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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssfzo12 12601 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 < 𝐿) → ((𝐾..^𝐿) ⊆ (𝑀..^𝑁) → (𝑀𝐾𝐿𝑁)))

Theoremssfzoulel 12602 If a half-open integer range is a subset of a half-open range of nonnegative integers, but its lower bound is greater than or equal to the upper bound of the containing range, or its upper bound is less than or equal to 0, then its upper bound is less than or equal to its lower bound (and therefore it is actually empty). (Contributed by Alexander van der Vekens, 24-May-2018.)
((𝑁 ∈ ℕ0𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁𝐴𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵𝐴)))

Theoremssfzo12bi 12603 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.)
(((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 < 𝐿) → ((𝐾..^𝐿) ⊆ (𝑀..^𝑁) ↔ (𝑀𝐾𝐿𝑁)))

Theoremubmelm1fzo 12604 The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)
(𝐾 ∈ (0..^𝑁) → ((𝑁𝐾) − 1) ∈ (0..^𝑁))

Theoremfzofzp1 12605 If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐶 ∈ (𝐴..^𝐵) → (𝐶 + 1) ∈ (𝐴...𝐵))

Theoremfzofzp1b 12606 If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐶 ∈ (ℤ𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))

Theoremelfzom1b 12607 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 12608 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 12609 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 12610 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 12611 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 12612 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 12613 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.) (Revised by Thierry Arnoux, 22-Dec-2021.)
((𝑀 ∈ (0...𝐾) ∧ ¬ 𝑀 ∈ (1..^𝐾)) → (𝑀 = 0 ∨ 𝑀 = 𝐾))

Theoremelfznelfzob 12614 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.) (Revised by Thierry Arnoux, 22-Dec-2021.)
(𝑀 ∈ (0...𝐾) → (¬ 𝑀 ∈ (1..^𝐾) ↔ (𝑀 = 0 ∨ 𝑀 = 𝐾)))

Theorempeano2fzor 12615 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 12616 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}))

Theoremfzosplitpr 12617 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
(𝐵 ∈ (ℤ𝐴) → (𝐴..^(𝐵 + 2)) = ((𝐴..^𝐵) ∪ {𝐵, (𝐵 + 1)}))

Theoremfzosplitprm1 12618 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 25-Jun-2022.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}))

Theoremfzosplitsni 12619 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 12620 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}))

Theoremelfzr 12621 A member of a finite interval of integers is either a member of the corresponding half-open integer range or the upper bound of the interval. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (𝑀...𝑁) → (𝐾 ∈ (𝑀..^𝑁) ∨ 𝐾 = 𝑁))

Theoremelfzlmr 12622 A member of a finite interval of integers is either its lower bound or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (𝑀...𝑁) → (𝐾 = 𝑀𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))

Theoremelfz0lmr 12623 A member of a finite interval of nonnegative integers is either 0 or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (0...𝑁) → (𝐾 = 0 ∨ 𝐾 ∈ (1..^𝑁) ∨ 𝐾 = 𝑁))

Theoremfzostep1 12624 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 12625* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 12466. (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 12626* 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 11513 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 12627 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 12628 Lemma for injresinj 12629. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Proof shortened by AV, 14-Feb-2021.) (Revised by Thierry Arnoux, 23-Dec-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 12629 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 12630 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 12631 Extend class notation with floor (greatest integer) function.
class

Syntaxcceil 12632 Extend class notation to include the ceiling function.
class

Definitiondf-fl 12633* Define the floor (greatest integer less than or equal to) function. See flval 12635 for its value, fllelt 12638 for its basic property, and flcl 12636 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 27434).

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 12634 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 12679 for its value, ceilge 12685 and ceilm1lt 12687 for its basic properties, and ceilcl 12683 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 27435).

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 12635* 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 12636 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 12637 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)

Theoremfllelt 12638 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 12639 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (⌊‘𝐴) ∈ ℤ)

Theoremflle 12640 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)

Theoremflltp1 12641 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))

Theoremfllep1 12642 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
(𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))

Theoremfraclt1 12643 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1)

Theoremfracle1 12644 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 12645 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴)))

Theoremflge 12646 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 12647 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 12648 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 12649 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
(𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)

Theoremflidm 12650 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
(𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))

Theoremflidz 12651 A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ → ((⌊‘𝐴) = 𝐴𝐴 ∈ ℤ))

Theoremflltnz 12652 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 12653 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 12654 Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (⌊‘𝐵) ∈ (ℤ‘(⌊‘𝐴)))

Theoremflval2 12655* 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 12656* 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 12657 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 12658 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))

Theoremadddivflid 12659 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 12660 The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 12735 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 12661 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 12662 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 12663 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 12664 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 12665 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 12666 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 12667 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 12668 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 12669 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 11825.) (Contributed by NM, 12-Mar-2005.)
(((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁𝐴𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))

Theorem2tnp1ge0ge0 12670 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 12671 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 12672 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 12673 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 12674 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 12675 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 12676 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 12677 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 12678 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 12679 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))

Theoremdfceil2 12680* Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
⌈ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))

Theoremceilval2 12681* The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = (𝑦 ∈ ℤ (𝐴𝑦𝑦 < (𝐴 + 1))))

Theoremceicl 12682 The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)
(𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ)

Theoremceilcl 12683 Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ)

Theoremceige 12684 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 12685 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 12686 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 12687 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 12688 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 12689 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 12690 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)

Theoremceilidz 12691 A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))

Theoremflleceil 12692 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 12693 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 12694 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 15173. (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 12695 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 12696 Alternate proof of quoremnn0 12695 not using quoremz 12694. TODO - Keep either quoremnn0ALT 12696 (if we don't keep quoremz 12694) or quoremnn0 12695. (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 12697 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 12725? (Contributed by NM, 16-Aug-2008.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴𝑍)       (𝐴 ∈ ℝ → (0 ≤ 𝐹𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))

Theoremintfracq 12698 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 12697. (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 12699 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 12700 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))))

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