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Theorem List for Metamath Proof Explorer - 40301-40400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.32.7.1  Inferior limit (lim inf)
 
Syntaxclsi 40301 Extend class notation to include the liminf function. (actually, it makes sense for any extended real function defined on a subset of RR which is not upper-bounded)
class lim inf
 
Definitiondf-liminf 40302* Define the inferior limit of a sequence of extended real numbers. (Contributed by GS, 2-Jan-2022.)
lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
 
Theoremlimsuplt2 40303* The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴))
 
Theoremliminfgord 40304 Ordering property of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → inf(((𝐹 “ (𝐴[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝐵[,)+∞)) ∩ ℝ*), ℝ*, < ))
 
Theoremlimsupvald 40305* The superior limit of a sequence 𝐹 of extended real numbers is the infimum of the set of suprema of all restrictions of 𝐹 to an upperset of reals . (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
 
Theoremlimsupresicompt 40306* The superior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)       (𝜑 → (lim sup‘(𝑥𝐴𝐵)) = (lim sup‘(𝑥 ∈ (𝐴𝑍) ↦ 𝐵)))
 
Theoremlimsupcli 40307 Closure of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹𝑉       (lim sup‘𝐹) ∈ ℝ*
 
Theoremliminfgf 40308 Closure of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       𝐺:ℝ⟶ℝ*
 
Theoremliminfval 40309* The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
 
Theoremclimlimsup 40310 A sequence of real numbers converges if and only if it converges to its superior limit. The first hypothesis is needed (see climlimsupcex 40319 for a counterexample) (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹)))
 
Theoremlimsupge 40311* The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
 
Theoremliminfgval 40312* Value of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝑀 ∈ ℝ → (𝐺𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < ))
 
Theoremliminfcl 40313 Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐹𝑉 → (lim inf‘𝐹) ∈ ℝ*)
 
Theoremliminfvald 40314* The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
 
Theoremliminfval5 40315* The inferior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))       (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
 
Theoremlimsupresxr 40316 The superior limit of a function only depends on the restriction of that function to the preimage of the set of extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → Fun 𝐹)    &   𝐴 = (𝐹 “ ℝ*)       (𝜑 → (lim sup‘(𝐹𝐴)) = (lim sup‘𝐹))
 
Theoremliminfresxr 40317 The inferior limit of a function only depends on the preimage of the extended real part. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → Fun 𝐹)    &   𝐴 = (𝐹 “ ℝ*)       (𝜑 → (lim inf‘(𝐹𝐴)) = (lim inf‘𝐹))
 
Theoremliminfval2 40318* The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   (𝜑𝐹𝑉)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (lim inf‘𝐹) = sup((𝐺𝐴), ℝ*, < ))
 
Theoremclimlimsupcex 40319 Counterexample for climlimsup 40310, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 9992 and its comment) (Contributed by Glauco Siliprandi, 2-Jan-2022.)
¬ 𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)    &   𝐹 = ∅       ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹:𝑍⟶ℝ ∧ 𝐹 ∈ dom ⇝ ∧ ¬ 𝐹 ⇝ (lim sup‘𝐹)))
 
Theoremliminfcld 40320 Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)       (𝜑 → (lim inf‘𝐹) ∈ ℝ*)
 
Theoremliminfresico 40321 The inferior limit doesn't change when a function is restricted to an upperset of reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)    &   (𝜑𝐹𝑉)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))
 
Theoremlimsup10exlem 40322* The range of the given function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))    &   (𝜑𝐾 ∈ ℝ)       (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1})
 
Theoremlimsup10ex 40323 The superior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))       (lim sup‘𝐹) = 1
 
Theoremliminf10ex 40324 The inferior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))       (lim inf‘𝐹) = 0
 
Theoremliminflelimsuplem 40325* The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
 
Theoremliminflelimsup 40326* The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 40347 for a counterexample). The inequality can be strict, see liminfltlimsupex 40331. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
 
Theoremlimsupgtlem 40327* For any positive real, the superior limit of F is larger than any of its values at large enough arguments, up to that positive real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
 
Theoremlimsupgt 40328* Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
 
Theoremliminfresre 40329 The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)       (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹))
 
Theoremliminfresicompt 40330* The inferior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)    &   (𝜑𝐴𝑉)       (𝜑 → (lim inf‘(𝑥 ∈ (𝐴𝑍) ↦ 𝐵)) = (lim inf‘(𝑥𝐴𝐵)))
 
Theoremliminfltlimsupex 40331 An example where the lim inf is strictly smaller than the lim sup. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))       (lim inf‘𝐹) < (lim sup‘𝐹)
 
Theoremliminfgelimsup 40332* The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
 
Theoremliminfvalxr 40333* Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝐹    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒(𝐹𝑥))))
 
Theoremliminfresuz 40334 If the real part of the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))
 
Theoremliminflelimsupuz 40335 The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
 
Theoremliminfvalxrmpt 40336* Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒𝐵)))
 
Theoremliminfresuz2 40337 If the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom 𝐹 ⊆ ℤ)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))
 
Theoremliminfgelimsupuz 40338 The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
 
Theoremliminfval4 40339* Alternate definition of lim inf when the given function is eventually real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝐵)))
 
Theoremliminfval3 40340* Alternate definition of lim inf when the given function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒𝐵)))
 
Theoremliminfequzmpt2 40341* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑗𝜑    &   𝑗𝐴    &   𝑗𝐵    &   𝐴 = (ℤ𝑀)    &   𝐵 = (ℤ𝑁)    &   (𝜑𝐾𝐴)    &   (𝜑𝐾𝐵)    &   ((𝜑𝑗 ∈ (ℤ𝐾)) → 𝐶𝑉)       (𝜑 → (lim inf‘(𝑗𝐴𝐶)) = (lim inf‘(𝑗𝐵𝐶)))
 
Theoremliminfvaluz 40342* 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.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑘𝑍𝐵)) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝑒𝐵)))
 
Theoremliminf0 40343 The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim inf‘∅) = +∞
 
Theoremlimsupval4 40344* Alternate definition of lim inf when the given a function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*)       (𝜑 → (lim sup‘(𝑥𝐴𝐵)) = -𝑒(lim inf‘(𝑥𝐴 ↦ -𝑒𝐵)))
 
Theoremliminfvaluz2 40345* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)       (𝜑 → (lim inf‘(𝑘𝑍𝐵)) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝐵)))
 
Theoremliminfvaluz3 40346* 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.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝑒(𝐹𝑘))))
 
Theoremliminflelimsupcex 40347 A counterexample for liminflelimsup 40326, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim sup‘∅) < (lim inf‘∅)
 
Theoremlimsupvaluz3 40348* 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.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)       (𝜑 → (lim sup‘(𝑘𝑍𝐵)) = -𝑒(lim inf‘(𝑘𝑍 ↦ -𝑒𝐵)))
 
Theoremliminfvaluz4 40349* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘𝑍 ↦ -(𝐹𝑘))))
 
Theoremlimsupvaluz4 40350* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)       (𝜑 → (lim sup‘(𝑘𝑍𝐵)) = -𝑒(lim inf‘(𝑘𝑍 ↦ -𝐵)))
 
Theoremclimliminflimsupd 40351 If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐹 ∈ dom ⇝ )       (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹))
 
Theoremliminfreuzlem 40352* 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.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
 
Theoremliminfreuz 40353* 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.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
 
Theoremliminfltlem 40354* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(lim inf‘𝐹) < ((𝐹𝑘) + 𝑋))
 
Theoremliminflt 40355* 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.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(lim inf‘𝐹) < ((𝐹𝑘) + 𝑋))
 
Theoremclimliminf 40356 A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹)))
 
Theoremliminflimsupclim 40357 A sequence of real numbers converges if its inferior limit is real, and it is greater or equal to the superior limit (in such a case, they are actually equal, see liminflelimsupuz 40335). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹))       (𝜑𝐹 ∈ dom ⇝ )
 
Theoremclimliminflimsup 40358 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 40338). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
 
Theoremclimliminflimsup2 40359 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 40338). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
 
Theoremclimliminflimsup3 40360 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))
 
Theoremclimliminflimsup4 40361 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))
 
20.32.7.2  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 40392.

 
Syntaxclsxlim 40362 Extend class notation with convergence relation for limits in the extended real numbers.
class ~~>*
 
Definitiondf-xlim 40363 Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
~~>* = (⇝𝑡‘(ordTop‘ ≤ ))
 
Theoremxlimrel 40364 The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Rel ~~>*
 
Theoremxlimres 40365 A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐹 ∈ (ℝ*pm ℂ))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ𝑀))~~>*𝐴))
 
Theoremxlimcl 40366 The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝐹~~>*𝐴𝐴 ∈ ℝ*)
 
Theoremrexlimddv2 40367* Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremxlimclim 40368 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 40163). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))
 
Theoremxlimconst 40369* A constant sequence converges to its value, w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹 Fn 𝑍)    &   (𝜑𝐴 ∈ ℝ*)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremclimxlim 40370 A converging sequence in the reals is a converging sequence in the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremxlimbr 40371* 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.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   𝐽 = (ordTop‘ ≤ )       (𝜑 → (𝐹~~>*𝑃 ↔ (𝑃 ∈ ℝ* ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
 
Theoremfuzxrpmcn 40372 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 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑𝐹 ∈ (ℝ*pm ℂ))
 
Theoremcnrefiisplem 40373* Lemma for cnrefiisp 40374 (some local definitions are used). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   𝐶 = (ℝ ∪ 𝐵)    &   𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})    &   𝑋 = inf(𝐷, ℝ*, < )       (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
 
Theoremcnrefiisp 40374* 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   𝐶 = (ℝ ∪ 𝐵)       (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
 
Theoremxlimxrre 40375* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐹~~>*𝐴)       (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
 
Theoremxlimmnfvlem1 40376* Lemma for xlimmnfv 40378: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*-∞)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑋)
 
Theoremxlimmnfvlem2 40377* Lemma for xlimmnf 40385: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) < 𝑥)       (𝜑𝐹~~>*-∞)
 
Theoremxlimmnfv 40378* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥))
 
Theoremxlimconst2 40379* 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.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝑁𝑍)    &   (𝜑𝐴 ∈ ℝ*)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐹𝑘) = 𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremxlimpnfvlem1 40380* Lemma for xlimpnfv 40382: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*+∞)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑋 ≤ (𝐹𝑘))
 
Theoremxlimpnfvlem2 40381* Lemma for xlimpnfv 40382: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 < (𝐹𝑘))       (𝜑𝐹~~>*+∞)
 
Theoremxlimpnfv 40382* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))
 
Theoremxlimclim2lem 40383* Lemma for xlimclim2 40384. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))
 
Theoremxlimclim2 40384 Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 40163), 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))
 
Theoremxlimmnf 40385* 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.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥))
 
Theoremxlimpnf 40386* 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.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))
 
Theoremxlimmnfmpt 40387* 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.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)    &   𝐹 = (𝑘𝑍𝐵)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝐵𝑥))
 
Theoremxlimpnfmpt 40388* 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.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)    &   𝐹 = (𝑘𝑍𝐵)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥𝐵))
 
Theoremclimxlim2lem 40389 In this lemma for climxlim2 40390 there is the additional assumption that the converging function is complex valued on the whole domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹:𝑍⟶ℂ)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremclimxlim2 40390 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremdfxlim2v 40391* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))))
 
Theoremdfxlim2 40392* 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.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))))
 
20.32.8  Trigonometry
 
Theoremcoseq0 40393 A complex number whose cosine is zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℂ → ((cos‘𝐴) = 0 ↔ ((𝐴 / π) + (1 / 2)) ∈ ℤ))
 
Theoremsinmulcos 40394 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‘(𝐴𝐵))) / 2))
 
Theoremcoskpi2 40395 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))
 
Theoremcosnegpi 40396 The cosine of negative π is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(cos‘-π) = -1
 
Theoremsinaover2ne0 40397 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)
 
Theoremcosknegpi 40398 The cosine of an integer multiple of negative π is either 1 ore negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ ℤ → (cos‘(𝐾 · -π)) = if(2 ∥ 𝐾, 1, -1))
 
20.32.9  Continuous Functions
 
Theoremmulcncff 40399 The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (𝑋cn→ℂ))    &   (𝜑𝐺 ∈ (𝑋cn→ℂ))       (𝜑 → (𝐹𝑓 · 𝐺) ∈ (𝑋cn→ℂ))
 
Theoremsubcncf 40400* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐵)) ∈ (𝑋cn→ℂ))
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