P. 456 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45501-45600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iocleub 45501	An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵)
Theorem	eliccd 45502	Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
Theorem	eliccre 45503	A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ)
Theorem	eliooshift 45504	Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷))))
Theorem	eliocd 45505	Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))
Theorem	icoltub 45506	An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
Theorem	eliocre 45507	A member of a left-open right-closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ)
Theorem	iooltub 45508	An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵)
Theorem	ioontr 45509	The interior of an interval in the standard topology on ℝ is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
Theorem	snunioo1 45510	The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
Theorem	lbioc 45511	A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ¬ 𝐴 ∈ (𝐴(,]𝐵)
Theorem	ioomidp 45512	The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
Theorem	iccdifioo 45513	If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵})
Theorem	iccdifprioo 45514	An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵))
Theorem	ioossioobi 45515	Biconditional form of ioossioo 13402. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 < 𝐷)    ⇒   ⊢ (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)))
Theorem	iccshift 45516*	A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
Theorem	iccsuble 45517	An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵))    ⇒   ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴))
Theorem	iocopn 45518	A left-open right-closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝐽 = (𝐾 ↾t (𝐴(,]𝐵))    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐶(,]𝐵) ∈ 𝐽)
Theorem	eliccelioc 45519	Membership in a closed interval and in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)))
Theorem	iooshift 45520*	An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)})
Theorem	iccintsng 45521	Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵})
Theorem	icoiccdif 45522	Left-closed right-open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ((𝐴[,]𝐵) ∖ {𝐵}))
Theorem	icoopn 45523	A left-closed right-open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝐽 = (𝐾 ↾t (𝐴[,)𝐵))    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴[,)𝐶) ∈ 𝐽)
Theorem	icoub 45524	A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ ℝ* → ¬ 𝐵 ∈ (𝐴[,)𝐵))
Theorem	eliccxrd 45525	Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
Theorem	pnfel0pnf 45526	+∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ +∞ ∈ (0[,]+∞)
Theorem	eliccnelico 45527	An element of a closed interval that is not a member of the left-closed right-open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵))    ⇒   ⊢ (𝜑 → 𝐶 = 𝐵)
Theorem	eliccelicod 45528	A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝐶 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))
Theorem	ge0xrre 45529	A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)
Theorem	ge0lere 45530	A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ℝ)
Theorem	elicores 45531*	Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))
Theorem	inficc 45532	The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    ⇒   ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵))
Theorem	qinioo 45533	The rational numbers are dense in ℝ. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	lenelioc 45534	A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐴)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))
Theorem	ioonct 45535	A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ 𝐶 = (𝐴(,)𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ≼ ω)
Theorem	xrgtnelicc 45536	A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Theorem	iccdificc 45537	The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴[,]𝐶) ∖ (𝐴[,]𝐵)) = (𝐵(,]𝐶))
Theorem	iocnct 45538	A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ 𝐶 = (𝐴(,]𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ≼ ω)
Theorem	iccnct 45539	A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ 𝐶 = (𝐴[,]𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ≼ ω)
Theorem	iooiinicc 45540*	A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ ((𝐴 − (1 / 𝑛))(,)(𝐵 + (1 / 𝑛))) = (𝐴[,]𝐵))
Theorem	iccgelbd 45541	An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐶)
Theorem	iooltubd 45542	An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → 𝐶 < 𝐵)
Theorem	icoltubd 45543	An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))    ⇒   ⊢ (𝜑 → 𝐶 < 𝐵)
Theorem	qelioo 45544*	The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℚ 𝑥 ∈ (𝐴(,)𝐵))
Theorem	tgqioo2 45545*	Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    ⇒   ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞))
Theorem	iccleubd 45546	An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    ⇒   ⊢ (𝜑 → 𝐶 ≤ 𝐵)
Theorem	elioored 45547	A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ (𝐵(,)𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	ioogtlbd 45548	An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	ioofun 45549	(,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Fun (,)
Theorem	icomnfinre 45550	A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴))
Theorem	sqrlearg 45551	The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴↑2) ≤ 𝐴 ↔ 𝐴 ∈ (0[,]1)))
Theorem	ressiocsup 45552	If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (-∞(,]𝑆)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
Theorem	ressioosup 45553	If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    &   ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (-∞(,)𝑆)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
Theorem	iooiinioc 45554*	A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))
Theorem	ressiooinf 45555	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.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = inf(𝐴, ℝ*, < )    &   ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (𝑆(,)+∞)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
Theorem	iocleubd 45556	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))    ⇒   ⊢ (𝜑 → 𝐶 ≤ 𝐵)
Theorem	uzinico 45557	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞)))
Theorem	preimaiocmnf 45558*	Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (◡𝐹 “ (-∞(,]𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≤ 𝐵})
Theorem	uzinico2 45559	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.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞)))
Theorem	uzinico3 45560	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞)))
Theorem	dmico 45561	The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ dom [,) = (ℝ* × ℝ*)
Theorem	ndmico 45562	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) = ∅)
Theorem	uzubioo 45563*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍)
Theorem	uzubico 45564*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘 ∈ 𝑍)
Theorem	uzubioo2 45565*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘 ∈ 𝑍)
Theorem	uzubico2 45566*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘 ∈ 𝑍)
Theorem	iocgtlbd 45567	An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	xrtgioo2 45568	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.)
⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ)
21.43.5  Finite sums
Theorem	fsummulc1f 45569*	Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 15751 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶))
Theorem	fsumnncl 45570*	Closure of a nonempty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ)
Theorem	fsumge0cl 45571*	The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞))
Theorem	fsumf1of 45572*	Re-index a finite sum using a bijection. Same as fsumf1o 15689, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷)
Theorem	fsumiunss 45573*	Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 15787, but here 𝐴 and 𝐵 need not be finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐷)𝐶 = Σ𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐵 ∩ 𝐷) ≠ ∅}Σ𝑘 ∈ (𝐵 ∩ 𝐷)𝐶)
Theorem	fsumreclf 45574*	Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ)
Theorem	fsumlessf 45575*	A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵)
Theorem	fsumsupp0 45576*	Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝐹 supp 0)(𝐹‘𝑘) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘))
Theorem	fsumsermpt 45577*	A finite sum expressed in terms of a partial sum of an infinite series. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴)    &   ⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
21.43.6  Finite multiplication of numbers and finite multiplication of functions
Theorem	fmul01 45578*	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.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ 𝐴 = seq𝐿( · , 𝐵)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐿...𝑀))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)    ⇒   ⊢ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1))
Theorem	fmulcl 45579*	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 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))    &   ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)    &   ⊢ (𝜑 → 𝑇 ∈ V)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑌)
Theorem	fmuldfeqlem1 45580*	induction step for the proof of fmuldfeq 45581. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑓𝜑    &   ⊢ Ⅎ𝑔𝜑    &   ⊢ Ⅎ𝑡𝑌    &   ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))    &   ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))    &   ⊢ (𝜑 → 𝑇 ∈ V)    &   ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)    &   ⊢ (𝜑 → 𝑁 ∈ (1...𝑀))    &   ⊢ (𝜑 → (𝑁 + 1) ∈ (1...𝑀))    &   ⊢ (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑁))    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)    ⇒   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1)))
Theorem	fmuldfeq 45581*	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.)
⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑡𝑌    &   ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))    &   ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)    &   ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))    &   ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))    &   ⊢ (𝜑 → 𝑇 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
Theorem	fmul01lt1lem1 45582*	Given a finite multiplication of values between 0 and 1, a value larger than its first element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ 𝐴 = seq𝐿( · , 𝐵)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐵‘𝐿) < 𝐸)    ⇒   ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)
Theorem	fmul01lt1lem2 45583*	Given a finite multiplication of values between 0 and 1, a value 𝐸 larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ 𝐴 = seq𝐿( · , 𝐵)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀))    &   ⊢ (𝜑 → (𝐵‘𝐽) < 𝐸)    ⇒   ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)
Theorem	fmul01lt1 45584*	Given a finite multiplication of values between 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ 𝐴 = seq1( · , 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → ∃𝑗 ∈ (1...𝑀)(𝐵‘𝑗) < 𝐸)    ⇒   ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)
Theorem	cncfmptss 45585*	A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))
Theorem	rrpsscn 45586	The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ℝ+ ⊆ ℂ
Theorem	mulc1cncfg 45587*	A version of mulc1cncf 24798 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ))
Theorem	infrglb 45588*	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 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ, < ) < 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝐵))
Theorem	expcnfg 45589*	If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 24820. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) ∈ (𝐴–cn→ℂ))
Theorem	prodeq2ad 45590*	Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)
Theorem	fprodsplit1 45591*	Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
Theorem	fprodexp 45592*	Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁))
Theorem	fprodabs2 45593*	The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))
Theorem	fprod0 45594*	A finite product with a zero term is zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐶    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 = 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0)
Theorem	mccllem 45595*	* Induction step for mccl 45596. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m (𝐶 ∪ {𝐷})))    &   ⊢ (𝜑 → ∀𝑏 ∈ (ℕ0 ↑m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ)    ⇒   ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ)
Theorem	mccl 45596*	A multinomial coefficient, in its standard domain, is a positive integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝐵    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m 𝐴))    ⇒   ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐴 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐴 (!‘(𝐵‘𝑘))) ∈ ℕ)
Theorem	fprodcnlem 45597*	A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.) Avoid ax-mulf 11148. (Revised by GG, 19-Apr-2025.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))
Theorem	fprodcn 45598*	A finite product of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
21.43.7  Limits
Theorem	clim1fr1 45599*	A class of sequences of fractions that converge to 1. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴 · 𝑛) + 𝐵) / (𝐴 · 𝑛)))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 1)
Theorem	isumneg 45600*	Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 -𝐴 = -Σ𝑘 ∈ 𝑍 𝐴)