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Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzodisjsn 12501 A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.)
((𝐴..^𝐵) ∩ {𝐵}) = ∅
 
Theoremprinfzo0 12502 The intersection of a half-open integer range and the pair of its outer left borders is empty. (Contributed by AV, 9-Jan-2021.)
(𝑀 ∈ ℤ → ({𝑀, 𝑁} ∩ ((𝑀 + 1)..^𝑁)) = ∅)
 
Theoremlbfzo0 12503 An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.)
(0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
 
Theoremelfzo0 12504 Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0𝐵 ∈ ℕ ∧ 𝐴 < 𝐵))
 
Theoremelfzo0z 12505 Membership in a half-open range of nonnegative integers, generalization of elfzo0 12504 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0𝐵 ∈ ℤ ∧ 𝐴 < 𝐵))
 
Theoremnn0p1elfzo 12506 A nonnegative integer increased by 1 which is less than or equal to another integer is an element of a half-open range of integers. (Contributed by AV, 27-Feb-2021.)
((𝐾 ∈ ℕ0𝑁 ∈ ℕ0 ∧ (𝐾 + 1) ≤ 𝑁) → 𝐾 ∈ (0..^𝑁))
 
Theoremelfzo0le 12507 A member in a half-open range of nonnegative integers is less than or equal to the upper bound of the range. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴 ∈ (0..^𝐵) → 𝐴𝐵)
 
Theoremelfzonn0 12508 A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐾 ∈ (0..^𝑁) → 𝐾 ∈ ℕ0)
 
Theoremfzonmapblen 12509 The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less than the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
((𝐴 ∈ (0..^𝑁) ∧ 𝐵 ∈ (0..^𝑁) ∧ 𝐵 < 𝐴) → (𝐵 + (𝑁𝐴)) < 𝑁)
 
Theoremfzofzim 12510 If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
((𝐾𝑀𝐾 ∈ (0...𝑀)) → 𝐾 ∈ (0..^𝑀))
 
Theoremfz1fzo0m1 12511 Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.)
(𝑀 ∈ (1...𝑁) → (𝑀 − 1) ∈ (0..^𝑁))
 
Theoremfzossnn 12512 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ
 
Theoremelfzo1 12513 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀))
 
Theoremfzo1fzo0n0 12514 An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
(𝐾 ∈ (1..^𝑁) ↔ (𝐾 ∈ (0..^𝑁) ∧ 𝐾 ≠ 0))
 
Theoremfzo0n0 12515 A half-open integer range based at 0 is nonempty precisely if the upper bound is a positive integer. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
((0..^𝐴) ≠ ∅ ↔ 𝐴 ∈ ℕ)
 
Theoremfzoaddel 12516 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)..^(𝐶 + 𝐷)))
 
Theoremfzo0addel 12517 Translate membership in a 0 based half-open integer range. (Contributed by AV, 30-Apr-2020.)
((𝐴 ∈ (0..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 + 𝐷) ∈ (𝐷..^(𝐶 + 𝐷)))
 
Theoremfzo0addelr 12518 Translate membership in a 0 based half-open integer range. (Contributed by AV, 30-Apr-2020.)
((𝐴 ∈ (0..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 + 𝐷) ∈ (𝐷..^(𝐷 + 𝐶)))
 
Theoremfzoaddel2 12519 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (0..^(𝐵𝐶)) ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 + 𝐶) ∈ (𝐶..^𝐵))
 
Theoremelfzoext 12520 Membership of an integer in an extended open range of integers. (Contributed by AV, 30-Apr-2020.)
((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → 𝑍 ∈ (𝑀..^(𝑁 + 𝐼)))
 
Theoremelincfzoext 12521 Membership of an increased integer in a correspondingly extended half-open range of integers. (Contributed by AV, 30-Apr-2020.)
((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → (𝑍 + 𝐼) ∈ (𝑀..^(𝑁 + 𝐼)))
 
Theoremfzosubel 12522 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴𝐷) ∈ ((𝐵𝐷)..^(𝐶𝐷)))
 
Theoremfzosubel2 12523 Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ ((𝐵 + 𝐶)..^(𝐵 + 𝐷)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴𝐵) ∈ (𝐶..^𝐷))
 
Theoremfzosubel3 12524 Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (𝐵..^(𝐵 + 𝐷)) ∧ 𝐷 ∈ ℤ) → (𝐴𝐵) ∈ (0..^𝐷))
 
Theoremeluzgtdifelfzo 12525 Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ∈ (ℤ𝐴) ∧ 𝐵 < 𝐴) → (𝑁𝐴) ∈ (0..^(𝑁𝐵))))
 
Theoremige2m2fzo 12526 Membership of an integer greater than 1 decreased by 2 in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑁 ∈ (ℤ‘2) → (𝑁 − 2) ∈ (0..^(𝑁 − 1)))
 
Theoremfzocatel 12527 Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴𝐵) ∈ (0..^𝐶))
 
Theoremubmelfzo 12528 If an integer in a 1 based finite set of sequential integers is subtracted from the upper bound of this finite set of sequential integers, the result is contained in a half-open range of nonnegative integers with the same upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV, 30-Oct-2018.)
(𝐾 ∈ (1...𝑁) → (𝑁𝐾) ∈ (0..^𝑁))
 
Theoremelfzodifsumelfzo 12529 If an integer is in a half-open range of nonnegative integers with a difference as upper bound, the sum of the integer with the subtrahend of the difference is in the a half-open range of nonnegative integers containing the minuend of the difference. (Contributed by AV, 13-Nov-2018.)
((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...𝑃)) → (𝐼 ∈ (0..^(𝑁𝑀)) → (𝐼 + 𝑀) ∈ (0..^𝑃)))
 
Theoremelfzom1elp1fzo 12530 Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.)
((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (0..^𝑁))
 
Theoremelfzom1elfzo 12531 Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.)
((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → 𝐼 ∈ (0..^𝑁))
 
Theoremfzval3 12532 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)))
 
Theoremfz0add1fz1 12533 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 ∈ ℕ0𝑋 ∈ (0..^𝑁)) → (𝑋 + 1) ∈ (1...𝑁))
 
Theoremfzosn 12534 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
 
Theoremelfzomin 12535 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)))
 
Theoremzpnn0elfzo 12536 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)))
 
Theoremzpnn0elfzo1 12537 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))))
 
Theoremfzosplitsnm1 12538 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)}))
 
Theoremelfzonlteqm1 12539 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))
 
Theoremfzonn0p1 12540 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 ∈ ℕ0𝑁 ∈ (0..^(𝑁 + 1)))
 
Theoremfzossfzop1 12541 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)))
 
Theoremfzonn0p1p1 12542 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)))
 
Theoremelfzom1p1elfzo 12543 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))
 
Theoremfzo0ssnn0 12544 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
 
Theoremfzo0ssnn0OLD 12545 Obsolete proof of fzo0ssnn0 12544 as of 1-Jun-2021. Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(0..^𝑁) ⊆ ℕ0
 
Theoremfzo01 12546 Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(0..^1) = {0}
 
Theoremfzo12sn 12547 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}
 
Theoremfzo13pr 12548 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}
 
Theoremfzo0to2pr 12549 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}
 
Theoremfzo0to3tp 12550 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}
 
Theoremfzo0to42pr 12551 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})
 
Theoremfzo1to4tp 12552 A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.)
(1..^4) = {1, 2, 3}
 
Theoremfzo0sn0fzo1 12553 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)))
 
Theoremelfzo0l 12554 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)))
 
Theoremfzoend 12555 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵))
 
Theoremfzo0end 12556 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))
 
Theoremssfzo12 12557 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) → (𝑀𝐾𝐿𝑁)))
 
Theoremssfzoulel 12558 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 ∈ ℕ0𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁𝐴𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵𝐴)))
 
Theoremssfzo12bi 12559 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) ↔ (𝑀𝐾𝐿𝑁)))
 
Theoremubmelm1fzo 12560 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) → ((𝑁𝐾) − 1) ∈ (0..^𝑁))
 
Theoremfzofzp1 12561 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))
 
Theoremfzofzp1b 12562 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 ∈ (ℤ𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))
 
Theoremelfzom1b 12563 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))))
 
Theoremelfzom1elp1fzo1 12564 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))
 
Theoremelfzo1elm1fzo0 12565 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)))
 
Theoremelfzonelfzo 12566 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)))
 
Theoremfzonfzoufzol 12567 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 < 𝑁𝐼 ∈ (0..^𝑁)) → (¬ 𝐼 ∈ ((𝑁𝑀)..^𝑁) → 𝐼 ∈ (0..^(𝑁𝑀))))
 
Theoremelfzomelpfzo 12568 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 ∈ ((𝑀𝐿)..^(𝑁𝐿)) ↔ (𝐾 + 𝐿) ∈ (𝑀..^𝑁)))
 
Theoremelfznelfzo 12569 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))
 
Theoremelfznelfzob 12570 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)))
 
Theorempeano2fzor 12571 A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
((𝐾 ∈ (ℤ𝑀) ∧ (𝐾 + 1) ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑀..^𝑁))
 
Theoremfzosplitsn 12572 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵}))
 
Theoremfzosplitprm1 12573 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}))
 
Theoremfzosplitsni 12574 Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (𝐶 ∈ (𝐴..^(𝐵 + 1)) ↔ (𝐶 ∈ (𝐴..^𝐵) ∨ 𝐶 = 𝐵)))
 
Theoremfzisfzounsn 12575 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 ∈ (ℤ𝐴) → (𝐴...𝐵) = ((𝐴..^𝐵) ∪ {𝐵}))
 
Theoremelfzr 12576 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))
 
Theoremelfzlmr 12577 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 = 𝑀𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))
 
Theoremelfz0lmr 12578 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))
 
Theoremfzostep1 12579 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))
 
Theoremfzoshftral 12580* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 12424. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘𝐾) / 𝑗]𝜑))
 
Theoremfzind2 12581* 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 11472 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
(𝑥 = 𝑀 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐾 → (𝜑𝜏))    &   (𝑁 ∈ (ℤ𝑀) → 𝜓)    &   (𝑦 ∈ (𝑀..^𝑁) → (𝜒𝜃))       (𝐾 ∈ (𝑀...𝑁) → 𝜏)
 
Theoremfvinim0ffz 12582 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)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
 
Theoreminjresinjlem 12583 Lemma for injresinj 12584. (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) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))))))
 
Theoreminjresinj 12584 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) ≠ (𝐹𝐾)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))
 
Theoremsubfzo0 12585 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 < (𝐼𝐽) ∧ (𝐼𝐽) < 𝑁))
 
5.6  Elementary integer functions
 
5.6.1  The floor and ceiling functions
 
Syntaxcfl 12586 Extend class notation with floor (greatest integer) function.
class
 
Syntaxcceil 12587 Extend class notation to include the ceiling function.
class
 
Definitiondf-fl 12588* Define the floor (greatest integer less than or equal to) function. See flval 12590 for its value, fllelt 12593 for its basic property, and flcl 12591 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 27288).

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 (𝑦𝑥𝑥 < (𝑦 + 1))))
 
Definitiondf-ceil 12589 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 12634 for its value, ceilge 12640 and ceilm1lt 12642 for its basic properties, and ceilcl 12638 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 27289).

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))
 
Theoremflval 12590* 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 (𝑥𝐴𝐴 < (𝑥 + 1))))
 
Theoremflcl 12591 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)
 
Theoremreflcl 12592 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
 
Theoremfllelt 12593 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) ≤ 𝐴𝐴 < ((⌊‘𝐴) + 1)))
 
Theoremflcld 12594 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (⌊‘𝐴) ∈ ℤ)
 
Theoremflle 12595 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
 
Theoremflltp1 12596 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
(𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
 
Theoremfllep1 12597 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
(𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
 
Theoremfraclt1 12598 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
(𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1)
 
Theoremfracle1 12599 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)
 
Theoremfracge0 12600 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴)))
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