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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elfzom1b 13801 | 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 13802 | 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 13803 | 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 13804 | 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 13805 | 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 13806 | 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 13807 | 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 13808 | 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 13809 | 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 13810 | 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 13811 | 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 13812 | 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 13813 | 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 13814 | 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 13815 | 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 13816 | 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 13817 | 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 13818 | 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 13819* | Shift the scanning order inside of a universal quantification restricted to a half-open integer range, analogous to fzshftral 13651. (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 13820* | 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 12713 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 13821 | 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 13822 | Lemma for injresinj 13823. (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 13823 | 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 13824 | 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 13825 | 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)))) | ||
Syntax | cfl 13826 | Extend class notation with floor (greatest integer) function. |
class ⌊ | ||
Syntax | cceil 13827 | Extend class notation to include the ceiling function. |
class ⌈ | ||
Definition | df-fl 13828* |
Define the floor (greatest integer less than or equal to) function. See
flval 13830 for its value, fllelt 13833 for its basic property, and flcl 13831
for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 30475).
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 13829 |
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 13874 for its value, ceilge 13881 and ceilm1lt 13884 for its basic
properties, and ceilcl 13878 for its closure. For example,
(⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1
(ex-ceil 30476).
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 13830* | 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 13831 | 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 13832 | The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | ||
Theorem | fllelt 13833 | 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 13834 | 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 13835 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | ||
Theorem | flltp1 13836 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) | ||
Theorem | fllep1 13837 | A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1)) | ||
Theorem | fraclt1 13838 | The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1) | ||
Theorem | fracle1 13839 | 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 13840 | The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴))) | ||
Theorem | flge 13841 | 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 13842 | 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 13843 | 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 13844 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) | ||
Theorem | flidm 13845 | The floor function is idempotent. (Contributed by NM, 17-Aug-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴)) | ||
Theorem | flidz 13846 | 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 13847 | 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 13848 | 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 13849 | Ordering relation for the floor function. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴))) | ||
Theorem | flval2 13850* | 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 13851* | 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 13852 | 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 13853 | A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1))) | ||
Theorem | adddivflid 13854 | 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 13855 | The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13932 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 13856 | 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 13857 | 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 13858 | 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 13859 | 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 13860 | 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 13861 | 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 13862 | 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 13863 | 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 13864 | 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 12986.) (Contributed by NM, 12-Mar-2005.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁)) | ||
Theorem | 2tnp1ge0ge0 13865 | 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 13866 | 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 13867 | 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 13868 | 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 13869 | 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 13870 | 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 13871 | 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 13872 | 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 13873 | 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 13874 | The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴)) | ||
Theorem | dfceil2 13875* | 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 13876* | 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 13877 | The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.) |
⊢ (𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ) | ||
Theorem | ceilcl 13878 | Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceilcld 13879 | Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceige 13880 | 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 13881 | 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 13882 | 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 13883 | 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 13884 | 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 13885 | 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 13886 | 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 13887 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴) | ||
Theorem | ceilidz 13888 | 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 13889 | 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 13890 | 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 13891 | 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 16436. (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 13892 | 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 13893 | Alternate proof of quoremnn0 13892 not using quoremz 13891. TODO - Keep either quoremnn0ALT 13893 (if we don't keep quoremz 13891) or quoremnn0 13892? (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 13894 | Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 13922? (Contributed by NM, 16-Aug-2008.) |
⊢ 𝑍 = (⌊‘𝐴) & ⊢ 𝐹 = (𝐴 − 𝑍) ⇒ ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) | ||
Theorem | intfracq 13895 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 13894. (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 13896 | 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 13897 | 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 13898 | 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 13899 | 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 13900 | An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞) |
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