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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fzval3 13101 | Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ (𝑁 ∈ ℤ → (𝑀...𝑁) = (𝑀..^(𝑁 + 1))) | ||
Theorem | fz0add1fz1 13102 | Translate membership in a 0-based half-open integer range into membership in a 1-based finite sequence of integers. (Contributed by Alexander van der Vekens, 23-Nov-2017.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ (0..^𝑁)) → (𝑋 + 1) ∈ (1...𝑁)) | ||
Theorem | fzosn 13103 | Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
⊢ (𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴}) | ||
Theorem | elfzomin 13104 | Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ (𝑍 ∈ ℤ → 𝑍 ∈ (𝑍..^(𝑍 + 1))) | ||
Theorem | zpnn0elfzo 13105 | 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))) | ||
Theorem | zpnn0elfzo1 13106 | 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)))) | ||
Theorem | fzosplitsnm1 13107 | 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)})) | ||
Theorem | elfzonlteqm1 13108 | 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)) | ||
Theorem | fzonn0p1 13109 | 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 ∈ ℕ0 → 𝑁 ∈ (0..^(𝑁 + 1))) | ||
Theorem | fzossfzop1 13110 | 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))) | ||
Theorem | fzonn0p1p1 13111 | 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))) | ||
Theorem | elfzom1p1elfzo 13112 | 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)) | ||
Theorem | fzo0ssnn0 13113 | 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 | ||
Theorem | fzo01 13114 | Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ (0..^1) = {0} | ||
Theorem | fzo12sn 13115 | 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} | ||
Theorem | fzo13pr 13116 | 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} | ||
Theorem | fzo0to2pr 13117 | 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} | ||
Theorem | fzo0to3tp 13118 | 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} | ||
Theorem | fzo0to42pr 13119 | 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}) | ||
Theorem | fzo1to4tp 13120 | A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.) |
⊢ (1..^4) = {1, 2, 3} | ||
Theorem | fzo0sn0fzo1 13121 | 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))) | ||
Theorem | elfzo0l 13122 | 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))) | ||
Theorem | fzoend 13123 | The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵)) | ||
Theorem | fzo0end 13124 | 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)) | ||
Theorem | ssfzo12 13125 | 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) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))) | ||
Theorem | ssfzoulel 13126 | 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 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴))) | ||
Theorem | ssfzo12bi 13127 | 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) ↔ (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))) | ||
Theorem | ubmelm1fzo 13128 | 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 13129 | 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 13130 | 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 13131 | 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 13132 | 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 13133 | 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 13134 | 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 13135 | 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 13136 | 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 13137 | 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 13138 | 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 13139 | 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 13140 | 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 13141 | 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 13142 | 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 13143 | 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 13144 | 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 13145 | 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 13146 | 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 13147 | 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 13148 | 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 13149* | Shift the scanning order inside of a universal quantification restricted to a half-open integer range, analogous to fzshftral 12990. (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 13150* | 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 12068 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 13151 | 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 13152 | Lemma for injresinj 13153. (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 13153 | 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 13154 | 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 13155 | Extend class notation with floor (greatest integer) function. |
class ⌊ | ||
Syntax | cceil 13156 | Extend class notation to include the ceiling function. |
class ⌈ | ||
Definition | df-fl 13157* |
Define the floor (greatest integer less than or equal to) function. See
flval 13159 for its value, fllelt 13162 for its basic property, and flcl 13160
for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 28232).
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 13158 |
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 13203 for its value, ceilge 13209 and ceilm1lt 13211 for its basic
properties, and ceilcl 13207 for its closure. For example,
(⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1
(ex-ceil 28233).
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 13159* | 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 13160 | 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 13161 | The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | ||
Theorem | fllelt 13162 | 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 13163 | 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 13164 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | ||
Theorem | flltp1 13165 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) | ||
Theorem | fllep1 13166 | A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1)) | ||
Theorem | fraclt1 13167 | The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1) | ||
Theorem | fracle1 13168 | 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 13169 | The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴))) | ||
Theorem | flge 13170 | 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 13171 | 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 13172 | 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 13173 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) | ||
Theorem | flidm 13174 | The floor function is idempotent. (Contributed by NM, 17-Aug-2008.) |
⊢ (𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴)) | ||
Theorem | flidz 13175 | 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 13176 | 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) | ||
Theorem | flwordi 13177 | 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 ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵)) | ||
Theorem | flword2 13178 | Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴))) | ||
Theorem | flval2 13179* | An alternate way to define the floor (greatest integer) function. (Contributed by NM, 16-Nov-2004.) |
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦 ≤ 𝐴 → 𝑦 ≤ 𝑥)))) | ||
Theorem | flval3 13180* | 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 ∣ 𝑥 ≤ 𝐴}, ℝ, < )) | ||
Theorem | flbi 13181 | 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 13182 | A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1))) | ||
Theorem | adddivflid 13183 | 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 13184 | The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13259 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 13185 | 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 13186 | 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 13187 | 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 13188 | 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 13189 | 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 13190 | 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 13191 | 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 13192 | 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 13193 | 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 12335.) (Contributed by NM, 12-Mar-2005.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁)) | ||
Theorem | 2tnp1ge0ge0 13194 | 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 13195 | 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 13196 | 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 13197 | 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 13198 | 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 13199 | 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 13200 | 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)) |
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