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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzpnn0elfzo1 13101 Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^(𝑍 + (𝑁 + 1))))
 
Theoremfzosplitsnm1 13102 Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ‘(𝐴 + 1))) → (𝐴..^𝐵) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)}))
 
Theoremelfzonlteqm1 13103 If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.)
((𝐴 ∈ (0..^𝐵) ∧ ¬ 𝐴 < (𝐵 − 1)) → 𝐴 = (𝐵 − 1))
 
Theoremfzonn0p1 13104 A nonnegative integer is element of the half-open range of nonnegative integers with the element increased by one as an upper bound. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(𝑁 ∈ ℕ0𝑁 ∈ (0..^(𝑁 + 1)))
 
Theoremfzossfzop1 13105 A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(𝑁 ∈ ℕ0 → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
 
Theoremfzonn0p1p1 13106 If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(𝐼 ∈ (0..^𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1)))
 
Theoremelfzom1p1elfzo 13107 Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Proof shortened by Thierry Arnoux, 14-Dec-2023.)
((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁))
 
Theoremfzo0ssnn0 13108 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) (Proof shortened by JJ, 1-Jun-2021.)
(0..^𝑁) ⊆ ℕ0
 
Theoremfzo01 13109 Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(0..^1) = {0}
 
Theoremfzo12sn 13110 A 1-based half-open integer interval up to, but not including, 2 is a singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(1..^2) = {1}
 
Theoremfzo13pr 13111 A 1-based half-open integer interval up to, but not including, 3 is a pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
(1..^3) = {1, 2}
 
Theoremfzo0to2pr 13112 A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(0..^2) = {0, 1}
 
Theoremfzo0to3tp 13113 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
(0..^3) = {0, 1, 2}
 
Theoremfzo0to42pr 13114 A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
(0..^4) = ({0, 1} ∪ {2, 3})
 
Theoremfzo1to4tp 13115 A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.)
(1..^4) = {1, 2, 3}
 
Theoremfzo0sn0fzo1 13116 A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018.)
(𝑁 ∈ ℕ → (0..^𝑁) = ({0} ∪ (1..^𝑁)))
 
Theoremelfzo0l 13117 A member of a half-open range of nonnegative integers is either 0 or a member of the corresponding half-open range of positive integers. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (0..^𝑁) → (𝐾 = 0 ∨ 𝐾 ∈ (1..^𝑁)))
 
Theoremfzoend 13118 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵))
 
Theoremfzo0end 13119 The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐵 ∈ ℕ → (𝐵 − 1) ∈ (0..^𝐵))
 
Theoremssfzo12 13120 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 13121 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 13122 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 13123 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 13124 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 13125 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 13126 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 13127 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 13128 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 13129 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 13130 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 13131 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 13132 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 13133 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 13134 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 13135 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 13136 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 13137 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 13138 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 13139 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 13140 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 13141 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 13142 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 13143 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 13144* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 12985. (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 13145* 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 12069 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 13146 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 13147 Lemma for injresinj 13148. (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 13148 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 13149 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 13150 Extend class notation with floor (greatest integer) function.
class
 
Syntaxcceil 13151 Extend class notation to include the ceiling function.
class
 
Definitiondf-fl 13152* Define the floor (greatest integer less than or equal to) function. See flval 13154 for its value, fllelt 13157 for its basic property, and flcl 13155 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28154).

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 13153 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 13198 for its value, ceilge 13204 and ceilm1lt 13206 for its basic properties, and ceilcl 13202 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 28155).

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 13154* 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 13155 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 13156 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
 
Theoremfllelt 13157 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 13158 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (⌊‘𝐴) ∈ ℤ)
 
Theoremflle 13159 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
 
Theoremflltp1 13160 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
 
Theoremfllep1 13161 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
(𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
 
Theoremfraclt1 13162 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1)
 
Theoremfracle1 13163 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 13164 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴)))
 
Theoremflge 13165 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 13166 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 13167 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 13168 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
(𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
 
Theoremflidm 13169 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
(𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
 
Theoremflidz 13170 A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ → ((⌊‘𝐴) = 𝐴𝐴 ∈ ℤ))
 
Theoremflltnz 13171 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 13172 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 13173 Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (⌊‘𝐵) ∈ (ℤ‘(⌊‘𝐴)))
 
Theoremflval2 13174* 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 13175* 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 13176 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 13177 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))
 
Theoremadddivflid 13178 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 13179 The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13254 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 13180 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 13181 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 13182 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 13183 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 13184 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 13185 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 13186 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 13187 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 13188 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 12336.) (Contributed by NM, 12-Mar-2005.)
(((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁𝐴𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))
 
Theorem2tnp1ge0ge0 13189 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))
 
Theoremflhalf 13190 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 13191 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 13192 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 13193 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 13194 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 13195 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 13196 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))
 
Theoremfldiv4lem1div2 13197 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 13198 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
 
Theoremdfceil2 13199* Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
⌈ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))
 
Theoremceilval2 13200* The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = (𝑦 ∈ ℤ (𝐴𝑦𝑦 < (𝐴 + 1))))
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