P. 136 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	snunioo 13501	The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
Theorem	snunico 13502	The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))
Theorem	snunioc 13503	The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ({𝐴} ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
Theorem	prunioo 13504	The closure of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
Theorem	ioodisj 13505	If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅)
Theorem	ioojoin 13506	Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
Theorem	difreicc 13507	The class difference of ℝ and a closed interval. (Contributed by FL, 18-Jun-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) = ((-∞(,)𝐴) ∪ (𝐵(,)+∞)))
Theorem	iccsplit 13508	Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝐴[,]𝐶) ∪ (𝐶[,]𝐵)))
Theorem	iccshftr 13509	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 + 𝑅) = 𝐶    &   ⊢ (𝐵 + 𝑅) = 𝐷    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷)))
Theorem	iccshftri 13510	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝑅 ∈ ℝ    &   ⊢ (𝐴 + 𝑅) = 𝐶    &   ⊢ (𝐵 + 𝑅) = 𝐷    ⇒   ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))
Theorem	iccshftl 13511	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 − 𝑅) = 𝐶    &   ⊢ (𝐵 − 𝑅) = 𝐷    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 − 𝑅) ∈ (𝐶[,]𝐷)))
Theorem	iccshftli 13512	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝑅 ∈ ℝ    &   ⊢ (𝐴 − 𝑅) = 𝐶    &   ⊢ (𝐵 − 𝑅) = 𝐷    ⇒   ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 − 𝑅) ∈ (𝐶[,]𝐷))
Theorem	iccdil 13513	Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 · 𝑅) = 𝐶    &   ⊢ (𝐵 · 𝑅) = 𝐷    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 · 𝑅) ∈ (𝐶[,]𝐷)))
Theorem	iccdili 13514	Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝑅 ∈ ℝ+    &   ⊢ (𝐴 · 𝑅) = 𝐶    &   ⊢ (𝐵 · 𝑅) = 𝐷    ⇒   ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))
Theorem	icccntr 13515	Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 / 𝑅) = 𝐶    &   ⊢ (𝐵 / 𝑅) = 𝐷    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷)))
Theorem	icccntri 13516	Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝑅 ∈ ℝ+    &   ⊢ (𝐴 / 𝑅) = 𝐶    &   ⊢ (𝐵 / 𝑅) = 𝐷    ⇒   ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))
Theorem	divelunit 13517	A condition for a ratio to be a member of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ (0[,]1) ↔ 𝐴 ≤ 𝐵))
Theorem	lincmb01cmp 13518	A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵)) ∈ (𝐴[,]𝐵))
Theorem	iccf1o 13519*	Describe a bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵]. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) ∧ ◡𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴)))))
Theorem	iccen 13520	Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ≈ (𝐴[,]𝐵))
Theorem	xov1plusxeqvd 13521	A complex number 𝑋 is positive real iff 𝑋 / (1 + 𝑋) is in (0(,)1). Deduction form. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ -1)    ⇒   ⊢ (𝜑 → (𝑋 ∈ ℝ+ ↔ (𝑋 / (1 + 𝑋)) ∈ (0(,)1)))
Theorem	unitssre 13522	(0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
⊢ (0[,]1) ⊆ ℝ
Theorem	unitsscn 13523	The closed unit interval is a subset of the set of the complex numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.)
⊢ (0[,]1) ⊆ ℂ
Theorem	supicc 13524	Supremum of a bounded set of real numbers. (Contributed by Thierry Arnoux, 17-May-2019.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ (𝐵[,]𝐶))
Theorem	supiccub 13525	The supremum of a bounded set of real numbers is an upper bound. (Contributed by Thierry Arnoux, 20-May-2019.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐷 ≤ sup(𝐴, ℝ, < ))
Theorem	supicclub 13526*	The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐷 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐷 < 𝑧))
Theorem	supicclub2 13527*	The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ 𝐷)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐷)
Theorem	zltaddlt1le 13528	The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (0(,)1)) → ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≤ 𝑁))
Theorem	xnn0xrge0 13529	An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ (0[,]+∞))
Theorem	nnge2recico01 13530	The reciprocal of an integer greater than 1 is in the right open interval between 0 and 1. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (1 / 𝑁) ∈ (0[,)1))
5.5.5  Finite intervals of integers
Syntax	cfz 13531	Extend class notation to include the notation for a contiguous finite set of integers. Read "𝑀...𝑁 " as "the set of integers from 𝑀 to 𝑁 inclusive". This symbol is also used informally in some comments to denote an ellipsis, e.g., 𝐴 + 𝐴↑2 + ... + 𝐴↑(𝑁 − 1).
class ...
Definition	df-fz 13532*	Define an operation that produces a finite set of sequential integers. Read "𝑀...𝑁 " as "the set of integers from 𝑀 to 𝑁 inclusive". See fzval 13533 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
Theorem	fzval 13533*	The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where ℕk means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)})
Theorem	fzval2 13534	An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
Theorem	fzf 13535	Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
Theorem	elfz1 13536	Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)))
Theorem	elfz 13537	Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)))
Theorem	elfz2 13538	Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show 𝑀 ∈ ℤ and 𝑁 ∈ ℤ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)))
Theorem	elfzd 13539	Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝐾)    &   ⊢ (𝜑 → 𝐾 ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
Theorem	elfz5 13540	Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁))
Theorem	elfz4 13541	Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁))
Theorem	elfzuzb 13542	Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)))
Theorem	eluzfz 13543	Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (𝑀...𝑁))
Theorem	elfzuz 13544	A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
Theorem	elfzuz3 13545	Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
Theorem	elfzel2 13546	Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
Theorem	elfzel1 13547	Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
Theorem	elfzelz 13548	A member of a finite set of sequential integers is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
Theorem	elfzelzd 13549	A member of a finite set of sequential integers is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))    ⇒   ⊢ (𝜑 → 𝐾 ∈ ℤ)
Theorem	fzssz 13550	A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝑀...𝑁) ⊆ ℤ
Theorem	elfzle1 13551	A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾)
Theorem	elfzle2 13552	A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁)
Theorem	elfzuz2 13553	Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	elfzle3 13554	Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑁)
Theorem	eluzfz1 13555	Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
Theorem	eluzfz2 13556	Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
Theorem	eluzfz2b 13557	Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))
Theorem	elfz3 13558	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))
Theorem	elfz1eq 13559	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)
Theorem	elfzubelfz 13560	If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (𝑀...𝑁))
Theorem	peano2fzr 13561	A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁))
Theorem	fzn0 13562	Properties of a finite interval of integers which is nonempty. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝑀...𝑁) ≠ ∅ ↔ 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	fz0 13563	A finite set of sequential integers is empty if its bounds are not integers. (Contributed by AV, 13-Oct-2018.)
⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) → (𝑀...𝑁) = ∅)
Theorem	fzn 13564	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
Theorem	fzen 13565	A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀...𝑁) ≈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))
Theorem	fz1n 13566	A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℕ0 → ((1...𝑁) = ∅ ↔ 𝑁 = 0))
Theorem	0nelfz1 13567	0 is not an element of a finite interval of integers starting at 1. (Contributed by AV, 27-Aug-2020.)
⊢ 0 ∉ (1...𝑁)
Theorem	0fz1 13568	Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0))
Theorem	fz10 13569	There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (1...0) = ∅
Theorem	uzsubsubfz 13570	Membership of an integer greater than L decreased by ( L - M ) in an M-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ ((𝐿 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − (𝐿 − 𝑀)) ∈ (𝑀...𝑁))
Theorem	uzsubsubfz1 13571	Membership of an integer greater than L decreased by ( L - 1 ) in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ ((𝐿 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁))
Theorem	ige3m2fz 13572	Membership of an integer greater than 2 decreased by 2 in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ (1...𝑁))
Theorem	fzsplit2 13573	Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
⊢ (((𝐾 + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))
Theorem	fzsplit 13574	Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))
Theorem	fzdisj 13575	Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)
Theorem	fz01en 13576	0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
⊢ (𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁))
Theorem	elfznn 13577	A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ)
Theorem	elfz1end 13578	A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
Theorem	fz1ssnn 13579	A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (1...𝐴) ⊆ ℕ
Theorem	fznn0sub 13580	Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝐾) ∈ ℕ0)
Theorem	fzmmmeqm 13581	Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
Theorem	fzaddel 13582	Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))))
Theorem	fzadd2 13583	Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑂 ∈ ℤ ∧ 𝑃 ∈ ℤ)) → ((𝐽 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑂...𝑃)) → (𝐽 + 𝐾) ∈ ((𝑀 + 𝑂)...(𝑁 + 𝑃))))
Theorem	fzsubel 13584	Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))))
Theorem	fzopth 13585	A finite set of sequential integers has the ordered pair property (compare opth 5456) under certain conditions. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
Theorem	fzass4 13586	Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐵 ∈ (𝐴...𝐷) ∧ 𝐶 ∈ (𝐵...𝐷)) ↔ (𝐵 ∈ (𝐴...𝐶) ∧ 𝐶 ∈ (𝐴...𝐷)))
Theorem	fzss1 13587	Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))
Theorem	fzss2 13588	Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁))
Theorem	fzssuz 13589	A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀)
Theorem	fzsn 13590	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
Theorem	fzssp1 13591	Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
Theorem	fzssnn 13592	Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ)
Theorem	ssfzunsnext 13593	A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 13-Nov-2021.)
⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
Theorem	ssfzunsn 13594	A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 8-Jun-2021.) (Proof shortened by AV, 13-Nov-2021.)
⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
Theorem	fzsuc 13595	Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))
Theorem	fzpred 13596	Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
Theorem	fzpreddisj 13597	A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
Theorem	elfzp1 13598	Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))
Theorem	fzp1ss 13599	Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
Theorem	fzelp1 13600	Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (𝑀...(𝑁 + 1)))