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	fzossrbm1 13601	Subset of a half-open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
Theorem	fzo0ss1 13602	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 13603	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 13604	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 13605	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 13606	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
Theorem	fzouzsplit 13607	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 13608	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 13609	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 13610	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 13611	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 13612	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 13613	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 13614	Membership in a half-open range of nonnegative integers, generalization of elfzo0 13613 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 13615	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 13616	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	elfzonn0 13617	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 13618	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 13619	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 13620	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 13621	Half-open integer ranges starting with 1 are subsets of ℕ. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ (1..^𝑁) ⊆ ℕ
Theorem	elfzo1 13622	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	fzo1fzo0n0 13623	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 13624	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 13625	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 13626	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 13627	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 13628	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	elfzoext 13629	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)))
Theorem	elincfzoext 13630	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 13631	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 13632	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 13633	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 13634	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 13635	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 13636	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 13637	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 13638	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 13639	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 13640	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 13641	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 13642	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 13643	Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
Theorem	elfzomin 13644	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 13645	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 13646	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 13647	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 13648	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 13649	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 ∈ ℕ0 → 𝑁 ∈ (0..^(𝑁 + 1)))
Theorem	fzossfzop1 13650	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 13651	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)))
Theorem	elfzom1p1elfzo 13652	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 13653	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 13654	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (0..^1) = {0}
Theorem	fzo12sn 13655	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}
Theorem	fzo13pr 13656	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}
Theorem	fzo0to2pr 13657	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}
Theorem	fzo0to3tp 13658	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}
Theorem	fzo0to42pr 13659	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})
Theorem	fzo1to4tp 13660	A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.)
⊢ (1..^4) = {1, 2, 3}
Theorem	fzo0sn0fzo1 13661	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)))
Theorem	elfzo0l 13662	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)))
Theorem	fzoend 13663	The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵))
Theorem	fzo0end 13664	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))
Theorem	ssfzo12 13665	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) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
Theorem	ssfzoulel 13666	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 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))
Theorem	ssfzo12bi 13667	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) ↔ (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
Theorem	ubmelm1fzo 13668	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) → ((𝑁 − 𝐾) − 1) ∈ (0..^𝑁))
Theorem	fzofzp1 13669	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))
Theorem	fzofzp1b 13670	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 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))
Theorem	elfzom1b 13671	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))))
Theorem	elfzom1elp1fzo1 13672	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))
Theorem	elfzo1elm1fzo0 13673	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)))
Theorem	elfzonelfzo 13674	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)))
Theorem	fzonfzoufzol 13675	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 < 𝑁 ∧ 𝐼 ∈ (0..^𝑁)) → (¬ 𝐼 ∈ ((𝑁 − 𝑀)..^𝑁) → 𝐼 ∈ (0..^(𝑁 − 𝑀))))
Theorem	elfzomelpfzo 13676	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 ∈ ((𝑀 − 𝐿)..^(𝑁 − 𝐿)) ↔ (𝐾 + 𝐿) ∈ (𝑀..^𝑁)))
Theorem	elfznelfzo 13677	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))
Theorem	elfznelfzob 13678	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)))
Theorem	peano2fzor 13679	A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑀..^𝑁))
Theorem	fzosplitsn 13680	Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	fzosplitpr 13681	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 2)) = ((𝐴..^𝐵) ∪ {𝐵, (𝐵 + 1)}))
Theorem	fzosplitprm1 13682	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 25-Jun-2022.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}))
Theorem	fzosplitsni 13683	Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^(𝐵 + 1)) ↔ (𝐶 ∈ (𝐴..^𝐵) ∨ 𝐶 = 𝐵)))
Theorem	fzisfzounsn 13684	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 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	elfzr 13685	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))
Theorem	elfzlmr 13686	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 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))
Theorem	elfz0lmr 13687	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))
Theorem	fzostep1 13688	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))
Theorem	fzoshftral 13689*	Shift the scanning order inside of a universal quantification restricted to a half-open integer range, analogous to fzshftral 13529. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑))
Theorem	fzind2 13690*	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 12601 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜓)    &   ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝜒 → 𝜃))    ⇒   ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝜏)
Theorem	fvinim0ffz 13691	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)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
Theorem	injresinjlem 13692	Lemma for injresinj 13693. (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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))))))
Theorem	injresinj 13693	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) ≠ (𝐹‘𝐾)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ◡𝐹)))
Theorem	subfzo0 13694	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 < (𝐼 − 𝐽) ∧ (𝐼 − 𝐽) < 𝑁))
5.6  Elementary integer functions
5.6.1  The floor and ceiling functions
Syntax	cfl 13695	Extend class notation with floor (greatest integer) function.
class ⌊
Syntax	cceil 13696	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 13697*	Define the floor (greatest integer less than or equal to) function. See flval 13699 for its value, fllelt 13702 for its basic property, and flcl 13700 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 29391). 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 (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
Definition	df-ceil 13698	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 13743 for its value, ceilge 13750 and ceilm1lt 13753 for its basic properties, and ceilcl 13747 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 29392). 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))
Theorem	flval 13699*	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 (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
Theorem	flcl 13700	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)