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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzshftral 9201* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))[(𝑘𝐾) / 𝑗]𝜑))
 
Theoremige2m1fz1 9202 Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers (Contributed by Alexander van der Vekens, 14-Sep-2018.)
(𝑁 ∈ (ℤ‘2) → (𝑁 − 1) ∈ (1...𝑁))
 
Theoremige2m1fz 9203 Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ (0...𝑁))
 
Theoremfz01or 9204 An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
(𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1))
 
3.5.5  Finite intervals of nonnegative integers

Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: (0...𝑁), usually abbreviated by "fz0".

 
Theoremelfz2nn0 9205 Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0𝑁 ∈ ℕ0𝐾𝑁))
 
Theoremfznn0 9206 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
(𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0𝐾𝑁)))
 
Theoremelfznn0 9207 A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
 
Theoremelfz3nn0 9208 The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
 
Theorem0elfz 9209 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 9210 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 9211 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))))
 
Theoremfz0tp 9212 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}
 
Theoremelfz0ubfz0 9213 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 9214 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 9215 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 9216 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 9217 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 9218 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 9219 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 9220 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 9221 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 9222 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 9223 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 9224 Express the set of nonnegative integers as the disjoint (see nn0disj 9225) 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 9225 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))) = ∅
 
Theorem1fv 9226 A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝑁𝑉𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁))
 
Theorem4fvwrd4 9227* 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 9228* 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))))
 
3.5.6  Half-open integer ranges
 
Syntaxcfzo 9229 Syntax for half-open integer ranges.
class ..^
 
Definitiondf-fzo 9230* 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 9106, which includes 𝑁. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 9263 with fzsplit 9146, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
 
Theoremfzof 9231 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^:(ℤ × ℤ)⟶𝒫 ℤ
 
Theoremelfzoel1 9232 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)
 
Theoremelfzoel2 9233 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)
 
Theoremelfzoelz 9234 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)
 
Theoremfzoval 9235 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 9236 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 9237 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))
 
Theoremelfzouz 9238 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ𝑀))
 
Theoremfzolb 9239 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 alternate 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 9240 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 alternate 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 9241 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 9242 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 9243 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 9244 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 9245 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 9246 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
¬ 𝐵 ∈ (𝐴..^𝐵)
 
Theoremelfzouz2 9247 The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝐾))
 
Theoremelfzofz 9248 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 9249 Express membership in a half-open integer interval in terms of the "less than or equal" 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)))
 
Theoremfzom 9250* A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
(∃𝑥 𝑥 ∈ (𝑀..^𝑁) ↔ 𝑀 ∈ (𝑀..^𝑁))
 
Theoremfzossfz 9251 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)
 
Theoremfzon 9252 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) = ∅))
 
Theoremfzonlt0 9253 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 9254 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 9255 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 9256 If 𝑀 < 𝑁 then 𝑁𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑁𝑀) ∈ ℕ)
 
Theoremfzoss1 9257 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 9258 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 9259 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
 
Theoremfzo0ss1 9260 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 9261 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 9262 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 9263 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 9264 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
 
Theoremfzouzsplit 9265 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 9266 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)) = ∅
 
Theoremlbfzo0 9267 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 9268 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))
 
Theoremfzo1fzo0n0 9269 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))
 
Theoremelfzo0z 9270 Membership in a half-open range of nonnegative integers, generalization of elfzo0 9268 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))
 
Theoremelfzo0le 9271 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 9272 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 9273 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 then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
((𝐴 ∈ (0..^𝑁) ∧ 𝐵 ∈ (0..^𝑁) ∧ 𝐵 < 𝐴) → (𝐵 + (𝑁𝐴)) < 𝑁)
 
Theoremfzofzim 9274 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))
 
Theoremfzossnn 9275 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ
 
Theoremelfzo1 9276 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))
 
Theoremfzo0m 9277* A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.)
(∃𝑥 𝑥 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
 
Theoremfzoaddel 9278 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)))
 
Theoremfzoaddel2 9279 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))
 
Theoremfzosubel 9280 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 9281 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 9282 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 9283 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 9284 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 9285 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 9286 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 9287 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 9288 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 9289 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 9290 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)))
 
Theoremfzosn 9291 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
 
Theoremelfzomin 9292 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 9293 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 9294 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 9295 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 9296 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 9297 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 9298 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 9299 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 9300 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))
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