P. 139 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13801-13900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzoopth 13801	A half-open integer range can represent an ordered pair, analogous to fzopth 13601. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → ((𝑀..^𝑁) = (𝐽..^𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
Theorem	ubmelm1fzo 13802	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) → ((𝑁 − 𝐾) − 1) ∈ (0..^𝑁))
Theorem	fzofzp1 13803	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))
Theorem	fzofzp1b 13804	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 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))
Theorem	elfzom1b 13805	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))))
Theorem	elfzom1elp1fzo1 13806	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))
Theorem	elfzo1elm1fzo0 13807	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)))
Theorem	elfzonelfzo 13808	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)))
Theorem	fzonfzoufzol 13809	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 < 𝑁 ∧ 𝐼 ∈ (0..^𝑁)) → (¬ 𝐼 ∈ ((𝑁 − 𝑀)..^𝑁) → 𝐼 ∈ (0..^(𝑁 − 𝑀))))
Theorem	elfzomelpfzo 13810	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 ∈ ((𝑀 − 𝐿)..^(𝑁 − 𝐿)) ↔ (𝐾 + 𝐿) ∈ (𝑀..^𝑁)))
Theorem	elfznelfzo 13811	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))
Theorem	elfznelfzob 13812	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)))
Theorem	peano2fzor 13813	A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑀..^𝑁))
Theorem	fzosplitsn 13814	Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	fzosplitpr 13815	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 2)) = ((𝐴..^𝐵) ∪ {𝐵, (𝐵 + 1)}))
Theorem	fzosplitprm1 13816	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}))
Theorem	fzosplitsni 13817	Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^(𝐵 + 1)) ↔ (𝐶 ∈ (𝐴..^𝐵) ∨ 𝐶 = 𝐵)))
Theorem	fzisfzounsn 13818	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 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	elfzr 13819	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))
Theorem	elfzlmr 13820	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 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))
Theorem	elfz0lmr 13821	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))
Theorem	fzostep1 13822	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))
Theorem	fzoshftral 13823*	Shift the scanning order inside of a universal quantification restricted to a half-open integer range, analogous to fzshftral 13655. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑))
Theorem	fzind2 13824*	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 12716 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜓)    &   ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝜒 → 𝜃))    ⇒   ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝜏)
Theorem	fvinim0ffz 13825	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)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
Theorem	injresinjlem 13826	Lemma for injresinj 13827. (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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
Theorem	injresinj 13827	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) ≠ (𝐹‘𝐾)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ◡𝐹)))
Theorem	subfzo0 13828	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))
Theorem	fvf1tp 13829	Values of a one-to-one function between two sets with three elements. Actually, such a function is a bijection. (Contributed by AV, 23-Jul-2025.)
⊢ (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
5.6  Elementary integer functions
5.6.1  The floor and ceiling functions
Syntax	cfl 13830	Extend class notation with floor (greatest integer) function.
class ⌊
Syntax	cceil 13831	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 13832*	Define the floor (greatest integer less than or equal to) function. See flval 13834 for its value, fllelt 13837 for its basic property, and flcl 13835 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30466). 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 13833	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 13878 for its value, ceilge 13885 and ceilm1lt 13888 for its basic properties, and ceilcl 13882 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 30467). 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 13834*	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 13835	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 13836	The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
Theorem	fllelt 13837	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 13838	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 13839	A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
Theorem	flltp1 13840	A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
Theorem	fllep1 13841	A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
Theorem	fraclt1 13842	The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1)
Theorem	fracle1 13843	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 13844	The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴)))
Theorem	flge 13845	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 13846	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 13847	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 13848	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
Theorem	flidm 13849	The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
⊢ (𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
Theorem	flidz 13850	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 13851	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 13852	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 13853	Ordering relation for the floor function. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴)))
Theorem	flval2 13854*	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 13855*	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 13856	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 13857	A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1)))
Theorem	adddivflid 13858	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 13859	The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13936 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 13860	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 13861	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 13862	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 13863	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 13864	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 13865	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 13866	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 13867	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 13868	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 12989.) (Contributed by NM, 12-Mar-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))
Theorem	2tnp1ge0ge0 13869	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 13870	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 13871	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 13872	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 13873	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 13874	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 13875	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 13876	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 13877	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 13878	The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Theorem	dfceil2 13879*	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 13880*	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 13881	The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)
⊢ (𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ)
Theorem	ceilcl 13882	Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ)
Theorem	ceilcld 13883	Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (⌈‘𝐴) ∈ ℤ)
Theorem	ceige 13884	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 13885	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 13886	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 13887	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 13888	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 13889	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 13890	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 13891	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
Theorem	ceilidz 13892	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 13893	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 13894	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 13895	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 16440. (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 13896	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 13897	Alternate proof of quoremnn0 13896 not using quoremz 13895. TODO - Keep either quoremnn0ALT 13897 (if we don't keep quoremz 13895) or quoremnn0 13896? (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 13898	Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 13926? (Contributed by NM, 16-Aug-2008.)
⊢ 𝑍 = (⌊‘𝐴)    &   ⊢ 𝐹 = (𝐴 − 𝑍)    ⇒   ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))
Theorem	intfracq 13899	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 13898. (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 13900	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)))