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Theorem List for Metamath Proof Explorer - 41601-41700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreminfxrpnf 41601 Adding plus infinity to a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ⊆ ℝ* → inf((𝐴 ∪ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < ))
 
Theoreminfxrrnmptcl 41602* The infimum of an arbitrary indexed set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ∈ ℝ*)
 
Theoremleneg2d 41603 Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 ≤ -𝐵𝐵 ≤ -𝐴))
 
Theoremsupxrltinfxr 41604 The supremum of the empty set is strictly smaller than the infimum of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
sup(∅, ℝ*, < ) < inf(∅, ℝ*, < )
 
Theoremmax1d 41605 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)       (𝜑𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
 
Theoremceilcld 41606 Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (⌈‘𝐴) ∈ ℤ)
 
Theoremsupxrleubrnmptf 41607 The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
 
Theoremnleltd 41608 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ¬ 𝐵𝐴)       (𝜑𝐴 < 𝐵)
 
Theoremzxrd 41609 An integer is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℝ*)
 
Theoreminfxrgelbrnmpt 41610* 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.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑥𝐴 𝐶𝐵))
 
Theoremrphalfltd 41611 Half of a positive real is less than the original number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 / 2) < 𝐴)
 
Theoremuzssz2 41612 An upper set of integers is a subset of all integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (ℤ𝑀)       𝑍 ⊆ ℤ
 
Theoremleneg3d 41613 Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (-𝐴𝐵 ↔ -𝐵𝐴))
 
Theoremmax2d 41614 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 41615 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 41616 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 41617 Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝐴       𝑥-𝑒𝐴
 
Theoremuzxrd 41618 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑍)       (𝜑𝐴 ∈ ℝ*)
 
Theoreminfxrpnf2 41619 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 41620* 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 41621* 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.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
 
Theoremxnegre 41622 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 41623 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 41624 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ∈ (ℤ𝑀) → 𝐴 ∈ ℝ*)
 
Theoremsupminfxr2 41625* 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 41626 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 41627* 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 41628 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 41629 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 41630 Plus infinity is an upper bound for extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℝ*)       (𝜑𝐴 ≤ +∞)
 
Theoremxrnpnfmnf 41631 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 41632 An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(ℤ𝑀) ⊆ ℂ
 
Theoremabsimnre 41633 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 41634 An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑍 = (ℤ𝑀)       𝑍 ⊆ ℂ
 
Theoremxrtgcntopre 41635 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 41636 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 41637 The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
+ ⊆ ℝ*
 
Theoremmonoordxrv 41638* 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 41639* 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 41640* 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 41641* 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 41642* 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))
 
Theoremxlenegcon1 41643 Extended real version of lenegcon1 11133. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴𝐵 ↔ -𝑒𝐵𝐴))
 
Theoremxlenegcon2 41644 Extended real version of lenegcon2 11134. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ≤ -𝑒𝐵𝐵 ≤ -𝑒𝐴))
 
20.36.4  Real intervals
 
Theoremgtnelioc 41645 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 41646 An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)
 
Theoremioondisj2 41647 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 41648 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 41649 An open interval is a set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ ℂ
 
Theoremioogtlb 41650 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 41651* 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 41652 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 41653 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 41654 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 41655 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 41656 An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅
 
Theoremiocgtlb 41657 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 41658 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)) → 𝐶𝐵)
 
Theoremeliccd 41659 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 41660 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 41661 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 41662 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 41663 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))
 
Theoremicoltub 41664 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)
 
Theoremeliocre 41665 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 41666 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 41667 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)
 
Theoremsnunioo1 41668 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 41669 A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
¬ 𝐴 ∈ (𝐴(,]𝐵)
 
Theoremioomidp 41670 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 41671 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 41672 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 41673 Biconditional form of ioossioo 12819. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐶 < 𝐷)       (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴𝐶𝐷𝐵)))
 
Theoremiccshift 41674* 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 41675 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 41676 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 41677 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 41678* 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 41679 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 41680 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 41681 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 41682 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 41683 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 41684 +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
+∞ ∈ (0[,]+∞)
 
Theoremeliccnelico 41685 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 41686 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 41687 A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)
 
Theoremge0lere 41688 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 41689* 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 41690 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 41691 The rational numbers are dense in . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵𝐴))
 
Theoremlenelioc 41692 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 41693 A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   𝐶 = (𝐴(,)𝐵)       (𝜑 → ¬ 𝐶 ≼ ω)
 
Theoremxrgtnelicc 41694 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 41695 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 41696 A nonempty 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 41697 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 41698* 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 41699 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 41700 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)
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