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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremello1mpt2 14101* 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 14102* 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 14103* 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 14104* Refine o1bdd2 14120 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 14105* 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 14106* 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 14107* 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 14108 An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)

Theoremo1dm 14109 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 14110* 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 14111 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 14112* 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 14113* 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 14114* 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 14115* 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 14116* 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 14117* 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 14118* 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 14119* 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 14120* 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 14121* Refine o1bdd2 14120 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 14122* 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 14123* A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ⇝𝑟 𝐵)

Theoremrlimclim1 14124 Forward direction of rlimclim 14125. (Contributed by Mario Carneiro, 16-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑟 𝐴)    &   (𝜑𝑍 ⊆ dom 𝐹)       (𝜑𝐹𝐴)

Theoremrlimclim 14125 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 14126* 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 14127 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 14128 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
(ℤ × {0}) ⇝ 0

Theoremrlimuni 14129 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 14130 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 14131 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 14132 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
⇝ :dom ⇝ ⟶ℂ

Theoremclimdm 14133 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 14134* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
(𝐹𝐴 → ∃!𝑥 𝐹𝑥)

Theoremclimreu 14135* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
(𝐹𝐴 → ∃!𝑥 ∈ ℂ 𝐹𝑥)

Theoremclimmo 14136* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
∃*𝑥 𝐹𝑥

Theoremrlimres 14137 The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴)

Theoremlo1res 14138 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 14139 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 14140* 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 14141* 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 14142* 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 14143 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 14144 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 14145 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 14146* 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 14147* 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 14148* 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 14149* 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 14150* 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 14151* 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 14152* 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 14153 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 14154 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 14155 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 14156 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 14157* 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 14158* 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 14159* 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 14160* 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 14161* 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 14162* 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 14163* 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 14164* 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 14165* 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 14166* 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 14167* 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))

Theoremrlimcn1b 14168* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)    &   (𝜑𝐹:𝑋⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))       (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)) ⇝𝑟 (𝐹𝐶))

Theoremrlimcn2 14169* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.)
((𝜑𝑧𝐴) → 𝐵𝑋)    &   ((𝜑𝑧𝐴) → 𝐶𝑌)    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑 → (𝑧𝐴𝐵) ⇝𝑟 𝑅)    &   (𝜑 → (𝑧𝐴𝐶) ⇝𝑟 𝑆)    &   (𝜑𝐹:(𝑋 × 𝑌)⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+𝑠 ∈ ℝ+𝑢𝑋𝑣𝑌 (((abs‘(𝑢𝑅)) < 𝑟 ∧ (abs‘(𝑣𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))       (𝜑 → (𝑧𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))

Theoremclimcn1 14170* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑧𝐵) → (𝐹𝑧) ∈ ℂ)    &   (𝜑𝐺𝐴)    &   (𝜑𝐻𝑊)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝐵 ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐴))) < 𝑥))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ 𝐵)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))       (𝜑𝐻 ⇝ (𝐹𝐴))

Theoremclimcn2 14171* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   ((𝜑 ∧ (𝑢𝐶𝑣𝐷)) → (𝑢𝐹𝑣) ∈ ℂ)    &   (𝜑𝐺𝐴)    &   (𝜑𝐻𝐵)    &   (𝜑𝐾𝑊)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑢𝐶𝑣𝐷 (((abs‘(𝑢𝐴)) < 𝑦 ∧ (abs‘(𝑣𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ 𝐶)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) ∈ 𝐷)    &   ((𝜑𝑘𝑍) → (𝐾𝑘) = ((𝐺𝑘)𝐹(𝐻𝑘)))       (𝜑𝐾 ⇝ (𝐴𝐹𝐵))

Theoremaddcn2 14172* Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn 20841 and df-cncf 22489 are not yet available to us. See addcn 22476 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢𝐵)) < 𝑦 ∧ (abs‘(𝑣𝐶)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝐵 + 𝐶))) < 𝐴))

Theoremsubcn2 14173* Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢𝐵)) < 𝑦 ∧ (abs‘(𝑣𝐶)) < 𝑧) → (abs‘((𝑢𝑣) − (𝐵𝐶))) < 𝐴))

Theoremmulcn2 14174* Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢𝐵)) < 𝑦 ∧ (abs‘(𝑣𝐶)) < 𝑧) → (abs‘((𝑢 · 𝑣) − (𝐵 · 𝐶))) < 𝐴))

Theoremreccn2 14175* The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)
𝑇 = (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2))       ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))

Theoremcn1lem 14176* A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
𝐹:ℂ⟶ℂ    &   ((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘((𝐹𝑧) − (𝐹𝐴))) ≤ (abs‘(𝑧𝐴)))       ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐴))) < 𝑥))

Theoremabscn2 14177* The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((abs‘𝑧) − (abs‘𝐴))) < 𝑥))

Theoremcjcn2 14178* The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((∗‘𝑧) − (∗‘𝐴))) < 𝑥))

Theoremrecn2 14179* The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((ℜ‘𝑧) − (ℜ‘𝐴))) < 𝑥))

Theoremimcn2 14180* The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((ℑ‘𝑧) − (ℑ‘𝐴))) < 𝑥))

Theoremclimcn1lem 14181* The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   𝐻:ℂ⟶ℂ    &   ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((𝐻𝑧) − (𝐻𝐴))) < 𝑥))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐻‘(𝐹𝑘)))       (𝜑𝐺 ⇝ (𝐻𝐴))

Theoremclimabs 14182* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (abs‘(𝐹𝑘)))       (𝜑𝐺 ⇝ (abs‘𝐴))

Theoremclimcj 14183* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (∗‘(𝐹𝑘)))       (𝜑𝐺 ⇝ (∗‘𝐴))

Theoremclimre 14184* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (ℜ‘(𝐹𝑘)))       (𝜑𝐺 ⇝ (ℜ‘𝐴))

Theoremclimim 14185* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (ℑ‘(𝐹𝑘)))       (𝜑𝐺 ⇝ (ℑ‘𝐴))

Theoremrlimmptrcl 14186* Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)       ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)

Theoremrlimabs 14187* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)       (𝜑 → (𝑘𝐴 ↦ (abs‘𝐵)) ⇝𝑟 (abs‘𝐶))

Theoremrlimcj 14188* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)       (𝜑 → (𝑘𝐴 ↦ (∗‘𝐵)) ⇝𝑟 (∗‘𝐶))

Theoremrlimre 14189* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)       (𝜑 → (𝑘𝐴 ↦ (ℜ‘𝐵)) ⇝𝑟 (ℜ‘𝐶))

Theoremrlimim 14190* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)       (𝜑 → (𝑘𝐴 ↦ (ℑ‘𝐵)) ⇝𝑟 (ℑ‘𝐶))

Theoremo1of2 14191* Show that a binary operation preserves eventual boundedness. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ) → 𝑀 ∈ ℝ)    &   ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥𝑅𝑦) ∈ ℂ)    &   (((𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (((abs‘𝑥) ≤ 𝑚 ∧ (abs‘𝑦) ≤ 𝑛) → (abs‘(𝑥𝑅𝑦)) ≤ 𝑀))       ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹𝑓 𝑅𝐺) ∈ 𝑂(1))

Theoremo1add 14192 The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹𝑓 + 𝐺) ∈ 𝑂(1))

Theoremo1mul 14193 The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹𝑓 · 𝐺) ∈ 𝑂(1))

Theoremo1sub 14194 The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹𝑓𝐺) ∈ 𝑂(1))

Theoremrlimo1 14195 Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐹𝑟 𝐴𝐹 ∈ 𝑂(1))

Theoremrlimdmo1 14196 A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.)
(𝐹 ∈ dom ⇝𝑟𝐹 ∈ 𝑂(1))

Theoremo1rlimmul 14197 The product of an eventually bounded function and a function of limit zero has limit zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
((𝐹 ∈ 𝑂(1) ∧ 𝐺𝑟 0) → (𝐹𝑓 · 𝐺) ⇝𝑟 0)

Theoremo1const 14198* A constant function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ∈ 𝑂(1))

Theoremlo1const 14199* A constant function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥𝐴𝐵) ∈ ≤𝑂(1))

Theoremlo1mptrcl 14200* Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))       ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)

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