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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | climlimsupcex 40001 | Counterexample for climlimsup 39992, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 9954 and its comment) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ ¬ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐹 = ∅ ⇒ ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹:𝑍⟶ℝ ∧ 𝐹 ∈ dom ⇝ ∧ ¬ 𝐹 ⇝ (lim sup‘𝐹))) | ||
Theorem | liminfcld 40002 | Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) | ||
Theorem | liminfresico 40003 | The inferior limit doesn't change when a function is restricted to an upperset of reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
Theorem | limsup10exlem 40004* | The range of the given function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) & ⊢ (𝜑 → 𝐾 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1}) | ||
Theorem | limsup10ex 40005 | 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 | ||
Theorem | liminf10ex 40006 | 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 | ||
Theorem | liminflelimsuplem 40007* | The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
Theorem | liminflelimsup 40008* | The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 40029 for a counterexample). The inequality can be strict, see liminfltlimsupex 40013. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
Theorem | limsupgtlem 40009* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) | ||
Theorem | limsupgt 40010* | 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.) |
⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) | ||
Theorem | liminfresre 40011 | The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹)) | ||
Theorem | liminfresicompt 40012* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) | ||
Theorem | liminfltlimsupex 40013 | 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) | ||
Theorem | liminfgelimsup 40014* | 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.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) | ||
Theorem | liminfvalxr 40015* | Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒(𝐹‘𝑥)))) | ||
Theorem | liminfresuz 40016 | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
Theorem | liminflelimsupuz 40017 | The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
Theorem | liminfvalxrmpt 40018* | Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
Theorem | liminfresuz2 40019 | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom 𝐹 ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
Theorem | liminfgelimsupuz 40020 | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) | ||
Theorem | liminfval4 40021* | Alternate definition of lim inf when the given function is eventually real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵))) | ||
Theorem | liminfval3 40022* | Alternate definition of lim inf when the given function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
Theorem | liminfequzmpt2 40023* | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑗𝐵 & ⊢ 𝐴 = (ℤ≥‘𝑀) & ⊢ 𝐵 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶))) | ||
Theorem | liminfvaluz 40024* | 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 inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) | ||
Theorem | liminf0 40025 | The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (lim inf‘∅) = +∞ | ||
Theorem | limsupval4 40026* | Alternate definition of lim inf when the given a function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
Theorem | liminfvaluz2 40027* | 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 inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝐵))) | ||
Theorem | liminfvaluz3 40028* | 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 40029 | A counterexample for liminflelimsup 40008, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (lim sup‘∅) < (lim inf‘∅) | ||
Theorem | limsupvaluz3 40030* | 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 40031* | 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 40032* | 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 40033 | 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 40034* | 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 40035* | 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 40036* | 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 40037* | 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 40038 | 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 40039 | 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 40017). (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 40040 | 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 40020). (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 40041 | 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 40020). (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 40042 | 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 40043 | 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)))) | ||
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 40074. | ||
Syntax | clsxlim 40044 | Extend class notation with convergence relation for limits in the extended real numbers. |
class ~~>* | ||
Definition | df-xlim 40045 | Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | ||
Theorem | xlimrel 40046 | The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Rel ~~>* | ||
Theorem | xlimres 40047 | 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 40048 | 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 40049* | 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 40050 | 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 39845). (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 40051* | 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 40052 | 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 40053* | 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 40054 | 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 40055* | Lemma for cnrefiisp 40056 (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 40056* | 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 40057* | 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 40058* | The "only if" part of the biconditional in xlimmnf 40067. (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 40059* | The "if" part of the biconditional in xlimpnf 40068. (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 40060* | 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 40061* | 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 40062* | The "only if" part of the biconditional in xlimmnf 40067. (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 40063* | The "if" part of the biconditional in xlimpnf 40068. (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 40064* | 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 40065* | Lemma for xlimclim2 40066. 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 40066 | Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 39845), 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 40067* | 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 40068* | 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 40069* | 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 40070* | 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 40071 | In this lemma for climxlim2 40072 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 40072 | 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 40073* | 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 40074* | 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 | coseq0 40075 | 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 40076 | 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 40077 | 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 40078 | The cosine of negative π is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (cos‘-π) = -1 | ||
Theorem | sinaover2ne0 40079 | 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 40080 | 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)) | ||
Theorem | mulcncff 40081 | 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 | subcncf 40082* | The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 − 𝐵)) ∈ (𝑋–cn→ℂ)) | ||
Theorem | cncfmptssg 40083* | A continuous complex function restricted to a subset is continuous, using "map to" notation. This theorem generalizes cncfmptss 39819 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 40084* | 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 40085* | 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 | addcncf 40086* | The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝑋–cn→ℂ)) | ||
Theorem | cncfshift 40087* | 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 40088 | sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (sin ↾ ℝ) ∈ (ℝ–cn→ℝ) | ||
Theorem | addccncf2 40089* | 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 40090 | The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ∅ ∈ ({∅} Cn {∅}) | ||
Theorem | fsumcncf 40091* | 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 40092* | 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 40093 | 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 40094* | 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 40095* | 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 40096* | Composition of continuous functions. A generalization of cncfmpt1f 22716 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 40097 | The addition 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 40098 | 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 40099* | A function is continuous if it's domain is the union of sets 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 40100* | 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)) |
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