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Theorem List for Metamath Proof Explorer - 39501-39600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmax2d 39501 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 39502 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 39503 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 39504 Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝐴       𝑥-𝑒𝐴
 
Theoremuzxrd 39505 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑍)       (𝜑𝐴 ∈ ℝ*)
 
Theoreminfxrpnf2 39506 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 39507* 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 39508* 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 39509 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 39510 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 39511 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ∈ (ℤ𝑀) → 𝐴 ∈ ℝ*)
 
Theoremsupminfxr2 39512* 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 39513 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 39514* 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*, < ))
 
20.31.4  Real intervals
 
Theoremgtnelioc 39515 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 39516 An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)
 
Theoremioondisj2 39517 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 39518 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 39519 An open interval is a set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ ℂ
 
Theoremioogtlb 39520 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 39521* 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 39522 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 39523 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 39524 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 39525 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 39526 An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅
 
Theoremiocgtlb 39527 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 39528 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 39529 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 39530 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 39531 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 39532 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 39533 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 39534 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 39535 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 39536 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 22588 instead in fourierdlem48 40134, fourierdlem49 40135, fourierdlem62 40148 then delete this.
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (topGen‘ran (,))       (𝐴 ⊆ ℝ → (𝐽t 𝐴) = (𝐾t 𝐴))
 
Theoremeliocre 39537 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 39538 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 39539 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 39540 A member of a closed interval is a an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*)
 
Theoremsnunioo1 39541 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 39542 An left open right closed interval doesn't contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
¬ 𝐴 ∈ (𝐴(,]𝐵)
 
Theoremioomidp 39543 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 39544 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 39545 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 39546 Biconditional form of ioossioo 12250. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐶 < 𝐷)       (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴𝐶𝐷𝐵)))
 
Theoremiccshift 39547* 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 39548 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 39549 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 39550 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 39551* 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 39552 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 39553 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 39554 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 39555 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 39556 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 39557 +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
+∞ ∈ (0[,]+∞)
 
Theoremge0nemnf2 39558 A nonnegative extended real is not -∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞)
 
Theoremeliccnelico 39559 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 39560 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 39561 A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)
 
Theoremge0lere 39562 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 39563* 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 39564 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 39565 The rational numbers are dense in . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵𝐴))
 
Theoremlenelioc 39566 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 39567 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 39568 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 39569 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 39570 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 39571 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 39572* 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 39573 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 39574 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 39575 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 39576* 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 39577* 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 39578 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 39579 A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ (𝐵(,)𝐶))       (𝜑𝐴 ∈ ℝ)
 
Theoremioogtlbd 39580 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 39581 (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Fun (,)
 
Theoremicomnfinre 39582 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 39583 The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((𝐴↑2) ≤ 𝐴𝐴 ∈ (0[,]1)))
 
Theoremressiocsup 39584 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 39585 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)
 
Theoremiooiinioc 39586* A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))
 
Theoremressiooinf 39587 If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑆 = inf(𝐴, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝐴)    &   𝐼 = (𝑆(,)+∞)       (𝜑𝐴𝐼)
 
Theoremicogelbd 39588 An element of a left closed right open interval is larger or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴[,)𝐵))       (𝜑𝐴𝐶)
 
Theoremiocleubd 39589 An element of a left open right closed interval is smaller or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝐴(,]𝐵))       (𝜑𝐶𝐵)
 
Theoremuzinico 39590 An upper interval of integers is the intersection of the integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑𝑍 = (ℤ ∩ (𝑀[,)+∞)))
 
Theorempreimaiocmnf 39591* Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (-∞(,]𝐵)) = {𝑥𝐴 ∣ (𝐹𝑥) ≤ 𝐵})
 
Theoremuzinico2 39592 An upper interval of integers is the intersection of a larger upper interval of integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (ℤ𝑁) = ((ℤ𝑀) ∩ (𝑁[,)+∞)))
 
Theoremuzinico3 39593 An upper interval of integers doesn't change when it's intersected with a left-closed, unbounded above interval, with the same lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑𝑍 = (𝑍 ∩ (𝑀[,)+∞)))
 
Theoremicossico2 39594 Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))
 
Theoremdmico 39595 The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
dom [,) = (ℝ* × ℝ*)
 
Theoremndmico 39596 The closed-below, open-above interval function's value is empty outside of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(¬ (𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ∅)
 
Theoremuzubioo 39597* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘𝑍)
 
Theoremuzubico 39598* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘𝑍)
 
Theoremuzubioo2 39599* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘𝑍)
 
Theoremuzubico2 39600* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘𝑍)
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