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Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmax2d 40001 A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑𝐵 ≤ if(𝐴𝐵, 𝐵, 𝐴))

Theoremuzn0bi 40002 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.)
((ℤ𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ)

Theoremxnegrecl2 40003 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)

Theoremnfxneg 40004 Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝐴       𝑥-𝑒𝐴

Theoremuzxrd 40005 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑍)       (𝜑𝐴 ∈ ℝ*)

Theoreminfxrpnf2 40006 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*, < ))

Theoremsupminfxr 40007* 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.)
(𝜑𝐴 ⊆ ℝ)       (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ ∣ -𝑥𝐴}, ℝ*, < ))

Theoreminfrpgernmpt 40008* The infimum of a non empty, 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.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))

Theoremxnegre 40009 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))

Theoremxnegrecl2d 40010 If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → -𝑒𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremuzxr 40011 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ∈ (ℤ𝑀) → 𝐴 ∈ ℝ*)

Theoremsupminfxr2 40012* 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.)
(𝜑𝐴 ⊆ ℝ*)       (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < ))

Theoremxnegred 40013 An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ))

Theoremsupminfxrrnmpt 40014* 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.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))

Theoremmin1d 40015 The minimum of two numbers is less than or equal to the first. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → if(𝐴𝐵, 𝐴, 𝐵) ≤ 𝐴)

Theoremmin2d 40016 The minimum of two numbers is less than or equal to the second. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → if(𝐴𝐵, 𝐴, 𝐵) ≤ 𝐵)

Theorempnfged 40017 Plus infinity is an upper bound for extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℝ*)       (𝜑𝐴 ≤ +∞)

Theoremxrnpnfmnf 40018 An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ +∞)       (𝜑𝐴 = -∞)

Theoremuzsscn 40019 An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(ℤ𝑀) ⊆ ℂ

Theoremabsimnre 40020 The absolute value of the imaginary part of a non real, complex number, is strictly positive. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)       (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)

Theoremuzsscn2 40021 An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑍 = (ℤ𝑀)       𝑍 ⊆ ℂ

Theoremxrtgcntopre 40022 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)

Theoremabsimlere 40023 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐵𝐴)))

Theoremrpssxr 40024 The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
+ ⊆ ℝ*

Theoremmonoordxrv 40025* Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ*)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))       (𝜑 → (𝐹𝑀) ≤ (𝐹𝑁))

Theoremmonoordxr 40026* Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ*)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))       (𝜑 → (𝐹𝑀) ≤ (𝐹𝑁))

Theoremmonoord2xrv 40027* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ*)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))       (𝜑 → (𝐹𝑁) ≤ (𝐹𝑀))

Theoremmonoord2xr 40028* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ*)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))       (𝜑 → (𝐹𝑁) ≤ (𝐹𝑀))

Theoremxrpnf 40029* 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 𝑥𝐴))

20.32.4  Real intervals

Theoremgtnelioc 40030 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)       (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Theoremioossioc 40031 An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)

Theoremioondisj2 40032 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 < 𝐷𝐷𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)

Theoremioondisj1 40033 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)) ∧ (𝐴𝐶𝐶 < 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)

Theoremioosscn 40034 An open interval is a set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ ℂ

Theoremioogtlb 40035 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)

Theoremevthiccabs 40036* Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑦)) ≤ (abs‘(𝐹𝑥)) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑧)) ≤ (abs‘(𝐹𝑤))))

Theoremltnelicc 40037 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐶 < 𝐴)       (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Theoremeliood 40038 Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐶 < 𝐵)       (𝜑𝐶 ∈ (𝐴(,)𝐵))

Theoremiooabslt 40039 An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ((𝐴𝐵)(,)(𝐴 + 𝐵)))       (𝜑 → (abs‘(𝐴𝐶)) < 𝐵)

Theoremgtnelicc 40040 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)       (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Theoremiooinlbub 40041 An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅

Theoremiocgtlb 40042 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)

Theoremiocleub 40043 An element of a left open right closed interval is smaller or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐶𝐵)

Theoremeliccd 40044 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴[,]𝐵))

Theoremiccssred 40045 A closed real interval is a set of reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)

Theoremeliccre 40046 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)

Theoremeliooshift 40047 Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷))))

Theoremeliocd 40048 Membership in a left open, right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴(,]𝐵))

Theoremsnunioo2 40049 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) ∪ {𝐵}) = (𝐴(,]𝐵))

Theoremicoltub 40050 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)

Theoremtgiooss 40051 The restriction of the complex topology to a subset of reals, is a restriction of the standard topology on reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): Use rerest 22654 instead in fourierdlem48 40689, fourierdlem49 40690, fourierdlem62 40703 then delete this.
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (topGen‘ran (,))       (𝐴 ⊆ ℝ → (𝐽t 𝐴) = (𝐾t 𝐴))

Theoremeliocre 40052 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)

Theoremiooltub 40053 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)

Theoremioontr 40054 The interior of an interval in the standard topology on is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)

Theoremeliccxr 40055 A member of a closed interval is a an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*)

Theoremsnunioo1 40056 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))

Theoremlbioc 40057 An left open right closed interval doesn't contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
¬ 𝐴 ∈ (𝐴(,]𝐵)

Theoremioomidp 40058 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))

Theoremiccdifioo 40059 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*𝐴𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵})

Theoremiccdifprioo 40060 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))

Theoremioossioobi 40061 Biconditional form of ioossioo 12303. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐶 < 𝐷)       (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴𝐶𝐷𝐵)))

Theoremiccshift 40062* A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑇 ∈ ℝ)       (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})

Theoremiccsuble 40063 An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑𝐷 ∈ (𝐴[,]𝐵))       (𝜑 → (𝐶𝐷) ≤ (𝐵𝐴))

Theoremiocopn 40064 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   𝐾 = (topGen‘ran (,))    &   𝐽 = (𝐾t (𝐴(,]𝐵))    &   (𝜑𝐴𝐶)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐶(,]𝐵) ∈ 𝐽)

Theoremeliccelioc 40065 Membership in a closed interval and in a left open right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶𝐴)))

Theoremiooshift 40066* An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑇 ∈ ℝ)       (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)})

Theoremiccintsng 40067 Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝐵𝐵𝐶)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵})

Theoremicoiccdif 40068 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}))

Theoremicoopn 40069 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   𝐾 = (topGen‘ran (,))    &   𝐽 = (𝐾t (𝐴[,)𝐵))    &   (𝜑𝐶𝐵)       (𝜑 → (𝐴[,)𝐶) ∈ 𝐽)

Theoremicoub 40070 A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ ℝ* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Theoremeliccxrd 40071 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴[,]𝐵))

Theorempnfel0pnf 40072 +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
+∞ ∈ (0[,]+∞)

Theoremge0nemnf2 40073 A nonnegative extended real is not -∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞)

Theoremeliccnelico 40074 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵))       (𝜑𝐶 = 𝐵)

Theoremeliccelicod 40075 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴[,)𝐵))

Theoremge0xrre 40076 A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)

Theoremge0lere 40077 A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑𝐵 ∈ ℝ)

Theoremelicores 40078* 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))

Theoreminficc 40079 The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝑆 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑆 ≠ ∅)       (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵))

Theoremqinioo 40080 The rational numbers are dense in . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵𝐴))

Theoremlenelioc 40081 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐶𝐴)       (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Theoremioonct 40082 C non empty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   𝐶 = (𝐴(,)𝐵)       (𝜑 → ¬ 𝐶 ≼ ω)

Theoremxrgtnelicc 40083 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)       (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Theoremiccdificc 40084 The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝐴[,]𝐶) ∖ (𝐴[,]𝐵)) = (𝐵(,]𝐶))

Theoremiocnct 40085 A non empty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   𝐶 = (𝐴(,]𝐵)       (𝜑 → ¬ 𝐶 ≼ ω)

Theoremiccnct 40086 A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   𝐶 = (𝐴[,]𝐵)       (𝜑 → ¬ 𝐶 ≼ ω)

Theoremiooiinicc 40087* A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 𝑛 ∈ ℕ ((𝐴 − (1 / 𝑛))(,)(𝐵 + (1 / 𝑛))) = (𝐴[,]𝐵))

Theoremiccgelbd 40088 An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))       (𝜑𝐴𝐶)

Theoremiooltubd 40089 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))       (𝜑𝐶 < 𝐵)

Theoremicoltubd 40090 An element of a left closed right open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,)𝐵))       (𝜑𝐶 < 𝐵)

Theoremqelioo 40091* The rational numbers are dense in *: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃𝑥 ∈ ℚ 𝑥 ∈ (𝐴(,)𝐵))

Theoremtgqioo2 40092* 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 (,))    &   (𝜑𝐴𝐽)       (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = 𝑞))

Theoremiccleubd 40093 An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))       (𝜑𝐶𝐵)

Theoremelioored 40094 A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ (𝐵(,)𝐶))       (𝜑𝐴 ∈ ℝ)

Theoremioogtlbd 40095 An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))       (𝜑𝐴 < 𝐶)

Theoremioofun 40096 (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Fun (,)

Theoremicomnfinre 40097 A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴))

Theoremsqrlearg 40098 The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((𝐴↑2) ≤ 𝐴𝐴 ∈ (0[,]1)))

Theoremressiocsup 40099 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.)
(𝜑𝐴 ⊆ ℝ)    &   𝑆 = sup(𝐴, ℝ*, < )    &   (𝜑𝑆𝐴)    &   𝐼 = (-∞(,]𝑆)       (𝜑𝐴𝐼)

Theoremressioosup 40100 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.)
(𝜑𝐴 ⊆ ℝ)    &   𝑆 = sup(𝐴, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝐴)    &   𝐼 = (-∞(,)𝑆)       (𝜑𝐴𝐼)

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