)))) | ||
Theorem | flval3 13827* | An alternate way to define the floor function, as the supremum of all integers less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 6-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < )) | ||
Theorem | flbi 13828 | A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)))) | ||
Theorem | flbi2 13829 | A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1))) | ||
Theorem | adddivflid 13830 | The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴)) | ||
Theorem | ico01fl0 13831 | The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13908 or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019.) |
⊢ (𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0) | ||
Theorem | flge0nn0 13832 | The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by NM, 26-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0) | ||
Theorem | flge1nn 13833 | The floor of a number greater than or equal to 1 is a positive integer. (Contributed by NM, 26-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ) | ||
Theorem | fldivnn0 13834 | The floor function of a division of a nonnegative integer by a positive integer is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ0) | ||
Theorem | refldivcl 13835 | The floor function of a division of a real number by a positive real number is a real number. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ+) → (⌊‘(𝐾 / 𝐿)) ∈ ℝ) | ||
Theorem | divfl0 13836 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0)) | ||
Theorem | fladdz 13837 | An integer can be moved in and out of the floor of a sum. (Contributed by NM, 27-Apr-2005.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁)) | ||
Theorem | flzadd 13838 | An integer can be moved in and out of the floor of a sum. (Contributed by NM, 2-Jan-2009.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℝ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴))) | ||
Theorem | flmulnn0 13839 | Move a nonnegative integer in and out of a floor. (Contributed by NM, 2-Jan-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℝ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴))) | ||
Theorem | btwnzge0 13840 | A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (For the first half see rebtwnz 12975.) (Contributed by NM, 12-Mar-2005.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁)) | ||
Theorem | 2tnp1ge0ge0 13841 | Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁)) | ||
Theorem | flhalf 13842 | Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (2 · (⌊‘((𝑁 + 1) / 2)))) | ||
Theorem | fldivle 13843 | The floor function of a division of a real number by a positive real number is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ+) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿)) | ||
Theorem | fldivnn0le 13844 | The floor function of a division of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿)) | ||
Theorem | flltdivnn0lt 13845 | The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿))) | ||
Theorem | ltdifltdiv 13846 | If the dividend of a division is less than the difference between a real number and the divisor, the floor function of the division plus 1 is less than the division of the real number by the divisor. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐶 − 𝐵) → ((⌊‘(𝐴 / 𝐵)) + 1) < (𝐶 / 𝐵))) | ||
Theorem | fldiv4p1lem1div2 13847 | The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.) |
⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2)) | ||
Theorem | fldiv4lem1div2uz2 13848 | The floor of an integer greater than 1, divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 5-Jul-2021.) (Proof shortened by AV, 9-Jul-2022.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | ||
Theorem | fldiv4lem1div2 13849 | The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | ||
Theorem | ceilval 13850 | The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴)) | ||
Theorem | dfceil2 13851* | Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.) |
⊢ ⌈ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) | ||
Theorem | ceilval2 13852* | The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.) |
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = (℩𝑦 ∈ ℤ (𝐴 ≤ 𝑦 ∧ 𝑦 < (𝐴 + 1)))) | ||
Theorem | ceicl 13853 | The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.) |
⊢ (𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ) | ||
Theorem | ceilcl 13854 | Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceilcld 13855 | Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceige 13856 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jeff Hankins, 10-Jun-2007.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ -(⌊‘-𝐴)) | ||
Theorem | ceilge 13857 | The ceiling of a real number is greater than or equal to that number. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (⌈‘𝐴)) | ||
Theorem | ceilged 13858 | The ceiling of a real number is greater than or equal to that number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ (⌈‘𝐴)) | ||
Theorem | ceim1l 13859 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeff Hankins, 10-Jun-2007.) |
⊢ (𝐴 ∈ ℝ → (-(⌊‘-𝐴) − 1) < 𝐴) | ||
Theorem | ceilm1lt 13860 | One less than the ceiling of a real number is strictly less than that number. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℝ → ((⌈‘𝐴) − 1) < 𝐴) | ||
Theorem | ceile 13861 | The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeff Hankins, 10-Jun-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵) | ||
Theorem | ceille 13862 | The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by AV, 30-Nov-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (⌈‘𝐴) ≤ 𝐵) | ||
Theorem | ceilid 13863 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴) | ||
Theorem | ceilidz 13864 | A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴)) | ||
Theorem | flleceil 13865 | The floor of a real number is less than or equal to its ceiling. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ (⌈‘𝐴)) | ||
Theorem | fleqceilz 13866 | A real number is an integer iff its floor equals its ceiling. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴))) | ||
Theorem | quoremz 13867 | Quotient and remainder of an integer divided by a positive integer. TODO - is this really needed for anything? Should we use mod to simplify it? Remark (AV): This is a special case of divalg 16398. (Contributed by NM, 14-Aug-2008.) |
⊢ 𝑄 = (⌊‘(𝐴 / 𝐵)) & ⊢ 𝑅 = (𝐴 − (𝐵 · 𝑄)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅)))) | ||
Theorem | quoremnn0 13868 | Quotient and remainder of a nonnegative integer divided by a positive integer. (Contributed by NM, 14-Aug-2008.) |
⊢ 𝑄 = (⌊‘(𝐴 / 𝐵)) & ⊢ 𝑅 = (𝐴 − (𝐵 · 𝑄)) ⇒ ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅)))) | ||
Theorem | quoremnn0ALT 13869 | Alternate proof of quoremnn0 13868 not using quoremz 13867. TODO - Keep either quoremnn0ALT 13869 (if we don't keep quoremz 13867) or quoremnn0 13868? (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑄 = (⌊‘(𝐴 / 𝐵)) & ⊢ 𝑅 = (𝐴 − (𝐵 · 𝑄)) ⇒ ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅)))) | ||
Theorem | intfrac2 13870 | Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 13898? (Contributed by NM, 16-Aug-2008.) |
⊢ 𝑍 = (⌊‘𝐴) & ⊢ 𝐹 = (𝐴 − 𝑍) ⇒ ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) | ||
Theorem | intfracq 13871 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 13870. (Contributed by NM, 16-Aug-2008.) |
⊢ 𝑍 = (⌊‘(𝑀 / 𝑁)) & ⊢ 𝐹 = ((𝑀 / 𝑁) − 𝑍) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝐹 ∧ 𝐹 ≤ ((𝑁 − 1) / 𝑁) ∧ (𝑀 / 𝑁) = (𝑍 + 𝐹))) | ||
Theorem | fldiv 13872 | Cancellation of the embedded floor of a real divided by an integer. (Contributed by NM, 16-Aug-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑁)) = (⌊‘(𝐴 / 𝑁))) | ||
Theorem | fldiv2 13873 | Cancellation of an embedded floor of a ratio. Generalization of Equation 2.4 in [CormenLeisersonRivest] p. 33 (where 𝐴 must be an integer). (Contributed by NM, 9-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘(𝐴 / 𝑀)) / 𝑁)) = (⌊‘(𝐴 / (𝑀 · 𝑁)))) | ||
Theorem | fznnfl 13874 | Finite set of sequential integers starting at 1 and ending at a real number. (Contributed by Mario Carneiro, 3-May-2016.) |
⊢ (𝑁 ∈ ℝ → (𝐾 ∈ (1...(⌊‘𝑁)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | uzsup 13875 | An upper set of integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞) | ||
Theorem | ioopnfsup 13876 | An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞) | ||
Theorem | icopnfsup 13877 | An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴[,)+∞), ℝ*, < ) = +∞) | ||
Theorem | rpsup 13878 | The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ sup(ℝ+, ℝ*, < ) = +∞ | ||
Theorem | resup 13879 | The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ sup(ℝ, ℝ*, < ) = +∞ | ||
Theorem | xrsup 13880 | The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ sup(ℝ*, ℝ*, < ) = +∞ | ||
Syntax | cmo 13881 | Extend class notation with the modulo operation. |
class mod | ||
Definition | df-mod 13882* | Define the modulo (remainder) operation. See modval 13883 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 30377). (Contributed by NM, 10-Nov-2008.) |
⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))) | ||
Theorem | modval 13883 | The value of the modulo operation. The modulo congruence notation of number theory, 𝐽≡𝐾 (modulo 𝑁), can be expressed in our notation as (𝐽 mod 𝑁) = (𝐾 mod 𝑁). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive reals to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) (Contributed by NM, 10-Nov-2008.) (Revised by Mario Carneiro, 3-Nov-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | ||
Theorem | modvalr 13884 | The value of the modulo operation (multiplication in reversed order). (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵))) | ||
Theorem | modcl 13885 | Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ ℝ) | ||
Theorem | flpmodeq 13886 | Partition of a division into its integer part and the remainder. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴) | ||
Theorem | modcld 13887 | Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) | ||
Theorem | mod0 13888 | 𝐴 mod 𝐵 is zero iff 𝐴 is evenly divisible by 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 0 ↔ (𝐴 / 𝐵) ∈ ℤ)) | ||
Theorem | mulmod0 13889 | The product of an integer and a positive real number is 0 modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) (Revised by AV, 5-Jul-2020.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · 𝑀) mod 𝑀) = 0) | ||
Theorem | negmod0 13890 | 𝐴 is divisible by 𝐵 iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0)) | ||
Theorem | modge0 13891 | The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 0 ≤ (𝐴 mod 𝐵)) | ||
Theorem | modlt 13892 | The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) < 𝐵) | ||
Theorem | modelico 13893 | Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ (0[,)𝐵)) | ||
Theorem | moddiffl 13894 | Value of the modulo operation rewritten to give two ways of expressing the quotient when "𝐴 is divided by 𝐵 using Euclidean division." Multiplying both sides by 𝐵, this implies that 𝐴 mod 𝐵 differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 6-Sep-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) | ||
Theorem | moddifz 13895 | The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Mario Carneiro, 15-Jul-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ) | ||
Theorem | modfrac 13896 | The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008.) |
⊢ (𝐴 ∈ ℝ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴))) | ||
Theorem | flmod 13897 | The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1))) | ||
Theorem | intfrac 13898 | Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008.) |
⊢ (𝐴 ∈ ℝ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1))) | ||
Theorem | zmod10 13899 | An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0) | ||
Theorem | zmod1congr 13900 | Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 mod 1) = (𝐵 mod 1)) |
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