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Theorem List for Metamath Proof Explorer - 43501-43600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiooltub 43501 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴(,)𝐡)) β†’ 𝐢 < 𝐡)
 
Theoremioontr 43502 The interior of an interval in the standard topology on ℝ is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((intβ€˜(topGenβ€˜ran (,)))β€˜(𝐴(,)𝐡)) = (𝐴(,)𝐡)
 
Theoremsnunioo1 43503 The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 < 𝐡) β†’ ((𝐴(,)𝐡) βˆͺ {𝐴}) = (𝐴[,)𝐡))
 
Theoremlbioc 43504 A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
¬ 𝐴 ∈ (𝐴(,]𝐡)
 
Theoremioomidp 43505 The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡) β†’ ((𝐴 + 𝐡) / 2) ∈ (𝐴(,)𝐡))
 
Theoremiccdifioo 43506 If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 ≀ 𝐡) β†’ ((𝐴[,]𝐡) βˆ– (𝐴(,)𝐡)) = {𝐴, 𝐡})
 
Theoremiccdifprioo 43507 An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((𝐴[,]𝐡) βˆ– {𝐴, 𝐡}) = (𝐴(,)𝐡))
 
Theoremioossioobi 43508 Biconditional form of ioossioo 13286. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐷 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 < 𝐷)    β‡’   (πœ‘ β†’ ((𝐢(,)𝐷) βŠ† (𝐴(,)𝐡) ↔ (𝐴 ≀ 𝐢 ∧ 𝐷 ≀ 𝐡)))
 
Theoremiccshift 43509* A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    β‡’   (πœ‘ β†’ ((𝐴 + 𝑇)[,](𝐡 + 𝑇)) = {𝑀 ∈ β„‚ ∣ βˆƒπ‘§ ∈ (𝐴[,]𝐡)𝑀 = (𝑧 + 𝑇)})
 
Theoremiccsuble 43510 An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ 𝐷 ∈ (𝐴[,]𝐡))    β‡’   (πœ‘ β†’ (𝐢 βˆ’ 𝐷) ≀ (𝐡 βˆ’ 𝐴))
 
Theoremiocopn 43511 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   πΎ = (topGenβ€˜ran (,))    &   π½ = (𝐾 β†Ύt (𝐴(,]𝐡))    &   (πœ‘ β†’ 𝐴 ≀ 𝐢)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐢(,]𝐡) ∈ 𝐽)
 
Theoremeliccelioc 43512 Membership in a closed interval and in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐢 ∈ (𝐴(,]𝐡) ↔ (𝐢 ∈ (𝐴[,]𝐡) ∧ 𝐢 β‰  𝐴)))
 
Theoremiooshift 43513* An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    β‡’   (πœ‘ β†’ ((𝐴 + 𝑇)(,)(𝐡 + 𝑇)) = {𝑀 ∈ β„‚ ∣ βˆƒπ‘§ ∈ (𝐴(,)𝐡)𝑀 = (𝑧 + 𝑇)})
 
Theoremiccintsng 43514 Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (𝐴 ≀ 𝐡 ∧ 𝐡 ≀ 𝐢)) β†’ ((𝐴[,]𝐡) ∩ (𝐡[,]𝐢)) = {𝐡})
 
Theoremicoiccdif 43515 Left-closed right-open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴[,)𝐡) = ((𝐴[,]𝐡) βˆ– {𝐡}))
 
Theoremicoopn 43516 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   πΎ = (topGenβ€˜ran (,))    &   π½ = (𝐾 β†Ύt (𝐴[,)𝐡))    &   (πœ‘ β†’ 𝐢 ≀ 𝐡)    β‡’   (πœ‘ β†’ (𝐴[,)𝐢) ∈ 𝐽)
 
Theoremicoub 43517 A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ ℝ* β†’ Β¬ 𝐡 ∈ (𝐴[,)𝐡))
 
Theoremeliccxrd 43518 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 ≀ 𝐢)    &   (πœ‘ β†’ 𝐢 ≀ 𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))
 
Theorempnfel0pnf 43519 +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
+∞ ∈ (0[,]+∞)
 
Theoremeliccnelico 43520 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴[,)𝐡))    β‡’   (πœ‘ β†’ 𝐢 = 𝐡)
 
Theoremeliccelicod 43521 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ 𝐢 β‰  𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴[,)𝐡))
 
Theoremge0xrre 43522 A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐴 β‰  +∞) β†’ 𝐴 ∈ ℝ)
 
Theoremge0lere 43523 A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ (0[,]+∞))    &   (πœ‘ β†’ 𝐡 ≀ 𝐴)    β‡’   (πœ‘ β†’ 𝐡 ∈ ℝ)
 
Theoremelicores 43524* Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ ran ([,) β†Ύ (ℝ Γ— ℝ)) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝐴 = (π‘₯[,)𝑦))
 
Theoreminficc 43525 The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝑆 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ 𝑆 β‰  βˆ…)    β‡’   (πœ‘ β†’ inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐡))
 
Theoremqinioo 43526 The rational numbers are dense in ℝ. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ ((β„š ∩ (𝐴(,)𝐡)) = βˆ… ↔ 𝐡 ≀ 𝐴))
 
Theoremlenelioc 43527 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ≀ 𝐴)    β‡’   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴(,]𝐡))
 
Theoremioonct 43528 A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   πΆ = (𝐴(,)𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐢 β‰Ό Ο‰)
 
Theoremxrgtnelicc 43529 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 < 𝐢)    β‡’   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴[,]𝐡))
 
Theoremiccdificc 43530 The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    β‡’   (πœ‘ β†’ ((𝐴[,]𝐢) βˆ– (𝐴[,]𝐡)) = (𝐡(,]𝐢))
 
Theoremiocnct 43531 A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   πΆ = (𝐴(,]𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐢 β‰Ό Ο‰)
 
Theoremiccnct 43532 A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   πΆ = (𝐴[,]𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐢 β‰Ό Ο‰)
 
Theoremiooiinicc 43533* A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ∩ 𝑛 ∈ β„• ((𝐴 βˆ’ (1 / 𝑛))(,)(𝐡 + (1 / 𝑛))) = (𝐴[,]𝐡))
 
Theoremiccgelbd 43534 An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    β‡’   (πœ‘ β†’ 𝐴 ≀ 𝐢)
 
Theoremiooltubd 43535 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))    β‡’   (πœ‘ β†’ 𝐢 < 𝐡)
 
Theoremicoltubd 43536 An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,)𝐡))    β‡’   (πœ‘ β†’ 𝐢 < 𝐡)
 
Theoremqelioo 43537* The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„š π‘₯ ∈ (𝐴(,)𝐡))
 
Theoremtgqioo2 43538* Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐴 ∈ 𝐽)    β‡’   (πœ‘ β†’ βˆƒπ‘ž(π‘ž βŠ† ((,) β€œ (β„š Γ— β„š)) ∧ 𝐴 = βˆͺ π‘ž))
 
Theoremiccleubd 43539 An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    β‡’   (πœ‘ β†’ 𝐢 ≀ 𝐡)
 
Theoremelioored 43540 A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ (𝐡(,)𝐢))    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremioogtlbd 43541 An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))    β‡’   (πœ‘ β†’ 𝐴 < 𝐢)
 
Theoremioofun 43542 (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Fun (,)
 
Theoremicomnfinre 43543 A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴))
 
Theoremsqrlearg 43544 The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ ((𝐴↑2) ≀ 𝐴 ↔ 𝐴 ∈ (0[,]1)))
 
Theoremressiocsup 43545 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.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   π‘† = sup(𝐴, ℝ*, < )    &   (πœ‘ β†’ 𝑆 ∈ 𝐴)    &   πΌ = (-∞(,]𝑆)    β‡’   (πœ‘ β†’ 𝐴 βŠ† 𝐼)
 
Theoremressioosup 43546 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.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   π‘† = sup(𝐴, ℝ*, < )    &   (πœ‘ β†’ Β¬ 𝑆 ∈ 𝐴)    &   πΌ = (-∞(,)𝑆)    β‡’   (πœ‘ β†’ 𝐴 βŠ† 𝐼)
 
Theoremiooiinioc 43547* A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ∩ 𝑛 ∈ β„• (𝐴(,)(𝐡 + (1 / 𝑛))) = (𝐴(,]𝐡))
 
Theoremressiooinf 43548 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.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   π‘† = inf(𝐴, ℝ*, < )    &   (πœ‘ β†’ Β¬ 𝑆 ∈ 𝐴)    &   πΌ = (𝑆(,)+∞)    β‡’   (πœ‘ β†’ 𝐴 βŠ† 𝐼)
 
Theoremicogelbd 43549 An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,)𝐡))    β‡’   (πœ‘ β†’ 𝐴 ≀ 𝐢)
 
Theoremiocleubd 43550 An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,]𝐡))    β‡’   (πœ‘ β†’ 𝐢 ≀ 𝐡)
 
Theoremuzinico 43551 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    β‡’   (πœ‘ β†’ 𝑍 = (β„€ ∩ (𝑀[,)+∞)))
 
Theorempreimaiocmnf 43552* Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (◑𝐹 β€œ (-∞(,]𝐡)) = {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) ≀ 𝐡})
 
Theoremuzinico2 43553 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.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ (β„€β‰₯β€˜π‘) = ((β„€β‰₯β€˜π‘€) ∩ (𝑁[,)+∞)))
 
Theoremuzinico3 43554 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    β‡’   (πœ‘ β†’ 𝑍 = (𝑍 ∩ (𝑀[,)+∞)))
 
Theoremicossico2 43555 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.)
(πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ≀ 𝐴)    β‡’   (πœ‘ β†’ (𝐴[,)𝐢) βŠ† (𝐡[,)𝐢))
 
Theoremdmico 43556 The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
dom [,) = (ℝ* Γ— ℝ*)
 
Theoremndmico 43557 The closed-below, open-above interval function's value is empty outside of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(Β¬ (𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴[,)𝐡) = βˆ…)
 
Theoremuzubioo 43558* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    β‡’   (πœ‘ β†’ βˆƒπ‘˜ ∈ (𝑋(,)+∞)π‘˜ ∈ 𝑍)
 
Theoremuzubico 43559* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    β‡’   (πœ‘ β†’ βˆƒπ‘˜ ∈ (𝑋[,)+∞)π‘˜ ∈ 𝑍)
 
Theoremuzubioo2 43560* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ βˆƒπ‘˜ ∈ (π‘₯(,)+∞)π‘˜ ∈ 𝑍)
 
Theoremuzubico2 43561* The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ βˆƒπ‘˜ ∈ (π‘₯[,)+∞)π‘˜ ∈ 𝑍)
 
Theoremiocgtlbd 43562 An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,]𝐡))    β‡’   (πœ‘ β†’ 𝐴 < 𝐢)
 
Theoremxrtgioo2 43563 The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(topGenβ€˜ran (,)) = ((ordTopβ€˜ ≀ ) β†Ύt ℝ)
 
Theoremtgioo4 43564 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(topGenβ€˜ran (,)) = ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)
 
21.38.5  Finite sums
 
Theoremfsummulc1f 43565* Closure of a finite sum of complex numbers 𝐴(π‘˜). A version of fsummulc1 15604 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (Ξ£π‘˜ ∈ 𝐴 𝐡 Β· 𝐢) = Ξ£π‘˜ ∈ 𝐴 (𝐡 Β· 𝐢))
 
Theoremfsumnncl 43566* Closure of a nonempty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝐴 β‰  βˆ…)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„•)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 ∈ β„•)
 
Theoremfsumge0cl 43567* The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ (0[,)+∞))    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 ∈ (0[,)+∞))
 
Theoremfsumf1of 43568* Re-index a finite sum using a bijection. Same as fsumf1o 15542, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
β„²π‘˜πœ‘    &   β„²π‘›πœ‘    &   (π‘˜ = 𝐺 β†’ 𝐡 = 𝐷)    &   (πœ‘ β†’ 𝐢 ∈ Fin)    &   (πœ‘ β†’ 𝐹:𝐢–1-1-onto→𝐴)    &   ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ (πΉβ€˜π‘›) = 𝐺)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 = Σ𝑛 ∈ 𝐢 𝐷)
 
Theoremfsumiunss 43569* Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 15640, but here 𝐴 and 𝐡 need not be finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ Disj π‘₯ ∈ 𝐴 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴 ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐷 ∈ Fin)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ βˆͺ π‘₯ ∈ 𝐴 (𝐡 ∩ 𝐷)𝐢 = Ξ£π‘₯ ∈ {π‘₯ ∈ 𝐴 ∣ (𝐡 ∩ 𝐷) β‰  βˆ…}Ξ£π‘˜ ∈ (𝐡 ∩ 𝐷)𝐢)
 
Theoremfsumreclf 43570* Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 ∈ ℝ)
 
Theoremfsumlessf 43571* A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 0 ≀ 𝐡)    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐢 𝐡 ≀ Ξ£π‘˜ ∈ 𝐴 𝐡)
 
Theoremfsumsupp0 43572* Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ (𝐹 supp 0)(πΉβ€˜π‘˜) = Ξ£π‘˜ ∈ 𝐴 (πΉβ€˜π‘˜))
 
Theoremfsumsermpt 43573* A finite sum expressed in terms of a partial sum of an infinite series. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐴 ∈ β„‚)    &   πΉ = (𝑛 ∈ 𝑍 ↦ Ξ£π‘˜ ∈ (𝑀...𝑛)𝐴)    &   πΊ = seq𝑀( + , (π‘˜ ∈ 𝑍 ↦ 𝐴))    β‡’   (πœ‘ β†’ 𝐹 = 𝐺)
 
21.38.6  Finite multiplication of numbers and finite multiplication of functions
 
Theoremfmul01 43574* Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑖𝐡    &   β„²π‘–πœ‘    &   π΄ = seq𝐿( Β· , 𝐡)    &   (πœ‘ β†’ 𝐿 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜πΏ))    &   (πœ‘ β†’ 𝐾 ∈ (𝐿...𝑀))    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ (π΅β€˜π‘–) ∈ ℝ)    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ 0 ≀ (π΅β€˜π‘–))    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ (π΅β€˜π‘–) ≀ 1)    β‡’   (πœ‘ β†’ (0 ≀ (π΄β€˜πΎ) ∧ (π΄β€˜πΎ) ≀ 1))
 
Theoremfmulcl 43575* If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑃 = (𝑓 ∈ π‘Œ, 𝑔 ∈ π‘Œ ↦ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))))    &   π‘‹ = (seq1(𝑃, π‘ˆ)β€˜π‘)    &   (πœ‘ β†’ 𝑁 ∈ (1...𝑀))    &   (πœ‘ β†’ π‘ˆ:(1...𝑀)βŸΆπ‘Œ)    &   ((πœ‘ ∧ 𝑓 ∈ π‘Œ ∧ 𝑔 ∈ π‘Œ) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ π‘Œ)    &   (πœ‘ β†’ 𝑇 ∈ V)    β‡’   (πœ‘ β†’ 𝑋 ∈ π‘Œ)
 
Theoremfmuldfeqlem1 43576* induction step for the proof of fmuldfeq 43577. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘“πœ‘    &   β„²π‘”πœ‘    &   β„²π‘‘π‘Œ    &   π‘ƒ = (𝑓 ∈ π‘Œ, 𝑔 ∈ π‘Œ ↦ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))))    &   πΉ = (𝑑 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((π‘ˆβ€˜π‘–)β€˜π‘‘)))    &   (πœ‘ β†’ 𝑇 ∈ V)    &   (πœ‘ β†’ π‘ˆ:(1...𝑀)βŸΆπ‘Œ)    &   ((πœ‘ ∧ 𝑓 ∈ π‘Œ ∧ 𝑔 ∈ π‘Œ) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ π‘Œ)    &   (πœ‘ β†’ 𝑁 ∈ (1...𝑀))    &   (πœ‘ β†’ (𝑁 + 1) ∈ (1...𝑀))    &   (πœ‘ β†’ ((seq1(𝑃, π‘ˆ)β€˜π‘)β€˜π‘‘) = (seq1( Β· , (πΉβ€˜π‘‘))β€˜π‘))    &   ((πœ‘ ∧ 𝑓 ∈ π‘Œ) β†’ 𝑓:π‘‡βŸΆβ„)    β‡’   ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ ((seq1(𝑃, π‘ˆ)β€˜(𝑁 + 1))β€˜π‘‘) = (seq1( Β· , (πΉβ€˜π‘‘))β€˜(𝑁 + 1)))
 
Theoremfmuldfeq 43577* X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘–πœ‘    &   β„²π‘‘π‘Œ    &   π‘ƒ = (𝑓 ∈ π‘Œ, 𝑔 ∈ π‘Œ ↦ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))))    &   π‘‹ = (seq1(𝑃, π‘ˆ)β€˜π‘€)    &   πΉ = (𝑑 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((π‘ˆβ€˜π‘–)β€˜π‘‘)))    &   π‘ = (𝑑 ∈ 𝑇 ↦ (seq1( Β· , (πΉβ€˜π‘‘))β€˜π‘€))    &   (πœ‘ β†’ 𝑇 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ π‘ˆ:(1...𝑀)βŸΆπ‘Œ)    &   ((πœ‘ ∧ 𝑓 ∈ π‘Œ) β†’ 𝑓:π‘‡βŸΆβ„)    &   ((πœ‘ ∧ 𝑓 ∈ π‘Œ ∧ 𝑔 ∈ π‘Œ) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ π‘Œ)    β‡’   ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ (π‘‹β€˜π‘‘) = (π‘β€˜π‘‘))
 
Theoremfmul01lt1lem1 43578* Given a finite multiplication of values betweeen 0 and 1, a value larger than its first element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑖𝐡    &   β„²π‘–πœ‘    &   π΄ = seq𝐿( Β· , 𝐡)    &   (πœ‘ β†’ 𝐿 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜πΏ))    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ (π΅β€˜π‘–) ∈ ℝ)    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ 0 ≀ (π΅β€˜π‘–))    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ (π΅β€˜π‘–) ≀ 1)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ (π΅β€˜πΏ) < 𝐸)    β‡’   (πœ‘ β†’ (π΄β€˜π‘€) < 𝐸)
 
Theoremfmul01lt1lem2 43579* Given a finite multiplication of values betweeen 0 and 1, a value 𝐸 larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑖𝐡    &   β„²π‘–πœ‘    &   π΄ = seq𝐿( Β· , 𝐡)    &   (πœ‘ β†’ 𝐿 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜πΏ))    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ (π΅β€˜π‘–) ∈ ℝ)    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ 0 ≀ (π΅β€˜π‘–))    &   ((πœ‘ ∧ 𝑖 ∈ (𝐿...𝑀)) β†’ (π΅β€˜π‘–) ≀ 1)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐽 ∈ (𝐿...𝑀))    &   (πœ‘ β†’ (π΅β€˜π½) < 𝐸)    β‡’   (πœ‘ β†’ (π΄β€˜π‘€) < 𝐸)
 
Theoremfmul01lt1 43580* Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑖𝐡    &   β„²π‘–πœ‘    &   β„²π‘—𝐴    &   π΄ = seq1( Β· , 𝐡)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝐡:(1...𝑀)βŸΆβ„)    &   ((πœ‘ ∧ 𝑖 ∈ (1...𝑀)) β†’ 0 ≀ (π΅β€˜π‘–))    &   ((πœ‘ ∧ 𝑖 ∈ (1...𝑀)) β†’ (π΅β€˜π‘–) ≀ 1)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ βˆƒπ‘— ∈ (1...𝑀)(π΅β€˜π‘—) < 𝐸)    β‡’   (πœ‘ β†’ (π΄β€˜π‘€) < 𝐸)
 
Theoremcncfmptss 43581* A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cn→𝐡))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐢 ↦ (πΉβ€˜π‘₯)) ∈ (𝐢–cn→𝐡))
 
Theoremrrpsscn 43582 The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
ℝ+ βŠ† β„‚
 
Theoremmulc1cncfg 43583* A version of mulc1cncf 24190 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
β„²π‘₯𝐹    &   β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ (𝐡 Β· (πΉβ€˜π‘₯))) ∈ (𝐴–cnβ†’β„‚))
 
Theoreminfrglb 43584* The infimum of a nonempty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
(((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 π‘₯ ≀ 𝑦) ∧ 𝐡 ∈ ℝ) β†’ (inf(𝐴, ℝ, < ) < 𝐡 ↔ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝐡))
 
Theoremexpcnfg 43585* If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 24211. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ ((πΉβ€˜π‘₯)↑𝑁)) ∈ (𝐴–cnβ†’β„‚))
 
Theoremprodeq2ad 43586* Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘˜ ∈ 𝐴 𝐢)
 
Theoremfprodsplit1 43587* Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ = 𝐢) β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = (𝐷 Β· βˆπ‘˜ ∈ (𝐴 βˆ– {𝐢})𝐡))
 
Theoremfprodexp 43588* Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 (𝐡↑𝑁) = (βˆπ‘˜ ∈ 𝐴 𝐡↑𝑁))
 
Theoremfprodabs2 43589* The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (absβ€˜βˆπ‘˜ ∈ 𝐴 𝐡) = βˆπ‘˜ ∈ 𝐴 (absβ€˜π΅))
 
Theoremfprod0 43590* A finite product with a zero term is zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   β„²π‘˜πΆ    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (π‘˜ = 𝐾 β†’ 𝐡 = 𝐢)    &   (πœ‘ β†’ 𝐾 ∈ 𝐴)    &   (πœ‘ β†’ 𝐢 = 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = 0)
 
Theoremmccllem 43591* * Induction step for mccl 43592. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    &   (πœ‘ β†’ 𝐷 ∈ (𝐴 βˆ– 𝐢))    &   (πœ‘ β†’ 𝐡 ∈ (β„•0 ↑m (𝐢 βˆͺ {𝐷})))    &   (πœ‘ β†’ βˆ€π‘ ∈ (β„•0 ↑m 𝐢)((!β€˜Ξ£π‘˜ ∈ 𝐢 (π‘β€˜π‘˜)) / βˆπ‘˜ ∈ 𝐢 (!β€˜(π‘β€˜π‘˜))) ∈ β„•)    β‡’   (πœ‘ β†’ ((!β€˜Ξ£π‘˜ ∈ (𝐢 βˆͺ {𝐷})(π΅β€˜π‘˜)) / βˆπ‘˜ ∈ (𝐢 βˆͺ {𝐷})(!β€˜(π΅β€˜π‘˜))) ∈ β„•)
 
Theoremmccl 43592* A multinomial coefficient, in its standard domain, is a positive integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜π΅    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 ∈ (β„•0 ↑m 𝐴))    β‡’   (πœ‘ β†’ ((!β€˜Ξ£π‘˜ ∈ 𝐴 (π΅β€˜π‘˜)) / βˆπ‘˜ ∈ 𝐴 (!β€˜(π΅β€˜π‘˜))) ∈ β„•)
 
Theoremfprodcnlem 43593* A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   πΎ = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ 𝑍 βŠ† 𝐴)    &   (πœ‘ β†’ π‘Š ∈ (𝐴 βˆ– 𝑍))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ βˆπ‘˜ ∈ 𝑍 𝐡) ∈ (𝐽 Cn 𝐾))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ βˆπ‘˜ ∈ (𝑍 βˆͺ {π‘Š})𝐡) ∈ (𝐽 Cn 𝐾))
 
Theoremfprodcn 43594* A finite product of functions to complex numbers from a common topological space is continuous. The class expression for 𝐡 normally contains free variables π‘˜ and π‘₯ to index it. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   πΎ = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ βˆπ‘˜ ∈ 𝐴 𝐡) ∈ (𝐽 Cn 𝐾))
 
21.38.7  Limits
 
Theoremclim1fr1 43595* A class of sequences of fractions that converge to 1. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐹 = (𝑛 ∈ β„• ↦ (((𝐴 Β· 𝑛) + 𝐡) / (𝐴 Β· 𝑛)))    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 β‰  0)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ 𝐹 ⇝ 1)
 
Theoremisumneg 43596* Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝑍 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) = 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ seq𝑀( + , 𝐹) ∈ dom ⇝ )    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝑍 -𝐴 = -Ξ£π‘˜ ∈ 𝑍 𝐴)
 
Theoremclimrec 43597* Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐺 ⇝ 𝐴)    &   (πœ‘ β†’ 𝐴 β‰  0)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) ∈ (β„‚ βˆ– {0}))    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (π»β€˜π‘˜) = (1 / (πΊβ€˜π‘˜)))    &   (πœ‘ β†’ 𝐻 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐻 ⇝ (1 / 𝐴))
 
Theoremclimmulf 43598* A version of climmul 15449 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
β„²π‘˜πœ‘    &   β„²π‘˜πΉ    &   β„²π‘˜πΊ    &   β„²π‘˜π»    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹 ⇝ 𝐴)    &   (πœ‘ β†’ 𝐻 ∈ 𝑋)    &   (πœ‘ β†’ 𝐺 ⇝ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) Β· (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ 𝐻 ⇝ (𝐴 Β· 𝐡))
 
Theoremclimexp 43599* The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
β„²π‘˜πœ‘    &   β„²π‘˜πΉ    &   β„²π‘˜π»    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„‚)    &   (πœ‘ β†’ 𝐹 ⇝ 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐻 ∈ 𝑉)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜)↑𝑁))    β‡’   (πœ‘ β†’ 𝐻 ⇝ (𝐴↑𝑁))
 
Theoremcliminf 43600* A bounded monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘˜ ∈ 𝑍 π‘₯ ≀ (πΉβ€˜π‘˜))    β‡’   (πœ‘ β†’ 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
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