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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elfzom1elp1fzo1 13601 | 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 13602 | 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 13603 | 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 13604 | 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 13605 | 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 13606 | 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 13607 | 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 13608 | 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 13609 | 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 13610 | 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 13611 | 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 13612 | 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 13613 | 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 13614 | 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 13615 | 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 13616 | 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 13617 | 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 13618* | Shift the scanning order inside of a universal quantification restricted to a half-open integer range, analogous to fzshftral 13458. (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 13619* | 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 12532 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 13620 | 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 13621 | Lemma for injresinj 13622. (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 13622 | 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 13623 | 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)) | ||
Syntax | cfl 13624 | Extend class notation with floor (greatest integer) function. |
class ⌊ | ||
Syntax | cceil 13625 | Extend class notation to include the ceiling function. |
class ⌈ | ||
Definition | df-fl 13626* |
Define the floor (greatest integer less than or equal to) function. See
flval 13628 for its value, fllelt 13631 for its basic property, and flcl 13629
for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 29177).
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 13627 |
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 13672 for its value, ceilge 13679 and ceilm1lt 13682 for its basic
properties, and ceilcl 13676 for its closure. For example,
(⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1
(ex-ceil 29178).
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 13628* | 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 13629 | 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 13630 | The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | ||
Theorem | fllelt 13631 | 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 13632 | 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 13633 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | ||
Theorem | flltp1 13634 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) | ||
Theorem | fllep1 13635 | A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1)) | ||
Theorem | fraclt1 13636 | The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1) | ||
Theorem | fracle1 13637 | 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 13638 | The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴))) | ||
Theorem | flge 13639 | 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 13640 | 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 13641 | 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 13642 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) | ||
Theorem | flidm 13643 | The floor function is idempotent. (Contributed by NM, 17-Aug-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴)) | ||
Theorem | flidz 13644 | 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 13645 | 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 13646 | 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 13647 | Ordering relation for the floor function. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴))) | ||
Theorem | flval2 13648* | 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 13649* | 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 13650 | 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 13651 | A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1))) | ||
Theorem | adddivflid 13652 | 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 13653 | The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13730 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 13654 | 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 13655 | 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 13656 | 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 13657 | 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 13658 | 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 13659 | 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 13660 | 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 13661 | 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 13662 | 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 12801.) (Contributed by NM, 12-Mar-2005.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁)) | ||
Theorem | 2tnp1ge0ge0 13663 | 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 13664 | 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 13665 | 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 13666 | 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 13667 | 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 13668 | 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 13669 | 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 13670 | 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 13671 | 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 13672 | The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴)) | ||
Theorem | dfceil2 13673* | 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 13674* | 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 13675 | The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.) |
⊢ (𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ) | ||
Theorem | ceilcl 13676 | Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) |
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceilcld 13677 | Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceige 13678 | 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 13679 | 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 13680 | 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 13681 | 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 13682 | 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 13683 | 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 13684 | 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 13685 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴) | ||
Theorem | ceilidz 13686 | 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 13687 | 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 13688 | 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 13689 | 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 16220. (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 13690 | 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 13691 | Alternate proof of quoremnn0 13690 not using quoremz 13689. TODO - Keep either quoremnn0ALT 13691 (if we don't keep quoremz 13689) or quoremnn0 13690? (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 13692 | Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 13720? (Contributed by NM, 16-Aug-2008.) |
⊢ 𝑍 = (⌊‘𝐴) & ⊢ 𝐹 = (𝐴 − 𝑍) ⇒ ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) | ||
Theorem | intfracq 13693 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 13692. (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 13694 | 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 13695 | 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 13696 | 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 13697 | 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 13698 | An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞) | ||
Theorem | icopnfsup 13699 | An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴[,)+∞), ℝ*, < ) = +∞) | ||
Theorem | rpsup 13700 | The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ sup(ℝ+, ℝ*, < ) = +∞ |
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