P. 135 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	0e0iccpnf 13401	0 is a member of (0[,]+∞). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ∈ (0[,]+∞)
Theorem	ge0addcl 13402	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 + 𝐵) ∈ (0[,)+∞))
Theorem	ge0mulcl 13403	The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 · 𝐵) ∈ (0[,)+∞))
Theorem	ge0xaddcl 13404	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))
Theorem	ge0xmulcl 13405	The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴 ·e 𝐵) ∈ (0[,]+∞))
Theorem	lbicc2 13406	The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
Theorem	ubicc2 13407	The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
Theorem	elicc01 13408	Membership in the closed real interval between 0 and 1, also called the closed unit interval. (Contributed by AV, 20-Aug-2022.)
⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
Theorem	elunitrn 13409	The closed unit interval is a subset of the set of the real numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 21-Dec-2016.)
⊢ (𝐴 ∈ (0[,]1) → 𝐴 ∈ ℝ)
Theorem	elunitcn 13410	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, 21-Dec-2016.)
⊢ (𝐴 ∈ (0[,]1) → 𝐴 ∈ ℂ)
Theorem	0elunit 13411	Zero is an element of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.)
⊢ 0 ∈ (0[,]1)
Theorem	1elunit 13412	One is an element of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.)
⊢ 1 ∈ (0[,]1)
Theorem	iooneg 13413	Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴)))
Theorem	iccneg 13414	Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ -𝐶 ∈ (-𝐵[,]-𝐴)))
Theorem	icoshft 13415	A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑋 ∈ (𝐴[,)𝐵) → (𝑋 + 𝐶) ∈ ((𝐴 + 𝐶)[,)(𝐵 + 𝐶))))
Theorem	icoshftf1o 13416*	Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (𝐴[,)𝐵) ↦ (𝑥 + 𝐶))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐹:(𝐴[,)𝐵)–1-1-onto→((𝐴 + 𝐶)[,)(𝐵 + 𝐶)))
Theorem	icoun 13417	The union of two adjacent left-closed right-open real intervals is a left-closed right-open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴[,)𝐶))
Theorem	icodisj 13418	Adjacent left-closed right-open real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)
Theorem	ioounsn 13419	The union of an open interval with its upper endpoint is a left-open right-closed interval. (Contributed by Jon Pennant, 8-Jun-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
Theorem	snunioo 13420	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 13421	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 13422	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 13423	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 13424	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 13425	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 13426	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 13427	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 13428	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 13429	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 13430	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 13431	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 13432	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 13433	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 13434	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 13435	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 13436	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 13437	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 13438*	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 13439	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 13440	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 13441	(0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
⊢ (0[,]1) ⊆ ℝ
Theorem	unitsscn 13442	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 13443	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 13444	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 13445*	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 13446*	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 13447	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 13448	An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ (0[,]+∞))
Theorem	nnge2recico01 13449	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 13450	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 13451*	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 13452 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
Theorem	fzval 13452*	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 13453	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 13454	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 13455	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 13456	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 13457	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 13458	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 13459	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 13460	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 13461	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 13462	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 13463	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 13464	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 13465	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 13466	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 13467	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 13468	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 13469	A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝑀...𝑁) ⊆ ℤ
Theorem	elfzle1 13470	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 13471	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 13472	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 13473	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 13474	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 13475	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 13476	Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))
Theorem	elfz3 13477	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))
Theorem	elfz1eq 13478	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)
Theorem	elfzubelfz 13479	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 13480	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 13481	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 13482	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 13483	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 13484	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 13485	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 13486	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 13487	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 13488	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 13489	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 13490	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 13491	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 13492	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 13493	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 13494	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 13495	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 13496	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 13497	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 13498	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 13499	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 13500	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))