P. 459 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45801-45900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	liminfvaluz3 45801*	Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘))))
Theorem	liminflelimsupcex 45802	A counterexample for liminflelimsup 45781, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (lim sup‘∅) < (lim inf‘∅)
Theorem	limsupvaluz3 45803*	Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵)))
Theorem	liminfvaluz4 45804*	Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))))
Theorem	limsupvaluz4 45805*	Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝐵)))
Theorem	climliminflimsupd 45806	If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹))
Theorem	liminfreuzlem 45807*	Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))
Theorem	liminfreuz 45808*	Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))
Theorem	liminfltlem 45809*	Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))
Theorem	liminflt 45810*	Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))
Theorem	climliminf 45811	A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹)))
Theorem	liminflimsupclim 45812	A sequence of real numbers converges if its inferior limit is real, and it is greater than or equal to the superior limit (in such a case, they are actually equal, see liminflelimsupuz 45790). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
Theorem	climliminflimsup 45813	A sequence of real numbers converges if and only if its inferior limit is real and it is greater than or equal to its superior limit (in such a case, they are actually equal, see liminfgelimsupuz 45793). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
Theorem	climliminflimsup2 45814	A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 45793). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
Theorem	climliminflimsup3 45815	A sequence of real numbers converges if and only if its inferior limit is real and equal to its superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))
Theorem	climliminflimsup4 45816	A sequence of real numbers converges if and only if its superior limit is real and equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))
Theorem	limsupub2 45817*	A extended real valued function, with limsup that is not +∞, is eventually less than +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    &   ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))
Theorem	limsupubuz2 45818*	A sequence with values in the extended reals, and with limsup that is not +∞, is eventually less than +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) < +∞)
Theorem	xlimpnfxnegmnf 45819*	A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ Ⅎ𝑗𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥))
Theorem	liminflbuz2 45820*	A sequence with values in the extended reals, and with liminf that is not -∞, is eventually greater than -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → (lim inf‘𝐹) ≠ -∞)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-∞ < (𝐹‘𝑗))
Theorem	liminfpnfuz 45821*	The inferior limit of a function is +∞ if and only if every real number is the lower bound of the restriction of the function to a set of upper integers. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim inf‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
Theorem	liminflimsupxrre 45822*	A sequence with values in the extended reals, and with real liminf and limsup, is eventually real. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)    &   ⊢ (𝜑 → (lim inf‘𝐹) ≠ -∞)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶ℝ)
21.43.7.2  Limits for sequences of extended real numbers Textbooks generally use a single symbol to denote the limit of a sequence of real numbers. But then, three distinct definitions are usually given: one for the case of convergence to a real number, one for the case of limit to +∞ and one for the case of limit to -∞. It turns out that these three definitions can be expressed as the limit w.r.t. to the standard topology on the extended reals. In this section, a relation ~~>* is defined that captures all three definitions (and can be applied to sequences of extended reals, also), see dfxlim2 45853.
Syntax	clsxlim 45823	Extend class notation with convergence relation for limits in the extended real numbers.
class ~~>*
Definition	df-xlim 45824	Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ ))
Theorem	xlimrel 45825	The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Rel ~~>*
Theorem	xlimres 45826	A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴))
Theorem	xlimcl 45827	The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝐹~~>*𝐴 → 𝐴 ∈ ℝ*)
Theorem	rexlimddv2 45828*	Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	xlimclim 45829	Given a sequence of reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals, if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals (see climreeq 45618). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	xlimconst 45830*	A constant sequence converges to its value, w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 Fn 𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐹~~>*𝐴)
Theorem	climxlim 45831	A converging sequence in the reals is a converging sequence in the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    ⇒   ⊢ (𝜑 → 𝐹~~>*𝐴)
Theorem	xlimbr 45832*	Express the binary relation "sequence 𝐹 converges to point 𝑃 " w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ 𝐽 = (ordTop‘ ≤ )    ⇒   ⊢ (𝜑 → (𝐹~~>*𝑃 ↔ (𝑃 ∈ ℝ* ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))))
Theorem	fuzxrpmcn 45833	A function mapping from an upper set of integers to the extended reals is a partial map on the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ))
Theorem	cnrefiisplem 45834*	Lemma for cnrefiisp 45835 (some local definitions are used). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ 𝐶 = (ℝ ∪ 𝐵)    &   ⊢ 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})    &   ⊢ 𝑋 = inf(𝐷, ℝ*, < )    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
Theorem	cnrefiisp 45835*	A non-real, complex number is an isolated point w.r.t. the union of the reals with any finite set (the extended reals is an example of such a union). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ 𝐶 = (ℝ ∪ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
Theorem	xlimxrre 45836*	If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹~~>*𝐴)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
Theorem	xlimmnfvlem1 45837*	Lemma for xlimmnfv 45839: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐹~~>*-∞)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑋)
Theorem	xlimmnfvlem2 45838*	Lemma for xlimmnf 45846: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) < 𝑥)    ⇒   ⊢ (𝜑 → 𝐹~~>*-∞)
Theorem	xlimmnfv 45839*	A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥))
Theorem	xlimconst2 45840*	A sequence that eventually becomes constant, converges to its constant value (w.r.t. the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐹~~>*𝐴)
Theorem	xlimpnfvlem1 45841*	Lemma for xlimpnfv 45843: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐹~~>*+∞)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑋 ≤ (𝐹‘𝑘))
Theorem	xlimpnfvlem2 45842*	Lemma for xlimpnfv 45843: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 𝐹~~>*+∞)
Theorem	xlimpnfv 45843*	A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)))
Theorem	xlimclim2lem 45844*	Lemma for xlimclim2 45845. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	xlimclim2 45845	Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 45618), if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals. In order for the first part of the statement to even make sense, the sequence will of course eventually become (and stay) real: showing this, is the key step of the proof. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	xlimmnf 45846*	A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥))
Theorem	xlimpnf 45847*	A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)))
Theorem	xlimmnfmpt 45848*	A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*)    &   ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥))
Theorem	xlimpnfmpt 45849*	A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*)    &   ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵))
Theorem	climxlim2lem 45850	In this lemma for climxlim2 45851 there is the additional assumption that the converging function is complex-valued on the whole domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    ⇒   ⊢ (𝜑 → 𝐹~~>*𝐴)
Theorem	climxlim2 45851	A sequence of extended reals, converging w.r.t. the standard topology on the complex numbers is a converging sequence w.r.t. the standard topology on the extended reals. This is non-trivial, because +∞ and -∞ could, in principle, be complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    ⇒   ⊢ (𝜑 → 𝐹~~>*𝐴)
Theorem	dfxlim2v 45852*	An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)))))
Theorem	dfxlim2 45853*	An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)))))
Theorem	climresd 45854	A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑀)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	climresdm 45855	A real function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ))
Theorem	dmclimxlim 45856	A real valued sequence that converges w.r.t. the topology on the complex numbers, converges w.r.t. the topology on the extended reals (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ~~>*)
Theorem	xlimmnflimsup2 45857	A sequence of extended reals converges to -∞ if and only if its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
Theorem	xlimuni 45858	An infinite sequence converges to at most one limit (w.r.t. to the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝐹~~>*𝐴)    &   ⊢ (𝜑 → 𝐹~~>*𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	xlimclimdm 45859	A sequence of extended reals that converges to a real w.r.t. the standard topology on the extended reals, also converges w.r.t. to the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐹~~>*𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
Theorem	xlimfun 45860	The convergence relation on the extended reals is a function. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ Fun ~~>*
Theorem	xlimmnflimsup 45861	If a sequence of extended reals converges to -∞ then its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐹~~>*-∞)    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = -∞)
Theorem	xlimdm 45862	Two ways to express that a function has a limit. (The expression (~~>*‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))
Theorem	xlimpnfxnegmnf2 45863*	A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞))
Theorem	xlimresdm 45864	A function converges in the extended reals iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*))
Theorem	xlimpnfliminf 45865	If a sequence of extended reals converges to +∞ then its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    &   ⊢ (𝜑 → 𝐹~~>*+∞)    ⇒   ⊢ (𝜑 → (lim inf‘𝐹) = +∞)
Theorem	xlimpnfliminf2 45866	A sequence of extended reals converges to +∞ if and only if its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
Theorem	xlimliminflimsup 45867	A sequence of extended reals converges if and only if its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
Theorem	xlimlimsupleliminf 45868	A sequence of extended reals converges if and only if its superior limit is smaller than or equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Dec-2023.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
21.43.8  Trigonometry
Theorem	coseq0 45869	A complex number whose cosine is zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) = 0 ↔ ((𝐴 / π) + (1 / 2)) ∈ ℤ))
Theorem	sinmulcos 45870	Multiplication formula for sine and cosine. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (cos‘𝐵)) = (((sin‘(𝐴 + 𝐵)) + (sin‘(𝐴 − 𝐵))) / 2))
Theorem	coskpi2 45871	The cosine of an integer multiple of negative π is either 1 or negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · π)) = if(2 ∥ 𝐾, 1, -1))
Theorem	cosnegpi 45872	The cosine of negative π is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (cos‘-π) = -1
Theorem	sinaover2ne0 45873	If 𝐴 in (0, 2π) then sin(𝐴 / 2) is not 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ (0(,)(2 · π)) → (sin‘(𝐴 / 2)) ≠ 0)
Theorem	cosknegpi 45874	The cosine of an integer multiple of negative π is either 1 or negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · -π)) = if(2 ∥ 𝐾, 1, -1))
21.43.9  Continuous Functions
Theorem	mulcncff 45875	The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ (𝑋–cn→ℂ))
Theorem	cncfmptssg 45876*	A continuous complex function restricted to a subset is continuous, using maps-to notation. This theorem generalizes cncfmptss 45592 because it allows to establish a subset for the codomain also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐸 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷))
Theorem	constcncfg 45877*	A constant function is a continuous function on ℂ. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐶 ⊆ ℂ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶))
Theorem	idcncfg 45878*	The identity function is a continuous function on ℂ. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ ℂ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→𝐵))
Theorem	cncfshift 45879*	A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    &   ⊢ 𝐵 = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 + 𝑇)}    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))    &   ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐹‘(𝑥 − 𝑇)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐵–cn→ℂ))
Theorem	resincncf 45880	sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (sin ↾ ℝ) ∈ (ℝ–cn→ℝ)
Theorem	addccncf2 45881*	Adding a constant is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝑥))    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	0cnf 45882	The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ∅ ∈ ({∅} Cn {∅})
Theorem	fsumcncf 45883*	The finite sum of continuous complex function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑋 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝑋–cn→ℂ))
Theorem	cncfperiod 45884*	A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ 𝐵 = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 + 𝑇)}    &   ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ⊆ dom 𝐹)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (𝐴–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (𝐵–cn→ℂ))
Theorem	subcncff 45885	The subtraction of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ (𝑋–cn→ℂ))
Theorem	negcncfg 45886*	The opposite of a continuous function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ (𝐴–cn→ℂ))
Theorem	cnfdmsn 45887*	A function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵}))
Theorem	cncfcompt 45888*	Composition of continuous functions. A generalization of cncfmpt1f 24814 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→𝐷))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷))
Theorem	addcncff 45889	The sum of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (𝑋–cn→ℂ))
Theorem	ioccncflimc 45890	Limit at the upper bound of a continuous function defined on a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵))
Theorem	cncfuni 45891*	A complex function on a subset of the complex numbers is continuous if its domain is the union of relatively open subsets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))    &   ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹 ↾ 𝑏) ∈ ((𝐴 ∩ 𝑏)–cn→ℂ))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	icccncfext 45892*	A continuous function on a closed interval can be extended to a continuous function on the whole real line. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴[,]𝐵), (𝐹‘𝑥), if(𝑥 < 𝐴, (𝐹‘𝐴), (𝐹‘𝐵))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ Top)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t (𝐴[,]𝐵)) Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝐺 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)) ∧ (𝐺 ↾ (𝐴[,]𝐵)) = 𝐹))
Theorem	cncficcgt0 45893*	A the absolute value of a continuous function on a closed interval, that is never 0, has a strictly positive lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ+ ∀𝑥 ∈ (𝐴[,]𝐵)𝑦 ≤ (abs‘𝐶))
Theorem	icocncflimc 45894	Limit at the lower bound, of a continuous function defined on a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴))
Theorem	cncfdmsn 45895*	A complex function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ ({𝐴}–cn→{𝐵}))
Theorem	divcncff 45896	The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→(ℂ ∖ {0})))    ⇒   ⊢ (𝜑 → (𝐹 ∘f / 𝐺) ∈ (𝑋–cn→ℂ))
Theorem	cncfshiftioo 45897*	A periodic continuous function stays continuous if the domain is an open interval that is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐶 = (𝐴(,)𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ 𝐷 = ((𝐴 + 𝑇)(,)(𝐵 + 𝑇))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ))    &   ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ))
Theorem	cncfiooicclem1 45898*	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 can be complex-valued. This lemma assumes 𝐴 < 𝐵, the invoking theorem drops this assumption. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Theorem	cncfiooicc 45899*	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 can be complex-valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Theorem	cncfiooiccre 45900*	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 is assumed to be real-valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))