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| 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)))) | ||
| 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))) | ||
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