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Type | Label | Description |
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Statement | ||
Theorem | liminflimsupclim 44201 | 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 44179). (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 44202 | 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 44182). (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 44203 | 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 44182). (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 44204 | 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 44205 | 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 44206* | 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 44207* | 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 44208* | 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 44209* | 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 44210* | 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 44211* | 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) | ||
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 44242. | ||
Syntax | clsxlim 44212 | Extend class notation with convergence relation for limits in the extended real numbers. |
class ~~>* | ||
Definition | df-xlim 44213 | Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | ||
Theorem | xlimrel 44214 | The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Rel ~~>* | ||
Theorem | xlimres 44215 | 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 44216 | 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 44217* | 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 44218 | 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 44007). (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 44219* | 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 44220 | 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 44221* | 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 44222 | 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 44223* | Lemma for cnrefiisp 44224 (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 44224* | 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 44225* | 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 44226* | Lemma for xlimmnfv 44228: 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 44227* | Lemma for xlimmnf 44235: 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 44228* | 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 44229* | 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 44230* | Lemma for xlimpnfv 44232: 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 44231* | Lemma for xlimpnfv 44232: 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 44232* | 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 44233* | Lemma for xlimclim2 44234. 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 44234 | Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 44007), 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 44235* | 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 44236* | 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 44237* | 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 44238* | 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 44239 | In this lemma for climxlim2 44240 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 44240 | 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 44241* | 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 44242* | 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 44243 | 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 44244 | 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 44245 | 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 44246 | 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 44247 | 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 44248 | 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 44249 | The convergence relation on the extended reals is a function. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ Fun ~~>* | ||
Theorem | xlimmnflimsup 44250 | 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 44251 | 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 44252* | 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 44253 | 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 44254 | 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 44255 | 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 44256 | 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 44257 | 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))) | ||
Theorem | coseq0 44258 | 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 44259 | 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 44260 | 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 44261 | The cosine of negative π is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (cos‘-π) = -1 | ||
Theorem | sinaover2ne0 44262 | 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 44263 | 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 | mulcncff 44264 | 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 44265* | A continuous complex function restricted to a subset is continuous, using maps-to notation. This theorem generalizes cncfmptss 43981 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 44266* | 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 44267* | 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 44268* | 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 44269 | sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (sin ↾ ℝ) ∈ (ℝ–cn→ℝ) | ||
Theorem | addccncf2 44270* | 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 44271 | The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ∅ ∈ ({∅} Cn {∅}) | ||
Theorem | fsumcncf 44272* | 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 44273* | 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 44274 | 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 44275* | 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 44276* | 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 44277* | Composition of continuous functions. A generalization of cncfmpt1f 24329 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 44278 | 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 44279 | 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 44280* | 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 44281* | 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 44282* | 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 44283 | 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 44284* | 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 44285 | 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 44286* | 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 44287* | 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 44288* | 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 44289* | 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)) | ||
Theorem | cncfioobdlem 44290* | 𝐺 actually extends 𝐹. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶𝑉) & ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → (𝐺‘𝐶) = (𝐹‘𝐶)) | ||
Theorem | cncfioobd 44291* | A continuous function 𝐹 on an open interval (𝐴(,)𝐵) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵 is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵)) & ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑥) | ||
Theorem | jumpncnp 44292 | Jump discontinuity or discontinuity of the first kind: if the left and the right limit don't match, the function is discontinuous at the point. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) & ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) & ⊢ (𝜑 → 𝐿 ≠ 𝑅) ⇒ ⊢ (𝜑 → ¬ 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵)) | ||
Theorem | cxpcncf2 44293* | The complex power function is continuous with respect to its second argument. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → (𝑥 ∈ ℂ ↦ (𝐴↑𝑐𝑥)) ∈ (ℂ–cn→ℂ)) | ||
Theorem | fprodcncf 44294* | The finite product of continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝐵 𝐶) ∈ (𝐴–cn→ℂ)) | ||
Theorem | add1cncf 44295* | Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | add2cncf 44296* | Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 + 𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | sub1cncfd 44297* | Subtracting a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 − 𝐴)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | sub2cncfd 44298* | Subtraction from a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 − 𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | fprodsub2cncf 44299* | 𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | fprodadd2cncf 44300* | 𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
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