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Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfz0ubfz0 13001 An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → 𝐾 ∈ (0...𝐿))
 
Theoremelfz0fzfz0 13002 A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → 𝑀 ∈ (0...𝑁))
 
Theoremfz0fzelfz0 13003 If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))
 
Theoremfznn0sub2 13004 Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ (0...𝑁))
 
Theoremuzsubfz0 13005 Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
((𝐿 ∈ ℕ0𝑁 ∈ (ℤ𝐿)) → (𝑁𝐿) ∈ (0...𝑁))
 
Theoremfz0fzdiffz0 13006 The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾𝑀) ∈ (0...𝑁))
 
Theoremelfzmlbm 13007 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐾 ∈ (𝑀...𝑁) → (𝐾𝑀) ∈ (0...(𝑁𝑀)))
 
Theoremelfzmlbp 13008 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
((𝑁 ∈ ℤ ∧ 𝐾 ∈ (𝑀...(𝑀 + 𝑁))) → (𝐾𝑀) ∈ (0...𝑁))
 
Theoremfzctr 13009 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
(𝑁 ∈ ℕ0𝑁 ∈ (0...(2 · 𝑁)))
 
Theoremdifelfzle 13010 The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
((𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁) ∧ 𝐾𝑀) → (𝑀𝐾) ∈ (0...𝑁))
 
Theoremdifelfznle 13011 The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
((𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁) ∧ ¬ 𝐾𝑀) → ((𝑀 + 𝑁) − 𝐾) ∈ (0...𝑁))
 
Theoremnn0split 13012 Express the set of nonnegative integers as the disjoint (see nn0disj 13013) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.)
(𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))
 
Theoremnn0disj 13013 The first 𝑁 + 1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅
 
Theoremfz0sn0fz1 13014 A finite set of sequential nonnegative integers is the union of the singleton containing 0 and a finite set of sequential positive integers. (Contributed by AV, 20-Mar-2021.)
(𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁)))
 
Theoremfvffz0 13015 The function value of a function from a finite interval of nonnegative integers. (Contributed by AV, 13-Feb-2021.)
(((𝑁 ∈ ℕ0𝐼 ∈ ℕ0𝐼 < 𝑁) ∧ 𝑃:(0...𝑁)⟶𝑉) → (𝑃𝐼) ∈ 𝑉)
 
Theorem1fv 13016 A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof shortened by AV, 18-Apr-2021.)
((𝑁𝑉𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁))
 
Theorem4fvwrd4 13017* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
((𝐿 ∈ (ℤ‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
 
Theorem2ffzeq 13018* Two functions over 0-based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
((𝑀 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
 
Theorempreduz 13019 The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)
(𝑁 ∈ (ℤ𝑀) → Pred( < , (ℤ𝑀), 𝑁) = (𝑀...(𝑁 − 1)))
 
Theoremprednn 13020 The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
(𝑁 ∈ ℕ → Pred( < , ℕ, 𝑁) = (1...(𝑁 − 1)))
 
Theoremprednn0 13021 The value of the predecessor class over 0. (Contributed by Scott Fenton, 9-May-2014.)
(𝑁 ∈ ℕ0 → Pred( < , ℕ0, 𝑁) = (0...(𝑁 − 1)))
 
Theorempredfz 13022 Calculate the predecessor of an integer under a finite set of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Proof shortened by Mario Carneiro, 3-May-2015.)
(𝐾 ∈ (𝑀...𝑁) → Pred( < , (𝑀...𝑁), 𝐾) = (𝑀...(𝐾 − 1)))
 
5.5.7  Half-open integer ranges
 
Syntaxcfzo 13023 Syntax for half-open integer ranges.
class ..^
 
Definitiondf-fzo 13024* Define a function generating sets of integers using a half-open range. Read (𝑀..^𝑁) as the integers from 𝑀 up to, but not including, 𝑁; contrast with (𝑀...𝑁) df-fz 12883, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 13060 with fzsplit 12923, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
 
Theoremfzof 13025 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^:(ℤ × ℤ)⟶𝒫 ℤ
 
Theoremelfzoel1 13026 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)
 
Theoremelfzoel2 13027 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)
 
Theoremelfzoelz 13028 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)
 
Theoremfzoval 13029 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
 
Theoremelfzo 13030 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀𝐾𝐾 < 𝑁)))
 
Theoremelfzo2 13031 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))
 
Theoremelfzouz 13032 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ𝑀))
 
Theoremnelfzo 13033 An integer not being a member of a half-open finite set of integers. (Contributed by AV, 29-Apr-2020.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∉ (𝑀..^𝑁) ↔ (𝐾 < 𝑀𝑁𝐾)))
 
Theoremfzolb 13034 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝑀 ∈ (𝑀..^𝑁) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁))
 
Theoremfzolb2 13035 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁))
 
Theoremelfzole1 13036 A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑀𝐾)
 
Theoremelfzolt2 13037 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁)
 
Theoremelfzolt3 13038 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑀 < 𝑁)
 
Theoremelfzolt2b 13039 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝐾..^𝑁))
 
Theoremelfzolt3b 13040 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ (𝑀..^𝑁))
 
Theoremfzonel 13041 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
¬ 𝐵 ∈ (𝐴..^𝐵)
 
Theoremelfzouz2 13042 The upper bound of a half-open range is greater than or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝐾))
 
Theoremelfzofz 13043 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀...𝑁))
 
Theoremelfzo3 13044 Express membership in a half-open integer interval in terms of the "less than or equal to" and "less than" predicates on integers, resp. 𝐾 ∈ (ℤ𝑀) ↔ 𝑀𝐾, 𝐾 ∈ (𝐾..^𝑁) ↔ 𝐾 < 𝑁. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝐾 ∈ (𝐾..^𝑁)))
 
Theoremfzon0 13045 A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
((𝑀..^𝑁) ≠ ∅ ↔ 𝑀 ∈ (𝑀..^𝑁))
 
Theoremfzossfz 13046 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐴..^𝐵) ⊆ (𝐴...𝐵)
 
Theoremfzossz 13047 A half-open integer interval is a set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑀..^𝑁) ⊆ ℤ
 
Theoremfzon 13048 A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑀 ↔ (𝑀..^𝑁) = ∅))
 
Theoremfzo0n 13049 A half-open range of nonnegative integers is empty iff the upper bound is not positive. (Contributed by AV, 2-May-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑀 ↔ (0..^(𝑁𝑀)) = ∅))
 
Theoremfzonlt0 13050 A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 < 𝑁 ↔ (𝑀..^𝑁) = ∅))
 
Theoremfzo0 13051 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐴..^𝐴) = ∅
 
Theoremfzonnsub 13052 If 𝐾 < 𝑁 then 𝑁𝐾 is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑁𝐾) ∈ ℕ)
 
Theoremfzonnsub2 13053 If 𝑀 < 𝑁 then 𝑁𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑁𝑀) ∈ ℕ)
 
Theoremfzoss1 13054 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (ℤ𝑀) → (𝐾..^𝑁) ⊆ (𝑀..^𝑁))
 
Theoremfzoss2 13055 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝑁 ∈ (ℤ𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁))
 
Theoremfzossrbm1 13056 Subset of a half-open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
 
Theoremfzo0ss1 13057 Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
(1..^𝑁) ⊆ (0..^𝑁)
 
Theoremfzossnn0 13058 A half-open integer range starting at a nonnegative integer is a subset of the nonnegative integers. (Contributed by Alexander van der Vekens, 13-May-2018.)
(𝑀 ∈ ℕ0 → (𝑀..^𝑁) ⊆ ℕ0)
 
Theoremfzospliti 13059 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 ∈ (𝐵..^𝐷) ∨ 𝐴 ∈ (𝐷..^𝐶)))
 
Theoremfzosplit 13060 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐷 ∈ (𝐵...𝐶) → (𝐵..^𝐶) = ((𝐵..^𝐷) ∪ (𝐷..^𝐶)))
 
Theoremfzodisj 13061 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
 
Theoremfzouzsplit 13062 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
(𝐵 ∈ (ℤ𝐴) → (ℤ𝐴) = ((𝐴..^𝐵) ∪ (ℤ𝐵)))
 
Theoremfzouzdisj 13063 A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
((𝐴..^𝐵) ∩ (ℤ𝐵)) = ∅
 
Theoremfzoun 13064 A half-open integer range as union of two half-open integer ranges. (Contributed by AV, 23-Apr-2022.)
((𝐵 ∈ (ℤ𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐴..^(𝐵 + 𝐶)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 𝐶))))
 
Theoremfzodisjsn 13065 A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.)
((𝐴..^𝐵) ∩ {𝐵}) = ∅
 
Theoremprinfzo0 13066 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 13067 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 13068 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 13069 Membership in a half-open range of nonnegative integers, generalization of elfzo0 13068 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 13070 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 13071 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 13072 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 13073 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 13074 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 13075 Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.)
(𝑀 ∈ (1...𝑁) → (𝑀 − 1) ∈ (0..^𝑁))
 
Theoremfzossnn 13076 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ
 
Theoremelfzo1 13077 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 13078 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 13079 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 13080 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 13081 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 13082 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 13083 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 13084 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 13085 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 13086 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 13087 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 13088 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 13089 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 13090 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 13091 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 13092 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 13093 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 13094 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 13095 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 13096 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 13097 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 13098 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
 
Theoremelfzomin 13099 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 13100 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)))
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