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

Theoremo1mptrcl 14201* Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))       ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)

Theoremo1add2 14202* 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 14203* 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 14204* 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 14205* 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 14206* 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 14207* 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 14208* 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 14209* 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 14205. (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 14210* 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 14211* 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 14212* 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 14213* 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 14214* 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 14215* 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 14216* 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 14217* 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 14218* 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 14219* 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 14220* 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 14221* 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 14222* 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 14223* 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 14224* 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 14225* 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 14226* 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 14227* Lemma for rlimsqz 14228 and rlimsqz2 14229. (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 14228* 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 14229* 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 14230* 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 14231* 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 14232* 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 14233* 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 14234* 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 14235* 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 14236* 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 14237* 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 14238* 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)

Theoremiserge0 14239* The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ (𝐹𝑘))       (𝜑 → 0 ≤ 𝐴)

Theoremclimub 14240* The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))       (𝜑 → (𝐹𝑁) ≤ 𝐴)

Theoremclimserle 14241* The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ (𝐹𝑘))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴)

Theoremisershft 14242 Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴))

Theoremisercolllem1 14243* Lemma for isercoll 14246. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:ℕ⟶𝑍)    &   ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))       ((𝜑𝑆 ⊆ ℕ) → (𝐺𝑆) Isom < , < (𝑆, (𝐺𝑆)))

Theoremisercolllem2 14244* Lemma for isercoll 14246. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:ℕ⟶𝑍)    &   ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))       ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (1...(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))) = (𝐺 “ (𝑀...𝑁)))

Theoremisercolllem3 14245* Lemma for isercoll 14246. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:ℕ⟶𝑍)    &   ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))    &   ((𝜑𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)    &   ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))       ((𝜑𝑁 ∈ (ℤ‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑁))))))

Theoremisercoll 14246* Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:ℕ⟶𝑍)    &   ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))    &   ((𝜑𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)    &   ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))       (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴))

Theoremisercoll2 14247* Generalize isercoll 14246 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐺:𝑍𝑊)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))    &   ((𝜑𝑛 ∈ (𝑊 ∖ ran 𝐺)) → (𝐹𝑛) = 0)    &   ((𝜑𝑛𝑊) → (𝐹𝑛) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))

Theoremclimsup 14248* A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)       (𝜑𝐹 ⇝ sup(ran 𝐹, ℝ, < ))

Theoremclimcau 14249* A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)

Theoremclimbdd 14250* A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) ≤ 𝑥)

Theoremcaucvgrlem 14251* Lemma for caurcvgr 14252. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by AV, 12-Sep-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝐴𝑘𝐴 (𝑗𝑘 → (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → ∃𝑗𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘𝐴 (𝑗𝑘 → (abs‘((𝐹𝑘) − (lim sup‘𝐹))) < (3 · 𝑅))))

Theoremcaurcvgr 14252* A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.) (Revised by AV, 12-Sep-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝐴𝑘𝐴 (𝑗𝑘 → (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))       (𝜑𝐹𝑟 (lim sup‘𝐹))

Theoremcaucvgrlem2 14253* Lemma for caucvgr 14254. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝐴𝑘𝐴 (𝑗𝑘 → (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))    &   𝐻:ℂ⟶ℝ    &   (((𝐹𝑘) ∈ ℂ ∧ (𝐹𝑗) ∈ ℂ) → (abs‘((𝐻‘(𝐹𝑘)) − (𝐻‘(𝐹𝑗)))) ≤ (abs‘((𝐹𝑘) − (𝐹𝑗))))       (𝜑 → (𝑛𝐴 ↦ (𝐻‘(𝐹𝑛))) ⇝𝑟 ( ⇝𝑟 ‘(𝐻𝐹)))

Theoremcaucvgr 14254* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝐴𝑘𝐴 (𝑗𝑘 → (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))       (𝜑𝐹 ∈ dom ⇝𝑟 )

Theoremcaurcvg 14255* A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by AV, 12-Sep-2020.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑚𝑍𝑘 ∈ (ℤ𝑚)(abs‘((𝐹𝑘) − (𝐹𝑚))) < 𝑥)       (𝜑𝐹 ⇝ (lim sup‘𝐹))

Theoremcaurcvg2 14256* A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℝ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))       (𝜑𝐹 ∈ dom ⇝ )

Theoremcaucvg 14257* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)
𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)    &   (𝜑𝐹𝑉)       (𝜑𝐹 ∈ dom ⇝ )

Theoremcaucvgb 14258* A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))

Theoremserf0 14259* If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑𝐹 ⇝ 0)

Theoremiseraltlem1 14260* Lemma for iseralt 14263. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺𝑘))    &   (𝜑𝐺 ⇝ 0)       ((𝜑𝑁𝑍) → 0 ≤ (𝐺𝑁))

Theoremiseraltlem2 14261* Lemma for iseralt 14263. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example 𝑆(1) ≤ 𝑆(3) ≤ 𝑆(5) ≤ ... and ... ≤ 𝑆(4) ≤ 𝑆(2) ≤ 𝑆(0) (assuming 𝑀 = 0 so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺𝑘))    &   (𝜑𝐺 ⇝ 0)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = ((-1↑𝑘) · (𝐺𝑘)))       ((𝜑𝑁𝑍𝐾 ∈ ℕ0) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))

Theoremiseraltlem3 14262* Lemma for iseralt 14263. From iseraltlem2 14261, we have (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) and (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1), and we also have (-1↑𝑛) · 𝑆(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) for each 𝑛 by the definition of the partial sum 𝑆, so combining the inequalities we get (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − 𝐺(𝑛 + 2𝑘 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) ≤ (-1↑𝑛) · 𝑆(𝑛) + 𝐺(𝑛 + 1), so ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) − (-1↑𝑛) · 𝑆(𝑛) ∣ = 𝑆(𝑛 + 2𝑘 + 1) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1) and ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − (-1↑𝑛) · 𝑆(𝑛) ∣ = 𝑆(𝑛 + 2𝑘) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1). Thus, both even and odd partial sums are Cauchy if 𝐺 converges to 0. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺𝑘))    &   (𝜑𝐺 ⇝ 0)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = ((-1↑𝑘) · (𝐺𝑘)))       ((𝜑𝑁𝑍𝐾 ∈ ℕ0) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1))))

Theoremiseralt 14263* The alternating series test. If 𝐺(𝑘) is a decreasing sequence that converges to 0, then Σ𝑘𝑍(-1↑𝑘) · 𝐺(𝑘) is a convergent series. (Note that the first term is positive if 𝑀 is even, and negative if 𝑀 is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by -1 using isermulc2 14236.) (Contributed by Mario Carneiro, 7-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺𝑘))    &   (𝜑𝐺 ⇝ 0)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = ((-1↑𝑘) · (𝐺𝑘)))       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

5.10.3  Finite and infinite sums

Syntaxcsu 14264 Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
class Σ𝑘𝐴 𝐵

Definitiondf-sum 14265* Define the sum of a series with an index set of integers 𝐴. 𝑘 is normally a free variable in 𝐵, i.e. 𝐵 can be thought of as 𝐵(𝑘). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers) by summo 14295. Examples: Σ𝑘 ∈ {1, 2, 4} 𝑘 means 1 + 2 + 4 = 7, and Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 14453). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))

Theoremsumex 14266 A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Σ𝑘𝐴 𝐵 ∈ V

Theoremsumeq1 14267 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
(𝐴 = 𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremnfsum1 14268 Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
𝑘𝐴       𝑘Σ𝑘𝐴 𝐵

Theoremnfsum 14269 Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘𝐴𝐵. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ𝑘𝐴 𝐵

Theoremsumeq2w 14270 Equality theorem for sum, when the class expressions 𝐵 and 𝐶 are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)
(∀𝑘 𝐵 = 𝐶 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2ii 14271* Equality theorem for sum, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Mario Carneiro, 13-Jun-2019.)
(∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2 14272* Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
(∀𝑘𝐴 𝐵 = 𝐶 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremcbvsum 14273* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
(𝑗 = 𝑘𝐵 = 𝐶)    &   𝑘𝐴    &   𝑗𝐴    &   𝑘𝐵    &   𝑗𝐶       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremcbvsumv 14274* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
(𝑗 = 𝑘𝐵 = 𝐶)       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremcbvsumi 14275* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
𝑘𝐵    &   𝑗𝐶    &   (𝑗 = 𝑘𝐵 = 𝐶)       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremsumeq1i 14276* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
𝐴 = 𝐵       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶

Theoremsumeq2i 14277* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
(𝑘𝐴𝐵 = 𝐶)       Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremsumeq12i 14278* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
𝐴 = 𝐵    &   (𝑘𝐴𝐶 = 𝐷)       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷

Theoremsumeq1d 14279* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremsumeq2d 14280* Equality deduction for sum. Note that unlike sumeq2dv 14281, 𝑘 may occur in 𝜑. (Contributed by NM, 1-Nov-2005.)
(𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2dv 14281* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2sdv 14282* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theorem2sumeq2dv 14283* Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝜑𝑗𝐴𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑗𝐴 Σ𝑘𝐵 𝐷)

Theoremsumeq12dv 14284* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)

Theoremsumeq12rdv 14285* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)

Theoremsum2id 14286* The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 ( I ‘𝐵)

Theoremsumfc 14287* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
Σ𝑗𝐴 ((𝑘𝐴𝐵)‘𝑗) = Σ𝑘𝐴 𝐵

Theoremfz1f1o 14288* A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)
(𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)))

Theoremsumrblem 14289* Lemma for sumrb 14291. (Contributed by Mario Carneiro, 12-Aug-2013.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ (ℤ𝑀))       ((𝜑𝐴 ⊆ (ℤ𝑁)) → (seq𝑀( + , 𝐹) ↾ (ℤ𝑁)) = seq𝑁( + , 𝐹))

Theoremfsumcvg 14290* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))

Theoremsumrb 14291* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (ℤ𝑁))       (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹) ⇝ 𝐶))

Theoremsummolem3 14292* Lemma for summo 14295. (Contributed by Mario Carneiro, 29-Mar-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)    &   𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)    &   (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)       (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Theoremsummolem2a 14293* Lemma for summo 14295. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)    &   𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)    &   (𝜑𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))

Theoremsummolem2 14294* Lemma for summo 14295. (Contributed by Mario Carneiro, 3-Apr-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)       ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Theoremsummo 14295* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)       (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))

Theoremzsum 14296* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))

Theoremisum 14297* Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))

Theoremfsum 14298* The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))

Theoremsum0 14299 Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
Σ𝑘 ∈ ∅ 𝐴 = 0

Theoremsumz 14300* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
((𝐴 ⊆ (ℤ𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘𝐴 0 = 0)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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