P. 431 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43001-43100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	infxrgelbrnmpt 43001*	The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))
Theorem	rphalfltd 43002	Half of a positive real is less than the original number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 2) < 𝐴)
Theorem	uzssz2 43003	An upper set of integers is a subset of all integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ 𝑍 ⊆ ℤ
Theorem	leneg3d 43004	Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴))
Theorem	max2d 43005	A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
Theorem	uzn0bi 43006	The upper integers function needs to be applied to an integer, in order to return a nonempty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ)
Theorem	xnegrecl2 43007	If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
Theorem	nfxneg 43008	Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥-𝑒𝐴
Theorem	uzxrd 43009	An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	infxrpnf2 43010	Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∖ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < ))
Theorem	supminfxr 43011*	The extended real suprema of a set of reals is the extended real negative of the extended real infima of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ*, < ))
Theorem	infrpgernmpt 43012*	The infimum of a nonempty, bounded below, indexed subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
Theorem	xnegre 43013	An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ))
Theorem	xnegrecl2d 43014	If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → -𝑒𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	uzxr 43015	An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ*)
Theorem	supminfxr2 43016*	The extended real suprema of a set of extended reals is the extended real negative of the extended real infima of that set's image under extended real negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ*)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥 ∈ 𝐴}, ℝ*, < ))
Theorem	xnegred 43017	An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ))
Theorem	supminfxrrnmpt 43018*	The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Theorem	min1d 43019	The minimum of two numbers is less than or equal to the first. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴)
Theorem	min2d 43020	The minimum of two numbers is less than or equal to the second. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)
Theorem	pnfged 43021	Plus infinity is an upper bound for extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → 𝐴 ≤ +∞)
Theorem	xrnpnfmnf 43022	An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ +∞)    ⇒   ⊢ (𝜑 → 𝐴 = -∞)
Theorem	uzsscn 43023	An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (ℤ≥‘𝑀) ⊆ ℂ
Theorem	absimnre 43024	The absolute value of the imaginary part of a non-real, complex number, is strictly positive. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
Theorem	uzsscn2 43025	An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ 𝑍 ⊆ ℂ
Theorem	xrtgcntopre 43026	The standard topologies on the extended reals and on the complex numbers, coincide when restricted to the reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ ((ordTop‘ ≤ ) ↾t ℝ) = ((TopOpen‘ℂfld) ↾t ℝ)
Theorem	absimlere 43027	The absolute value of the imaginary part of a complex number is a lower bound of the distance to any real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	rpssxr 43028	The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ ℝ+ ⊆ ℝ*
Theorem	monoordxrv 43029*	Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))
Theorem	monoordxr 43030*	Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))
Theorem	monoord2xrv 43031*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))
Theorem	monoord2xr 43032*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))
Theorem	xrpnf 43033*	An extended real is plus infinity iff it's larger than all real numbers. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ 𝐴))
Theorem	xlenegcon1 43034	Extended real version of lenegcon1 11488. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ 𝐴))
Theorem	xlenegcon2 43035	Extended real version of lenegcon2 11489. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ -𝑒𝐵 ↔ 𝐵 ≤ -𝑒𝐴))
20.37.4  Real intervals
Theorem	gtnelioc 43036	A real number larger than the upper bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))
Theorem	ioossioc 43037	An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)
Theorem	ioondisj2 43038	A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐶 < 𝐷)) ∧ (𝐴 < 𝐷 ∧ 𝐷 ≤ 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)
Theorem	ioondisj1 43039	A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐶 < 𝐷)) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)
Theorem	ioogtlb 43040	An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
Theorem	evthiccabs 43041*	Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤))))
Theorem	ltnelicc 43042	A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 < 𝐴)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Theorem	eliood 43043	Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐶 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
Theorem	iooabslt 43044	An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵)))    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < 𝐵)
Theorem	gtnelicc 43045	A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Theorem	iooinlbub 43046	An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅
Theorem	iocgtlb 43047	An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
Theorem	iocleub 43048	An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵)
Theorem	eliccd 43049	Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
Theorem	eliccre 43050	A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ)
Theorem	eliooshift 43051	Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷))))
Theorem	eliocd 43052	Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))
Theorem	icoltub 43053	An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
Theorem	eliocre 43054	A member of a left-open right-closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ)
Theorem	iooltub 43055	An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵)
Theorem	ioontr 43056	The interior of an interval in the standard topology on ℝ is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
Theorem	snunioo1 43057	The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
Theorem	lbioc 43058	A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ¬ 𝐴 ∈ (𝐴(,]𝐵)
Theorem	ioomidp 43059	The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
Theorem	iccdifioo 43060	If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵})
Theorem	iccdifprioo 43061	An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵))
Theorem	ioossioobi 43062	Biconditional form of ioossioo 13182. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 < 𝐷)    ⇒   ⊢ (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)))
Theorem	iccshift 43063*	A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
Theorem	iccsuble 43064	An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵))    ⇒   ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴))
Theorem	iocopn 43065	A left-open right-closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝐽 = (𝐾 ↾t (𝐴(,]𝐵))    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐶(,]𝐵) ∈ 𝐽)
Theorem	eliccelioc 43066	Membership in a closed interval and in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)))
Theorem	iooshift 43067*	An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)})
Theorem	iccintsng 43068	Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵})
Theorem	icoiccdif 43069	Left-closed right-open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ((𝐴[,]𝐵) ∖ {𝐵}))
Theorem	icoopn 43070	A left-closed right-open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝐽 = (𝐾 ↾t (𝐴[,)𝐵))    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴[,)𝐶) ∈ 𝐽)
Theorem	icoub 43071	A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ ℝ* → ¬ 𝐵 ∈ (𝐴[,)𝐵))
Theorem	eliccxrd 43072	Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
Theorem	pnfel0pnf 43073	+∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ +∞ ∈ (0[,]+∞)
Theorem	eliccnelico 43074	An element of a closed interval that is not a member of the left-closed right-open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵))    ⇒   ⊢ (𝜑 → 𝐶 = 𝐵)
Theorem	eliccelicod 43075	A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝐶 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))
Theorem	ge0xrre 43076	A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)
Theorem	ge0lere 43077	A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ℝ)
Theorem	elicores 43078*	Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))
Theorem	inficc 43079	The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    ⇒   ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵))
Theorem	qinioo 43080	The rational numbers are dense in ℝ. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	lenelioc 43081	A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐴)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))
Theorem	ioonct 43082	A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ 𝐶 = (𝐴(,)𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ≼ ω)
Theorem	xrgtnelicc 43083	A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Theorem	iccdificc 43084	The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴[,]𝐶) ∖ (𝐴[,]𝐵)) = (𝐵(,]𝐶))
Theorem	iocnct 43085	A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ 𝐶 = (𝐴(,]𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ≼ ω)
Theorem	iccnct 43086	A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ 𝐶 = (𝐴[,]𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ≼ ω)
Theorem	iooiinicc 43087*	A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ ((𝐴 − (1 / 𝑛))(,)(𝐵 + (1 / 𝑛))) = (𝐴[,]𝐵))
Theorem	iccgelbd 43088	An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐶)
Theorem	iooltubd 43089	An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → 𝐶 < 𝐵)
Theorem	icoltubd 43090	An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))    ⇒   ⊢ (𝜑 → 𝐶 < 𝐵)
Theorem	qelioo 43091*	The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℚ 𝑥 ∈ (𝐴(,)𝐵))
Theorem	tgqioo2 43092*	Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    ⇒   ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞))
Theorem	iccleubd 43093	An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    ⇒   ⊢ (𝜑 → 𝐶 ≤ 𝐵)
Theorem	elioored 43094	A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ (𝐵(,)𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	ioogtlbd 43095	An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	ioofun 43096	(,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Fun (,)
Theorem	icomnfinre 43097	A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴))
Theorem	sqrlearg 43098	The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴↑2) ≤ 𝐴 ↔ 𝐴 ∈ (0[,]1)))
Theorem	ressiocsup 43099	If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (-∞(,]𝑆)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
Theorem	ressioosup 43100	If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    &   ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (-∞(,)𝑆)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)