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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-rlim 14201* Define the limit relation for partial functions on the reals. See rlim 14207 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}

Definitiondf-o1 14202* Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of 𝑂(1) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}

Definitiondf-lo1 14203* Define the set of eventually upper bounded real functions. This fills a gap in 𝑂(1) coverage, to express statements like 𝑓(𝑥) ≤ 𝑔(𝑥) + 𝑂(𝑥) via (𝑥 ∈ ℝ+ ↦ (𝑓(𝑥) − 𝑔(𝑥)) / 𝑥) ∈ ≤𝑂(1). (Contributed by Mario Carneiro, 25-May-2016.)
≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚}

Theoremclimrel 14204 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
Rel ⇝

Theoremrlimrel 14205 The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
Rel ⇝𝑟

Theoremclim 14206* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑗 such that the absolute difference of any later complex number in the sequence and the limit is less than 𝑥. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝜑𝐹𝑉)    &   ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremrlim 14207* Express the predicate: The limit of complex number function 𝐹 is 𝐶, or 𝐹 converges to 𝐶, in the real sense. This means that for any real 𝑥, no matter how small, there always exists a number 𝑦 such that the absolute difference of any number in the function beyond 𝑦 and the limit is less than 𝑥. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)       (𝜑 → (𝐹𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥))))

Theoremrlim2 14208* Rewrite rlim 14207 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))

Theoremrlim2lt 14209* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦 < 𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))

Theoremrlim3 14210* Restrict the range of the domain bound to reals greater than some 𝐷 ∈ ℝ. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ (𝐷[,)+∞)∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))

Theoremclimcl 14211 Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹𝐴𝐴 ∈ ℂ)

Theoremrlimpm 14212 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))

Theoremrlimf 14213 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)

Theoremrlimss 14214 Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)

Theoremrlimcl 14215 Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹𝑟 𝐴𝐴 ∈ ℂ)

Theoremclim2 14216* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 14206. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremclim2c 14217* Express the predicate 𝐹 converges to 𝐴. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹𝐴 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝑥))

Theoremclim0 14218* Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘𝐵) < 𝑥)))

Theoremclim0c 14219* Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝑥))

Theoremrlim0 14220* Express the predicate 𝐵(𝑧) converges to 0. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 0 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘𝐵) < 𝑥)))

Theoremrlim0lt 14221* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 0 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦 < 𝑧 → (abs‘𝐵) < 𝑥)))

Theoremclimi 14222* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹𝐴)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶))

Theoremclimi2 14223* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹𝐴)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝐶)

Theoremclimi0 14224* Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹 ⇝ 0)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝐶)

Theoremrlimi 14225* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → (𝑧𝐴𝐵) ⇝𝑟 𝐶)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑅))

Theoremrlimi2 14226* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.)
(𝜑 → ∀𝑧𝐴 𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → (𝑧𝐴𝐵) ⇝𝑟 𝐶)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → ∃𝑦 ∈ (𝐷[,)+∞)∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑅))

Theoremello1 14227* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))

Theoremello12 14228* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚)))

Theoremello12r 14229* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (𝐹𝑥) ≤ 𝑀)) → 𝐹 ∈ ≤𝑂(1))

Theoremlo1f 14230 An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)

Theoremlo1dm 14231 An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)

Theoremlo1bdd 14232* The defining property of an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:𝐴⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))

Theoremello1mpt 14233* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))

Theoremello1mpt2 14234* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))

Theoremello1d 14235* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐶𝑥)) → 𝐵𝑀)       (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))

Theoremlo1bdd2 14236* If an eventually bounded function is bounded on every interval 𝐴 ∩ (-∞, 𝑦) by a function 𝑀(𝑦), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → 𝐵𝑀)       (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥𝐴 𝐵𝑚)

Theoremlo1bddrp 14237* Refine o1bdd2 14253 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → 𝐵𝑀)       (𝜑 → ∃𝑚 ∈ ℝ+𝑥𝐴 𝐵𝑚)

Theoremelo1 14238* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))

Theoremelo12 14239* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (abs‘(𝐹𝑦)) ≤ 𝑚)))

Theoremelo12r 14240* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
(((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1))

Theoremo1f 14241 An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)

Theoremo1dm 14242 An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)

Theoremo1bdd 14243* The defining property of an eventually bounded function. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐹 ∈ 𝑂(1) ∧ 𝐹:𝐴⟶ℂ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (abs‘(𝐹𝑦)) ≤ 𝑚))

Theoremlo1o1 14244 A function is eventually bounded iff its absolute value is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹:𝐴⟶ℂ → (𝐹 ∈ 𝑂(1) ↔ (abs ∘ 𝐹) ∈ ≤𝑂(1)))

Theoremlo1o12 14245* A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about ≤𝑂(1) to 𝑂(1).) (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1)))

Theoremelo1mpt 14246* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘𝐵) ≤ 𝑚)))

Theoremelo1mpt2 14247* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘𝐵) ≤ 𝑚)))

Theoremelo1d 14248* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐶𝑥)) → (abs‘𝐵) ≤ 𝑀)       (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))

Theoremo1lo1 14249* A real function is eventually bounded iff it is eventually lower bounded and eventually upper bounded. (Contributed by Mario Carneiro, 25-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ∧ (𝑥𝐴 ↦ -𝐵) ∈ ≤𝑂(1))))

Theoremo1lo12 14250* A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝑀𝐵)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ (𝑥𝐴𝐵) ∈ ≤𝑂(1)))

Theoremo1lo1d 14251* A real eventually bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))

Theoremicco1 14252* Derive eventual boundedness from separate upper and lower eventual bounds. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑𝑁 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐶𝑥)) → 𝐵 ∈ (𝑀[,]𝑁))       (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))

Theoremo1bdd2 14253* If an eventually bounded function is bounded on every interval 𝐴 ∩ (-∞, 𝑦) by a function 𝑀(𝑦), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → (abs‘𝐵) ≤ 𝑀)       (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥𝐴 (abs‘𝐵) ≤ 𝑚)

Theoremo1bddrp 14254* Refine o1bdd2 14253 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → (abs‘𝐵) ≤ 𝑀)       (𝜑 → ∃𝑚 ∈ ℝ+𝑥𝐴 (abs‘𝐵) ≤ 𝑚)

Theoremclimconst 14255* An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)       (𝜑𝐹𝐴)

Theoremrlimconst 14256* A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ⇝𝑟 𝐵)

Theoremrlimclim1 14257 Forward direction of rlimclim 14258. (Contributed by Mario Carneiro, 16-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑟 𝐴)    &   (𝜑𝑍 ⊆ dom 𝐹)       (𝜑𝐹𝐴)

Theoremrlimclim 14258 A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℂ)       (𝜑 → (𝐹𝑟 𝐴𝐹𝐴))

Theoremclimrlim2 14259* Produce a real limit from an integer limit, where the real function is only dependent on the integer part of 𝑥. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ𝑀)    &   (𝑛 = (⌊‘𝑥) → 𝐵 = 𝐶)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → (𝑛𝑍𝐵) ⇝ 𝐷)    &   ((𝜑𝑛𝑍) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝑀𝑥)       (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)

Theoremclimconst2 14260 A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
(ℤ𝑀) ⊆ 𝑍    &   𝑍 ∈ V       ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ⇝ 𝐴)

Theoremclimz 14261 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
(ℤ × {0}) ⇝ 0

Theoremrlimuni 14262 A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑𝐹𝑟 𝐵)    &   (𝜑𝐹𝑟 𝐶)       (𝜑𝐵 = 𝐶)

Theoremrlimdm 14263 Two ways to express that a function has a limit. (The expression ( ⇝𝑟𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 8-May-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟𝐹)))

Theoremclimuni 14264 An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
((𝐹𝐴𝐹𝐵) → 𝐴 = 𝐵)

Theoremfclim 14265 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
⇝ :dom ⇝ ⟶ℂ

Theoremclimdm 14266 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 Mario Carneiro, 18-Mar-2014.)
(𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))

Theoremclimeu 14267* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
(𝐹𝐴 → ∃!𝑥 𝐹𝑥)

Theoremclimreu 14268* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
(𝐹𝐴 → ∃!𝑥 ∈ ℂ 𝐹𝑥)

Theoremclimmo 14269* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
∃*𝑥 𝐹𝑥

Theoremrlimres 14270 The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴)

Theoremlo1res 14271 The restriction of an eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ ≤𝑂(1) → (𝐹𝐴) ∈ ≤𝑂(1))

Theoremo1res 14272 The restriction of an eventually bounded function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝐹 ∈ 𝑂(1) → (𝐹𝐴) ∈ 𝑂(1))

Theoremrlimres2 14273* The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝜑𝐴𝐵)    &   (𝜑 → (𝑥𝐵𝐶) ⇝𝑟 𝐷)       (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)

Theoremlo1res2 14274* The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐴𝐵)    &   (𝜑 → (𝑥𝐵𝐶) ∈ ≤𝑂(1))       (𝜑 → (𝑥𝐴𝐶) ∈ ≤𝑂(1))

Theoremo1res2 14275* The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 21-May-2016.)
(𝜑𝐴𝐵)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴𝐶) ∈ 𝑂(1))

Theoremlo1resb 14276 The restriction of a function to an unbounded-above interval is eventually upper bounded iff the original is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1)))

Theoremrlimresb 14277 The restriction of a function to an unbounded-above interval converges iff the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐹𝑟 𝐶 ↔ (𝐹 ↾ (𝐵[,)+∞)) ⇝𝑟 𝐶))

Theoremo1resb 14278 The restriction of a function to an unbounded-above interval is eventually bounded iff the original is eventually bounded. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐹 ∈ 𝑂(1) ↔ (𝐹 ↾ (𝐵[,)+∞)) ∈ 𝑂(1)))

Theoremclimeq 14279* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (𝐹𝐴𝐺𝐴))

Theoremlo1eq 14280* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐷𝑥)) → 𝐵 = 𝐶)       (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ (𝑥𝐴𝐶) ∈ ≤𝑂(1)))

Theoremrlimeq 14281* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐷𝑥)) → 𝐵 = 𝐶)       (𝜑 → ((𝑥𝐴𝐵) ⇝𝑟 𝐸 ↔ (𝑥𝐴𝐶) ⇝𝑟 𝐸))

Theoremo1eq 14282* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐷𝑥)) → 𝐵 = 𝐶)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ (𝑥𝐴𝐶) ∈ 𝑂(1)))

Theoremclimmpt 14283* Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   𝐺 = (𝑘𝑍 ↦ (𝐹𝑘))       ((𝑀 ∈ ℤ ∧ 𝐹𝑉) → (𝐹𝐴𝐺𝐴))

Theorem2clim 14284* If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺𝑉)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐺𝑘))) < 𝑥)    &   (𝜑𝐹𝐴)       (𝜑𝐺𝐴)

Theoremclimmpt2 14285* Relate an integer limit on a not-quite-function to a real limit. (Contributed by Mario Carneiro, 17-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → (𝐹𝐴 ↔ (𝑛𝑍 ↦ (𝐹𝑛)) ⇝𝑟 𝐴))

Theoremclimshftlem 14286 A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       (𝑀 ∈ ℤ → (𝐹𝐴 → (𝐹 shift 𝑀) ⇝ 𝐴))

Theoremclimres 14287 A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝑀 ∈ ℤ ∧ 𝐹𝑉) → ((𝐹 ↾ (ℤ𝑀)) ⇝ 𝐴𝐹𝐴))

Theoremclimshft 14288 A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝑀 ∈ ℤ ∧ 𝐹𝑉) → ((𝐹 shift 𝑀) ⇝ 𝐴𝐹𝐴))

Theoremserclim0 14289 The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
(𝑀 ∈ ℤ → seq𝑀( + , ((ℤ𝑀) × {0})) ⇝ 0)

Theoremrlimcld2 14290* If 𝐷 is a closed set in the topology of the complex numbers (stated here in basic form), and all the elements of the sequence lie in 𝐷, then the limit of the sequence also lies in 𝐷. (Contributed by Mario Carneiro, 10-May-2016.)
(𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐶)    &   (𝜑𝐷 ⊆ ℂ)    &   ((𝜑𝑦 ∈ (ℂ ∖ 𝐷)) → 𝑅 ∈ ℝ+)    &   (((𝜑𝑦 ∈ (ℂ ∖ 𝐷)) ∧ 𝑧𝐷) → 𝑅 ≤ (abs‘(𝑧𝑦)))    &   ((𝜑𝑥𝐴) → 𝐵𝐷)       (𝜑𝐶𝐷)

Theoremrlimrege0 14291* The limit of a sequence of complex numbers with nonnegative real part has nonnegative real part. (Contributed by Mario Carneiro, 10-May-2016.)
(𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 0 ≤ (ℜ‘𝐵))       (𝜑 → 0 ≤ (ℜ‘𝐶))

Theoremrlimrecl 14292* The limit of a real sequence is real. (Contributed by Mario Carneiro, 9-May-2016.)
(𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑𝐶 ∈ ℝ)

Theoremrlimge0 14293* The limit of a sequence of nonnegative reals is nonnegative. (Contributed by Mario Carneiro, 10-May-2016.)
(𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → 0 ≤ 𝐶)

Theoremclimshft2 14294* A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝐹𝑊)    &   (𝜑𝐺𝑋)    &   ((𝜑𝑘𝑍) → (𝐺‘(𝑘 + 𝐾)) = (𝐹𝑘))       (𝜑 → (𝐹𝐴𝐺𝐴))

Theoremclimrecl 14295* The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremclimge0 14296* A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ (𝐹𝑘))       (𝜑 → 0 ≤ 𝐴)

Theoremclimabs0 14297* Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (abs‘(𝐹𝑘)))       (𝜑 → (𝐹 ⇝ 0 ↔ 𝐺 ⇝ 0))

Theoremo1co 14298* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐹 ∈ 𝑂(1))    &   (𝜑𝐺:𝐵𝐴)    &   (𝜑𝐵 ⊆ ℝ)    &   ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)))       (𝜑 → (𝐹𝐺) ∈ 𝑂(1))

Theoremo1compt 14299* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐹 ∈ 𝑂(1))    &   ((𝜑𝑦𝐵) → 𝐶𝐴)    &   (𝜑𝐵 ⊆ ℝ)    &   ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶))       (𝜑 → (𝐹 ∘ (𝑦𝐵𝐶)) ∈ 𝑂(1))

Theoremrlimcn1 14300* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 17-Sep-2014.)
(𝜑𝐺:𝐴𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐺𝑟 𝐶)    &   (𝜑𝐹:𝑋⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))       (𝜑 → (𝐹𝐺) ⇝𝑟 (𝐹𝐶))

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