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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremo1res 14001 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 14002* 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 14003* 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 14004* 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 14005 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 14006 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 14007 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 14008* 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 14009* 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 14010* 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 14011* 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 14012* 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 14013* 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 14014* 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 14015 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 14016 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 14017 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 14018 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 14019* 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 14020* 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 14021* 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 14022* 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 14023* 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 14024* 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 14025* 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 14026* 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 14027* 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 14028* 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 14029* 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 14030* 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 14031* 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 14032* 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 14033* 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 14034* 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 20765 and df-cncf 22433 are not yet available to us. See addcn 22420 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 14035* 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 14036* 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 14037* 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 14038* 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 14039* 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 14040* 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 14041* 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 14042* 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 14043* 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 14044* 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 14045* 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 14046* 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 14047* 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 14048* Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)       ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
 
Theoremrlimabs 14049* 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 14050* 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 14051* 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 14052* 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 14053* 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 14054 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 14055 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 14056 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 14057 Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐹𝑟 𝐴𝐹 ∈ 𝑂(1))
 
Theoremrlimdmo1 14058 A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.)
(𝐹 ∈ dom ⇝𝑟𝐹 ∈ 𝑂(1))
 
Theoremo1rlimmul 14059 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 14060* 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 14061* A constant function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥𝐴𝐵) ∈ ≤𝑂(1))
 
Theoremlo1mptrcl 14062* Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))       ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
 
Theoremo1mptrcl 14063* Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))       ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
 
Theoremo1add2 14064* The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝑂(1))
 
Theoremo1mul2 14065* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴 ↦ (𝐵 · 𝐶)) ∈ 𝑂(1))
 
Theoremo1sub2 14066* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴 ↦ (𝐵𝐶)) ∈ 𝑂(1))
 
Theoremlo1add 14067* The sum of two eventually upper bounded functions is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   (𝜑 → (𝑥𝐴𝐶) ∈ ≤𝑂(1))       (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ ≤𝑂(1))
 
Theoremlo1mul 14068* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   (𝜑 → (𝑥𝐴𝐶) ∈ ≤𝑂(1))    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → (𝑥𝐴 ↦ (𝐵 · 𝐶)) ∈ ≤𝑂(1))
 
Theoremlo1mul2 14069* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   (𝜑 → (𝑥𝐴𝐶) ∈ ≤𝑂(1))    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ ≤𝑂(1))
 
Theoremo1dif 14070* If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → (𝑥𝐴 ↦ (𝐵𝐶)) ∈ 𝑂(1))       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ (𝑥𝐴𝐶) ∈ 𝑂(1)))
 
Theoremlo1sub 14071* The difference of an eventually upper bounded function and an eventually bounded function is eventually upper bounded. The "correct" sharp result here takes the second function to be eventually lower bounded instead of just bounded, but our notation for this is simply (𝑥𝐴 ↦ -𝐶) ∈ ≤𝑂(1), so it is just a special case of lo1add 14067. (Contributed by Mario Carneiro, 31-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴 ↦ (𝐵𝐶)) ∈ ≤𝑂(1))
 
Theoremclimadd 14072* Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐻𝑋)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑𝐻 ⇝ (𝐴 + 𝐵))
 
Theoremclimmul 14073* Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐻𝑋)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) · (𝐺𝑘)))       (𝜑𝐻 ⇝ (𝐴 · 𝐵))
 
Theoremclimsub 14074* Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐻𝑋)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) − (𝐺𝑘)))       (𝜑𝐻 ⇝ (𝐴𝐵))
 
Theoremclimaddc1 14075* Limit of a constant 𝐶 added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = ((𝐹𝑘) + 𝐶))       (𝜑𝐺 ⇝ (𝐴 + 𝐶))
 
Theoremclimaddc2 14076* Limit of a constant 𝐶 added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐶 + (𝐹𝑘)))       (𝜑𝐺 ⇝ (𝐶 + 𝐴))
 
Theoremclimmulc2 14077* Limit of a sequence multiplied by a constant 𝐶. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐶 · (𝐹𝑘)))       (𝜑𝐺 ⇝ (𝐶 · 𝐴))
 
Theoremclimsubc1 14078* Limit of a constant 𝐶 subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = ((𝐹𝑘) − 𝐶))       (𝜑𝐺 ⇝ (𝐴𝐶))
 
Theoremclimsubc2 14079* Limit of a constant 𝐶 minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐶 − (𝐹𝑘)))       (𝜑𝐺 ⇝ (𝐶𝐴))
 
Theoremclimle 14080* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐺𝑘))       (𝜑𝐴𝐵)
 
Theoremclimsqz 14081* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐺𝑘))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ≤ 𝐴)       (𝜑𝐺𝐴)
 
Theoremclimsqz2 14082* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ≤ (𝐹𝑘))    &   ((𝜑𝑘𝑍) → 𝐴 ≤ (𝐺𝑘))       (𝜑𝐺𝐴)
 
Theoremrlimadd 14083* Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐸)       (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ⇝𝑟 (𝐷 + 𝐸))
 
Theoremrlimsub 14084* Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐸)       (𝜑 → (𝑥𝐴 ↦ (𝐵𝐶)) ⇝𝑟 (𝐷𝐸))
 
Theoremrlimmul 14085* Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐸)       (𝜑 → (𝑥𝐴 ↦ (𝐵 · 𝐶)) ⇝𝑟 (𝐷 · 𝐸))
 
Theoremrlimdiv 14086* Limit of the quotient of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐸)    &   (𝜑𝐸 ≠ 0)    &   ((𝜑𝑥𝐴) → 𝐶 ≠ 0)       (𝜑 → (𝑥𝐴 ↦ (𝐵 / 𝐶)) ⇝𝑟 (𝐷 / 𝐸))
 
Theoremrlimneg 14087* Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016.)
((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)       (𝜑 → (𝑘𝐴 ↦ -𝐵) ⇝𝑟 -𝐶)
 
Theoremrlimle 14088* Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.)
(𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐸)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑𝐷𝐸)
 
Theoremrlimsqzlem 14089* Lemma for rlimsqz 14090 and rlimsqz2 14091. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
(𝜑𝑀 ∈ ℝ)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑 ∧ (𝑥𝐴𝑀𝑥)) → (abs‘(𝐶𝐸)) ≤ (abs‘(𝐵𝐷)))       (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐸)
 
Theoremrlimsqz 14090* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
(𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝑀𝑥)) → 𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐴𝑀𝑥)) → 𝐶𝐷)       (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
 
Theoremrlimsqz2 14091* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 3-Feb-2014.) (Revised by Mario Carneiro, 20-May-2016.)
(𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝑀𝑥)) → 𝐶𝐵)    &   ((𝜑 ∧ (𝑥𝐴𝑀𝑥)) → 𝐷𝐶)       (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
 
Theoremlo1le 14092* Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝑀 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝑀𝑥)) → 𝐶𝐵)       (𝜑 → (𝑥𝐴𝐶) ∈ ≤𝑂(1))
 
Theoremo1le 14093* Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝑀 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑 ∧ (𝑥𝐴𝑀𝑥)) → (abs‘𝐶) ≤ (abs‘𝐵))       (𝜑 → (𝑥𝐴𝐶) ∈ 𝑂(1))
 
Theoremrlimno1 14094* A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → (𝑥𝐴 ↦ (1 / 𝐵)) ⇝𝑟 0)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ 0)       (𝜑 → ¬ (𝑥𝐴𝐵) ∈ 𝑂(1))
 
Theoremclim2ser 14095* The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)       (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ (𝐴 − (seq𝑀( + , 𝐹)‘𝑁)))
 
Theoremclim2ser2 14096* The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ 𝐴)       (𝜑 → seq𝑀( + , 𝐹) ⇝ (𝐴 + (seq𝑀( + , 𝐹)‘𝑁)))
 
Theoremiserex 14097* An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
 
Theoremisermulc2 14098* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐶 · (𝐹𝑘)))       (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴))
 
Theoremclimlec2 14099* Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐹𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → 𝐴 ≤ (𝐹𝑘))       (𝜑𝐴𝐵)
 
Theoremiserle 14100* Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐺𝑘))       (𝜑𝐴𝐵)
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