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Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfzom1b 12301 An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (1..^𝑁) ↔ (𝐾 − 1) ∈ (0..^(𝑁 − 1))))
 
Theoremelfzom1elp1fzo1 12302 Membership of a nonnegative integer incremented by one in a half-open range of positive integers. (Contributed by AV, 20-Mar-2021.)
((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (1..^𝑁))
 
Theoremelfzo1elm1fzo0 12303 Membership of a positive integer decremented by one in a half-open range of nonnegative integers. (Contributed by AV, 20-Mar-2021.)
(𝐼 ∈ (1..^𝑁) → (𝐼 − 1) ∈ (0..^(𝑁 − 1)))
 
Theoremelfzonelfzo 12304 If an element of a half-open integer range is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
(𝑁 ∈ ℤ → ((𝐾 ∈ (𝑀..^𝑅) ∧ ¬ 𝐾 ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑁..^𝑅)))
 
Theoremfzonfzoufzol 12305 If an element of a half-open integer range is not in the upper part of the range, it is in the lower part of the range. (Contributed by Alexander van der Vekens, 29-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑀 < 𝑁𝐼 ∈ (0..^𝑁)) → (¬ 𝐼 ∈ ((𝑁𝑀)..^𝑁) → 𝐼 ∈ (0..^(𝑁𝑀))))
 
Theoremelfzomelpfzo 12306 An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → (𝐾 ∈ ((𝑀𝐿)..^(𝑁𝐿)) ↔ (𝐾 + 𝐿) ∈ (𝑀..^𝑁)))
 
Theoremelfznelfzo 12307 A value in a finite set of sequential integers is a border value if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
((𝑦 ∈ (0...𝐾) ∧ ¬ 𝑦 ∈ (1..^𝐾)) → (𝑦 = 0 ∨ 𝑦 = 𝐾))
 
Theoremelfznelfzob 12308 A value in a finite set of sequential integers is a border value if and only if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
(𝑦 ∈ (0...𝐾) → (¬ 𝑦 ∈ (1..^𝐾) ↔ (𝑦 = 0 ∨ 𝑦 = 𝐾)))
 
Theorempeano2fzor 12309 A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
((𝐾 ∈ (ℤ𝑀) ∧ (𝐾 + 1) ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑀..^𝑁))
 
Theoremfzosplitsn 12310 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵}))
 
Theoremfzosplitprm1 12311 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}))
 
Theoremfzosplitsni 12312 Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (𝐶 ∈ (𝐴..^(𝐵 + 1)) ↔ (𝐶 ∈ (𝐴..^𝐵) ∨ 𝐶 = 𝐵)))
 
Theoremfzisfzounsn 12313 A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
(𝐵 ∈ (ℤ𝐴) → (𝐴...𝐵) = ((𝐴..^𝐵) ∪ {𝐵}))
 
Theoremfzostep1 12314 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶))
 
Theoremfzoshftral 12315* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 12165. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘𝐾) / 𝑗]𝜑))
 
Theoremfzind2 12316* 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. Version of fzind 11215 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
(𝑥 = 𝑀 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐾 → (𝜑𝜏))    &   (𝑁 ∈ (ℤ𝑀) → 𝜓)    &   (𝑦 ∈ (𝑀..^𝑁) → (𝜒𝜃))       (𝐾 ∈ (𝑀...𝑁) → 𝜏)
 
Theoremfvinim0ffz 12317 The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
 
Theoreminjresinjlem 12318 Lemma for injresinj 12319. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Proof shortened by AV, 14-Feb-2021.)
𝑦 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
 
Theoreminjresinj 12319 A function whose restriction is injective and the values of the remaining arguments are different from all other values is injective itself. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
(𝐾 ∈ ℕ0 → ((𝐹:(0...𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹‘0) ≠ (𝐹𝐾)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))
 
Theoremsubfzo0 12320 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 < (𝐼𝐽) ∧ (𝐼𝐽) < 𝑁))
 
5.6  Elementary integer functions
 
5.6.1  The floor and ceiling functions
 
Syntaxcfl 12321 Extend class notation with floor (greatest integer) function.
class
 
Syntaxcceil 12322 Extend class notation to include the ceiling function.
class
 
Definitiondf-fl 12323* Define the floor (greatest integer less than or equal to) function. See flval 12325 for its value, fllelt 12328 for its basic property, and flcl 12326 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 26434).

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 (𝑦𝑥𝑥 < (𝑦 + 1))))
 
Definitiondf-ceil 12324 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 12369 for its value, ceilge 12375 and ceilm1lt 12377 for its basic properties, and ceilcl 12373 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 26435).

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))
 
Theoremflval 12325* 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 (𝑥𝐴𝐴 < (𝑥 + 1))))
 
Theoremflcl 12326 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)
 
Theoremreflcl 12327 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
 
Theoremfllelt 12328 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) ≤ 𝐴𝐴 < ((⌊‘𝐴) + 1)))
 
Theoremflcld 12329 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (⌊‘𝐴) ∈ ℤ)
 
Theoremflle 12330 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
 
Theoremflltp1 12331 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
 
Theoremfllep1 12332 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
(𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
 
Theoremfraclt1 12333 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1)
 
Theoremfracle1 12334 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)
 
Theoremfracge0 12335 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴)))
 
Theoremflge 12336 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) → (𝐵𝐴𝐵 ≤ (⌊‘𝐴)))
 
Theoremfllt 12337 The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵))
 
Theoremflflp1 12338 Move floor function between strict and non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((⌊‘𝐴) ≤ 𝐵𝐴 < ((⌊‘𝐵) + 1)))
 
Theoremflid 12339 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
(𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
 
Theoremflidm 12340 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
(𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
 
Theoremflidz 12341 A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ → ((⌊‘𝐴) = 𝐴𝐴 ∈ ℤ))
 
Theoremflltnz 12342 If A is not an integer, then the floor of A is less than A. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴)
 
Theoremflwordi 12343 Ordering relationship for the greatest integer function. (Contributed by NM, 31-Dec-2005.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))
 
Theoremflword2 12344 Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (⌊‘𝐵) ∈ (ℤ‘(⌊‘𝐴)))
 
Theoremflval2 12345* An alternate way to define the floor (greatest integer) function. (Contributed by NM, 16-Nov-2004.)
(𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦𝐴𝑦𝑥))))
 
Theoremflval3 12346* An alternate way to define the floor (greatest integer) 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 ∣ 𝑥𝐴}, ℝ, < ))
 
Theoremflbi 12347 A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵𝐴𝐴 < (𝐵 + 1))))
 
Theoremflbi2 12348 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))
 
Theoremadddivflid 12349 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 ∈ ℕ0𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))
 
Theoremico01fl0 12350 The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 12425 or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019.)
(𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0)
 
Theoremflge0nn0 12351 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)
 
Theoremflge1nn 12352 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)
 
Theoremfldivnn0 12353 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 ∈ ℕ0𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ0)
 
Theoremrefldivcl 12354 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)
 
Theoremdivfl0 12355 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))
 
Theoremfladdz 12356 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))
 
Theoremflzadd 12357 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)))
 
Theoremflmulnn0 12358 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 ∈ ℕ0𝐴 ∈ ℝ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
 
Theorembtwnzge0 12359 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 11529.) (Contributed by NM, 12-Mar-2005.)
(((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁𝐴𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))
 
Theorem2tnp1ge0ge0 12360 Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
(𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁))
 
Theoremflhalf 12361 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))))
 
Theoremfldivle 12362 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))
 
Theoremfldivnn0le 12363 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 ∈ ℕ0𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿))
 
Theoremflltdivnn0lt 12364 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 ∈ ℕ0𝑁 ∈ ℕ0𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿)))
 
Theoremltdifltdiv 12365 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 < (𝐶𝐵) → ((⌊‘(𝐴 / 𝐵)) + 1) < (𝐶 / 𝐵)))
 
Theoremfldiv4p1lem1div2 12366 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))
 
Theoremfldiv4lem1div2uz2 12367 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.)
(𝑁 ∈ (ℤ‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2))
 
Theoremfldiv4lem1div2 12368 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))
 
Theoremceilval 12369 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
 
Theoremdfceil2 12370* Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
⌈ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))
 
Theoremceilval2 12371* The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = (𝑦 ∈ ℤ (𝐴𝑦𝑦 < (𝐴 + 1))))
 
Theoremceicl 12372 The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)
(𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ)
 
Theoremceilcl 12373 Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ)
 
Theoremceige 12374 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))
 
Theoremceilge 12375 The ceiling of a real number is greater than or equal to that number. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → 𝐴 ≤ (⌈‘𝐴))
 
Theoremceim1l 12376 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)
 
Theoremceilm1lt 12377 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)
 
Theoremceile 12378 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 ∧ 𝐴𝐵) → -(⌊‘-𝐴) ≤ 𝐵)
 
Theoremceille 12379 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 ∧ 𝐴𝐵) → (⌈‘𝐴) ≤ 𝐵)
 
Theoremceilid 12380 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
 
Theoremceilidz 12381 A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
 
Theoremflleceil 12382 The floor of a real number is less than or equal to its ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ≤ (⌈‘𝐴))
 
Theoremfleqceilz 12383 A real number is an integer iff its floor equals its ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))
 
Theoremquoremz 12384 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 14836. (Contributed by NM, 14-Aug-2008.)
𝑄 = (⌊‘(𝐴 / 𝐵))    &   𝑅 = (𝐴 − (𝐵 · 𝑄))       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵𝐴 = ((𝐵 · 𝑄) + 𝑅))))
 
Theoremquoremnn0 12385 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 ∈ ℕ0𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵𝐴 = ((𝐵 · 𝑄) + 𝑅))))
 
Theoremquoremnn0ALT 12386 Alternate proof of quoremnn0 12385 not using quoremz 12384. TODO - Keep either quoremnn0ALT 12386 (if we don't keep quoremz 12384) or quoremnn0 12385. (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑄 = (⌊‘(𝐴 / 𝐵))    &   𝑅 = (𝐴 − (𝐵 · 𝑄))       ((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵𝐴 = ((𝐵 · 𝑄) + 𝑅))))
 
Theoremintfrac2 12387 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 12415? (Contributed by NM, 16-Aug-2008.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴𝑍)       (𝐴 ∈ ℝ → (0 ≤ 𝐹𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))
 
Theoremintfracq 12388 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 12387. (Contributed by NM, 16-Aug-2008.)
𝑍 = (⌊‘(𝑀 / 𝑁))    &   𝐹 = ((𝑀 / 𝑁) − 𝑍)       ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝐹𝐹 ≤ ((𝑁 − 1) / 𝑁) ∧ (𝑀 / 𝑁) = (𝑍 + 𝐹)))
 
Theoremfldiv 12389 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)))
 
Theoremfldiv2 12390 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))))
 
Theoremfznnfl 12391 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 ∧ 𝐾𝑁)))
 
Theoremuzsup 12392 An upper set of integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
 
Theoremioopnfsup 12393 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞)
 
Theoremicopnfsup 12394 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → sup((𝐴[,)+∞), ℝ*, < ) = +∞)
 
Theoremrpsup 12395 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ+, ℝ*, < ) = +∞
 
Theoremresup 12396 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ, ℝ*, < ) = +∞
 
Theoremxrsup 12397 The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ*, ℝ*, < ) = +∞
 
5.6.2  The modulo (remainder) operation
 
Syntaxcmo 12398 Extend class notation with the modulo operation.
class mod
 
Definitiondf-mod 12399* Define the modulo (remainder) operation. See modval 12400 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 26436). (Contributed by NM, 10-Nov-2008.)
mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
 
Theoremmodval 12400 The value of the modulo operation. The modulo congruence notation of number theory, 𝐽𝐾 (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)))))
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