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Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0elfz 13001 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.)
(𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))

Theoremnn0fz0 13002 A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.)
(𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))

Theoremelfz0add 13003 An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵))))

Theoremfz0sn 13004 An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
(0...0) = {0}

Theoremfz0tp 13005 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
(0...2) = {0, 1, 2}

Theoremfz0to3un2pr 13006 An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.)
(0...3) = ({0, 1} ∪ {2, 3})

Theoremfz0to4untppr 13007 An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
(0...4) = ({0, 1, 2} ∪ {3, 4})

Theoremelfz0ubfz0 13008 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 13009 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 13010 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 13011 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 13012 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 13013 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 13014 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 13015 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 13016 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 13017 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 13018 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 13019 Express the set of nonnegative integers as the disjoint (see nn0disj 13020) 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 13020 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 13021 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 13022 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 13023 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 13024* 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 13025* 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 13026 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 13027 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 13028 The value of the predecessor class over 0. (Contributed by Scott Fenton, 9-May-2014.)
(𝑁 ∈ ℕ0 → Pred( < , ℕ0, 𝑁) = (0...(𝑁 − 1)))

Theorempredfz 13029 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 13030 Syntax for half-open integer ranges.
class ..^

Definitiondf-fzo 13031* 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 12888, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 13067 with fzsplit 12930, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))

Theoremfzof 13032 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^:(ℤ × ℤ)⟶𝒫 ℤ

Theoremelfzoel1 13033 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)

Theoremelfzoel2 13034 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)

Theoremelfzoelz 13035 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)

Theoremfzoval 13036 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 13037 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 13038 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))

Theoremelfzouz 13039 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ𝑀))

Theoremnelfzo 13040 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 13041 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 13042 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 13043 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 13044 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 13045 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 13046 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 13047 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 13048 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
¬ 𝐵 ∈ (𝐴..^𝐵)

Theoremelfzouz2 13049 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 13050 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 13051 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 13052 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 13053 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 13054 A half-open integer interval is a set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑀..^𝑁) ⊆ ℤ

Theoremfzon 13055 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 13056 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 13057 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 13058 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 13059 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 13060 If 𝑀 < 𝑁 then 𝑁𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑁𝑀) ∈ ℕ)

Theoremfzoss1 13061 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 13062 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 13063 Subset of a half-open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))

Theoremfzo0ss1 13064 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 13065 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 13066 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 13067 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 13068 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅

Theoremfzouzsplit 13069 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 13070 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 13071 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 13072 A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.)
((𝐴..^𝐵) ∩ {𝐵}) = ∅

Theoremprinfzo0 13073 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 13074 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 13075 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 13076 Membership in a half-open range of nonnegative integers, generalization of elfzo0 13075 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 13077 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 13078 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 13079 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 13080 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 13081 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 13082 Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.)
(𝑀 ∈ (1...𝑁) → (𝑀 − 1) ∈ (0..^𝑁))

Theoremfzossnn 13083 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ

Theoremelfzo1 13084 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 13085 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 13086 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 13087 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 13088 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 13089 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 13090 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 13091 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 13092 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 13093 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 13094 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 13095 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 13096 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 13097 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 13098 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 13099 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 13100 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)))

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