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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fzossfzop1 10001 | 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 ∈ ℕ_{0} → (0..^𝑁) ⊆ (0..^(𝑁 + 1))) | ||
Theorem | fzonn0p1p1 10002 | 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 10003 | 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.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁)) | ||
Theorem | fzo0ssnn0 10004 | Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) |
⊢ (0..^𝑁) ⊆ ℕ_{0} | ||
Theorem | fzo01 10005 | Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ (0..^1) = {0} | ||
Theorem | fzo12sn 10006 | 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 | fzo0to2pr 10007 | 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 10008 | 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 10009 | 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 | fzo0sn0fzo1 10010 | 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 | fzoend 10011 | The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵)) | ||
Theorem | fzo0end 10012 | 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 10013 | 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 | ssfzo12bi 10014 | 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 10015 | 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 10016 | 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 10017 | 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 10018 | 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 | elfzonelfzo 10019 | 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 | elfzomelpfzo 10020 | 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 | peano2fzor 10021 | 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 10022 | 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 | fzosplitprm1 10023 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵})) | ||
Theorem | fzosplitsni 10024 | 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 10025 | 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 | fzostep1 10026 | 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 10027* | Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9900. (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 10028* | 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 9178 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 | exfzdc 10029* | Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓) ⇒ ⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓) | ||
Theorem | fvinim0ffz 10030 | 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 ∈ ℕ_{0}) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾))))) | ||
Theorem | subfzo0 10031 | 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 | qtri3or 10032 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
⊢ ((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀)) | ||
Theorem | qletric 10033 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | ||
Theorem | qlelttric 10034 | Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | qltnle 10035 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | ||
Theorem | qdceq 10036 | Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵) | ||
Theorem | exbtwnzlemstep 10037* | Lemma for exbtwnzlemex 10039. Induction step. (Contributed by Jim Kingdon, 10-May-2022.) |
⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛)) ⇒ ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐾))) | ||
Theorem | exbtwnzlemshrink 10038* | Lemma for exbtwnzlemex 10039. Shrinking the range around 𝐴. (Contributed by Jim Kingdon, 10-May-2022.) |
⊢ (𝜑 → 𝐽 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛)) ⇒ ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | ||
Theorem | exbtwnzlemex 10039* |
Existence of an integer so that a given real number is between the
integer and its successor. The real number must satisfy the
𝑛
≤ 𝐴 ∨ 𝐴 < 𝑛 hypothesis. For example either a
rational number or
a number which is irrational (in the sense of being apart from any
rational number) will meet this condition.
The proof starts by finding two integers which are less than and greater than 𝐴. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the 𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛 hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | ||
Theorem | exbtwnz 10040* | If a real number is between an integer and its successor, there is a unique greatest integer less than or equal to the real number. (Contributed by Jim Kingdon, 10-May-2022.) |
⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | ||
Theorem | qbtwnz 10041* | There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | ||
Theorem | rebtwn2zlemstep 10042* | Lemma for rebtwn2z 10044. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.) |
⊢ ((𝐾 ∈ (ℤ_{≥}‘2) ∧ 𝐴 ∈ ℝ ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐾))) | ||
Theorem | rebtwn2zlemshrink 10043* | Lemma for rebtwn2z 10044. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐽 ∈ (ℤ_{≥}‘2) ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2))) | ||
Theorem | rebtwn2z 10044* |
A real number can be bounded by integers above and below which are two
apart.
The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.) |
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2))) | ||
Theorem | qbtwnrelemcalc 10045 | Lemma for qbtwnre 10046. Calculations involved in showing the constructed rational number is less than 𝐵. (Contributed by Jim Kingdon, 14-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑀 < (𝐴 · (2 · 𝑁))) & ⊢ (𝜑 → (1 / 𝑁) < (𝐵 − 𝐴)) ⇒ ⊢ (𝜑 → ((𝑀 + 2) / (2 · 𝑁)) < 𝐵) | ||
Theorem | qbtwnre 10046* | The rational numbers are dense in ℝ: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) | ||
Theorem | qbtwnxr 10047* | The rational numbers are dense in ℝ^{*}: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*} ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) | ||
Theorem | qavgle 10048 | The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵)) | ||
Theorem | ioo0 10049 | An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*}) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | ioom 10050* | An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*}) → (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 < 𝐵)) | ||
Theorem | ico0 10051 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*}) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | ioc0 10052 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*}) → ((𝐴(,]𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Syntax | cfl 10053 | Extend class notation with floor (greatest integer) function. |
class ⌊ | ||
Syntax | cceil 10054 | Extend class notation to include the ceiling function. |
class ⌈ | ||
Definition | df-fl 10055* |
Define the floor (greatest integer less than or equal to) function. See
flval 10057 for its value, flqlelt 10061 for its basic property, and flqcl 10058 for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 12996).
Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision. 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 10056 |
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 ceilqval 10091 for its value, ceilqge 10095 and ceilqm1lt 10097 for its basic
properties, and ceilqcl 10093 for its closure. For example,
(⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1
(ex-ceil 12997).
As described in df-fl 10055 most theorems are only for rationals, not reals. 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 10057* | 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 | flqcl 10058 | The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers apart from any integer, see flapcl 10060. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | apbtwnz 10059* | There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.) |
⊢ ((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | ||
Theorem | flapcl 10060* | The floor (greatest integer) function yields an integer when applied to a real number apart from any integer. For example, an irrational number (see for example sqrt2irrap 11869) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.) |
⊢ ((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | flqlelt 10061 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | ||
Theorem | flqcld 10062 | The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | flqle 10063 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴) | ||
Theorem | flqltp1 10064 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1)) | ||
Theorem | qfraclt1 10065 | The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1) | ||
Theorem | qfracge0 10066 | The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴))) | ||
Theorem | flqge 10067 | The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴))) | ||
Theorem | flqlt 10068 | The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵)) | ||
Theorem | flid 10069 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) | ||
Theorem | flqidm 10070 | The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴)) | ||
Theorem | flqidz 10071 | A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ)) | ||
Theorem | flqltnz 10072 | If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴) | ||
Theorem | flqwordi 10073 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵)) | ||
Theorem | flqword2 10074 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ_{≥}‘(⌊‘𝐴))) | ||
Theorem | flqbi 10075 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)))) | ||
Theorem | flqbi2 10076 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1))) | ||
Theorem | adddivflid 10077 | 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 ∈ ℕ_{0} ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴)) | ||
Theorem | flqge0nn0 10078 | The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ_{0}) | ||
Theorem | flqge1nn 10079 | The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ) | ||
Theorem | fldivnn0 10080 | 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 ∈ ℕ_{0} ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ_{0}) | ||
Theorem | divfl0 10081 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0)) | ||
Theorem | flqaddz 10082 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁)) | ||
Theorem | flqzadd 10083 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℚ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴))) | ||
Theorem | flqmulnn0 10084 | Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴))) | ||
Theorem | btwnzge0 10085 | 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. (Contributed by NM, 12-Mar-2005.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁)) | ||
Theorem | 2tnp1ge0ge0 10086 | Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) |
⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁)) | ||
Theorem | flhalf 10087 | 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 | fldivnn0le 10088 | 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 ∈ ℕ_{0} ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿)) | ||
Theorem | flltdivnn0lt 10089 | 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 ∈ ℕ_{0} ∧ 𝑁 ∈ ℕ_{0} ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿))) | ||
Theorem | fldiv4p1lem1div2 10090 | 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 | ceilqval 10091 | The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴)) | ||
Theorem | ceiqcl 10092 | The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ) | ||
Theorem | ceilqcl 10093 | Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceiqge 10094 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ -(⌊‘-𝐴)) | ||
Theorem | ceilqge 10095 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴)) | ||
Theorem | ceiqm1l 10096 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) < 𝐴) | ||
Theorem | ceilqm1lt 10097 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → ((⌈‘𝐴) − 1) < 𝐴) | ||
Theorem | ceiqle 10098 | The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵) | ||
Theorem | ceilqle 10099 | The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (⌈‘𝐴) ≤ 𝐵) | ||
Theorem | ceilid 10100 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴) |
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