P. 137 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13601-13700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2ffzeq 13601*	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 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
Theorem	preduz 13602	The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → Pred( < , (ℤ≥‘𝑀), 𝑁) = (𝑀...(𝑁 − 1)))
Theorem	prednn 13603	The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
⊢ (𝑁 ∈ ℕ → Pred( < , ℕ, 𝑁) = (1...(𝑁 − 1)))
Theorem	prednn0 13604	The value of the predecessor class over ℕ0. (Contributed by Scott Fenton, 9-May-2014.)
⊢ (𝑁 ∈ ℕ0 → Pred( < , ℕ0, 𝑁) = (0...(𝑁 − 1)))
Theorem	predfz 13605	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
Syntax	cfzo 13606	Syntax for half-open integer ranges.
class ..^
Definition	df-fzo 13607*	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 13460, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 13645 with fzsplit 13502, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
Theorem	fzof 13608	Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ
Theorem	elfzoel1 13609	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)
Theorem	elfzoel2 13610	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)
Theorem	elfzoelz 13611	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)
Theorem	fzoval 13612	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)))
Theorem	elfzo 13613	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) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁)))
Theorem	elfzo2 13614	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))
Theorem	elfzod 13615	Membership in a half-open integer interval. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 < 𝑁)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝑀..^𝑁))
Theorem	elfzouz 13616	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
Theorem	nelfzo 13617	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 < 𝑀 ∨ 𝑁 ≤ 𝐾)))
Theorem	fzolb 13618	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 ∈ (ℤ≥‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝑀 ∈ (𝑀..^𝑁) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁))
Theorem	fzolb2 13619	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 ∈ (ℤ≥‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁))
Theorem	elfzole1 13620	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) → 𝑀 ≤ 𝐾)
Theorem	elfzolt2 13621	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)
Theorem	elfzolt3 13622	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)
Theorem	elfzolt2b 13623	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))
Theorem	elfzolt3b 13624	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))
Theorem	elfzop1le2 13625	A member in a half-open integer interval plus 1 is less than or equal to the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 + 1) ≤ 𝑁)
Theorem	fzonel 13626	A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ ¬ 𝐵 ∈ (𝐴..^𝐵)
Theorem	elfzouz2 13627	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 ∈ (ℤ≥‘𝐾))
Theorem	elfzofz 13628	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))
Theorem	elfzo3 13629	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 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐾, 𝐾 ∈ (𝐾..^𝑁) ↔ 𝐾 < 𝑁. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ (𝐾..^𝑁)))
Theorem	fzon0 13630	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))
Theorem	fzossfz 13631	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)
Theorem	fzossz 13632	A half-open integer interval is a set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝑀..^𝑁) ⊆ ℤ
Theorem	fzon 13633	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) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
Theorem	fzo0n 13634	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) → (𝑁 ≤ 𝑀 ↔ (0..^(𝑁 − 𝑀)) = ∅))
Theorem	fzonlt0 13635	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) = ∅))
Theorem	fzo0 13636	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) = ∅
Theorem	fzonnsub 13637	If 𝐾 < 𝑁 then 𝑁 − 𝐾 is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑁 − 𝐾) ∈ ℕ)
Theorem	fzonnsub2 13638	If 𝑀 < 𝑁 then 𝑁 − 𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑁 − 𝑀) ∈ ℕ)
Theorem	fzoss1 13639	Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾..^𝑁) ⊆ (𝑀..^𝑁))
Theorem	fzoss2 13640	Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁))
Theorem	fzossrbm1 13641	Subset of a half-open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
Theorem	fzo0ss1 13642	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)
Theorem	fzossnn0 13643	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)
Theorem	fzospliti 13644	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)))
Theorem	fzosplit 13645	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)))
Theorem	fzodisj 13646	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
Theorem	fzouzsplit 13647	Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (ℤ≥‘𝐴) = ((𝐴..^𝐵) ∪ (ℤ≥‘𝐵)))
Theorem	fzouzdisj 13648	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) ∩ (ℤ≥‘𝐵)) = ∅
Theorem	fzoun 13649	A half-open integer range as union of two half-open integer ranges. (Contributed by AV, 23-Apr-2022.)
⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐴..^(𝐵 + 𝐶)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 𝐶))))
Theorem	fzodisjsn 13650	A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.)
⊢ ((𝐴..^𝐵) ∩ {𝐵}) = ∅
Theorem	prinfzo0 13651	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)) = ∅)
Theorem	lbfzo0 13652	An integer is strictly greater than zero iff it is a member of ℕ. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
Theorem	elfzo0 13653	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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵))
Theorem	elfzo0z 13654	Membership in a half-open range of nonnegative integers, generalization of elfzo0 13653 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵))
Theorem	nn0p1elfzo 13655	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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ (𝐾 + 1) ≤ 𝑁) → 𝐾 ∈ (0..^𝑁))
Theorem	elfzo0le 13656	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) → 𝐴 ≤ 𝐵)
Theorem	elfzolem1 13657	A member in a half-open integer interval is less than or equal to the upper bound minus 1 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1))
Theorem	elfzo0subge1 13658	The difference of the upper bound of a half-open range of nonnegative integers and an element of this range is greater than or equal to 1. (Contributed by AV, 1-Sep-2025.) (Proof shortened by SN, 18-Sep-2025.)
⊢ (𝐴 ∈ (0..^𝐵) → 1 ≤ (𝐵 − 𝐴))
Theorem	elfzo0suble 13659	The difference of the upper bound of a half-open range of nonnegative integers and an element of this range is less than or equal to the upper bound. (Contributed by AV, 1-Sep-2025.) (Proof shortened by SN, 18-Sep-2025.)
⊢ (𝐴 ∈ (0..^𝐵) → (𝐵 − 𝐴) ≤ 𝐵)
Theorem	elfzonn0 13660	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)
Theorem	fzonmapblen 13661	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 + (𝑁 − 𝐴)) < 𝑁)
Theorem	fzofzim 13662	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.)
⊢ ((𝐾 ≠ 𝑀 ∧ 𝐾 ∈ (0...𝑀)) → 𝐾 ∈ (0..^𝑀))
Theorem	fz1fzo0m1 13663	Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.)
⊢ (𝑀 ∈ (1...𝑁) → (𝑀 − 1) ∈ (0..^𝑁))
Theorem	fzossnn 13664	Half-open integer ranges starting with 1 are subsets of ℕ. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ (1..^𝑁) ⊆ ℕ
Theorem	elfzo1 13665	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))
Theorem	fzo1lb 13666	1 is the left endpoint of a half-open integer range based at 1 iff the right endpoint is an integer greater than 1. (Contributed by AV, 4-Sep-2025.)
⊢ (1 ∈ (1..^𝑁) ↔ 𝑁 ∈ (ℤ≥‘2))
Theorem	1elfzo1 13667	1 is in a half-open range of positive integers iff its upper bound is greater than 1. (Contributed by AV, 22-Nov-2022.)
⊢ (1 ∈ (1..^𝑀) ↔ (𝑀 ∈ ℕ ∧ 1 < 𝑀))
Theorem	fzo1fzo0n0 13668	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))
Theorem	fzo0n0 13669	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)
Theorem	fzoaddel 13670	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)))
Theorem	fzo0addel 13671	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)))
Theorem	fzo0addelr 13672	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)))
Theorem	fzoaddel2 13673	Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐴 ∈ (0..^(𝐵 − 𝐶)) ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 + 𝐶) ∈ (𝐶..^𝐵))
Theorem	elfzoextl 13674	Membership of an integer in an extended open range of integers, extension added to the left. (Contributed by AV, 31-Aug-2025.) Generalized by replacing the left border of the ranges. (Revised by SN, 18-Sep-2025.)
⊢ ((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → 𝑍 ∈ (𝑀..^(𝐼 + 𝑁)))
Theorem	elfzoext 13675	Membership of an integer in an extended open range of integers, extension added to the right. (Contributed by AV, 30-Apr-2020.) (Proof shortened by AV, 23-Sep-2025.)
⊢ ((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → 𝑍 ∈ (𝑀..^(𝑁 + 𝐼)))
Theorem	elincfzoext 13676	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)))
Theorem	fzosubel 13677	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)))
Theorem	fzosubel2 13678	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)) → (𝐴 − 𝐵) ∈ (𝐶..^𝐷))
Theorem	fzosubel3 13679	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) → (𝐴 − 𝐵) ∈ (0..^𝐷))
Theorem	eluzgtdifelfzo 13680	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 ∈ (ℤ≥‘𝐴) ∧ 𝐵 < 𝐴) → (𝑁 − 𝐴) ∈ (0..^(𝑁 − 𝐵))))
Theorem	ige2m2fzo 13681	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)))
Theorem	fzocatel 13682	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)) → (𝐴 − 𝐵) ∈ (0..^𝐶))
Theorem	ubmelfzo 13683	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) → (𝑁 − 𝐾) ∈ (0..^𝑁))
Theorem	elfzodifsumelfzo 13684	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 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..^(𝑁 − 𝑀)) → (𝐼 + 𝑀) ∈ (0..^𝑃)))
Theorem	elfzom1elp1fzo 13685	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))
Theorem	elfzom1elfzo 13686	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))
Theorem	fzval3 13687	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)))
Theorem	fz0add1fz1 13688	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 ∈ ℕ0 ∧ 𝑋 ∈ (0..^𝑁)) → (𝑋 + 1) ∈ (1...𝑁))
Theorem	fzosn 13689	Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
Theorem	elfzomin 13690	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)))
Theorem	zpnn0elfzo 13691	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)))
Theorem	zpnn0elfzo1 13692	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))))
Theorem	fzosplitsnm1 13693	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)}))
Theorem	elfzonlteqm1 13694	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))
Theorem	fzonn0p1 13695	A nonnegative integer is an 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 ∈ ℕ0 → 𝑁 ∈ (0..^(𝑁 + 1)))
Theorem	fzossfzop1 13696	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)))
Theorem	fzonn0p1p1 13697	If a nonnegative integer is an 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 13698	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.) (Proof shortened by Thierry Arnoux, 14-Dec-2023.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁))
Theorem	fzo0ssnn0 13699	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
Theorem	fzo01 13700	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (0..^1) = {0}