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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrlimno1 15001* 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 15002* 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 15003* 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 15004* 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 15005* 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 15006* 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 15007* 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 15008* 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 15009* 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 15010* 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 15011 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 15012* Lemma for isercoll 15015. (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 15013* Lemma for isercoll 15015. (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 15014* Lemma for isercoll 15015. (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 15015* 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 15016* Generalize isercoll 15015 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 15017* 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 15018* 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 15019* 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 15020* Lemma for caurcvgr 15021. (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 15021* 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 15022* Lemma for caucvgr 15023. (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 15023* 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 15024* 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 15025* 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 15026* 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 15027* 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 15028* 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 15029* Lemma for iseralt 15032. 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 15030* Lemma for iseralt 15032. 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 15031* Lemma for iseralt 15032. From iseraltlem2 15030, 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 15032* 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 15005.) (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 9-Jul-2022.)
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 15033 Extend class notation to include finite and infinite 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 15034* 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 sets of integers) by summo 15065. 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 15229). (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 15035 A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Σ𝑘𝐴 𝐵 ∈ V

Theoremsumeq1 15036 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 15037 Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
𝑘𝐴       𝑘Σ𝑘𝐴 𝐵

Theoremnfsum 15038* Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘𝐴𝐵. Version of nfsum 15038 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by NM, 11-Dec-2005.) (Revised by Gino Giotto, 24-Feb-2024.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ𝑘𝐴 𝐵

TheoremnfsumOLD 15039 Obsolete version of nfsum 15038 as of 24-Feb-2024. 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.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ𝑘𝐴 𝐵

Theoremsumeq2w 15040 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 15041* 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 15042* 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 15043* 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 15044* 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 15045* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
𝑘𝐵    &   𝑗𝐶    &   (𝑗 = 𝑘𝐵 = 𝐶)       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremsumeq1i 15046* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
𝐴 = 𝐵       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶

Theoremsumeq2i 15047* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
(𝑘𝐴𝐵 = 𝐶)       Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶

Theoremsumeq12i 15048* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
𝐴 = 𝐵    &   (𝑘𝐴𝐶 = 𝐷)       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷

Theoremsumeq1d 15049* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremsumeq2d 15050* Equality deduction for sum. Note that unlike sumeq2dv 15051, 𝑘 may occur in 𝜑. (Contributed by NM, 1-Nov-2005.)
(𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theoremsumeq2dv 15051* 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 15052* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Proof shortened by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)

Theorem2sumeq2dv 15053* 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 15054* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)

Theoremsumeq12rdv 15055* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)

Theoremsum2id 15056* 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 15057* 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 15058* 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 15059* Lemma for sumrb 15061. (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 15060* 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 15061* 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 15062* Lemma for summo 15065. (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 15063* Lemma for summo 15065. (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 15064* Lemma for summo 15065. (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 15065* 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 15066* 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 15067* 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 15068* 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 15069 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 15070* 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)

Theoremfsumf1o 15071* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
(𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑛𝐶 𝐷)

Theoremsumss 15072* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑𝐵 ⊆ (ℤ𝑀))       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremfsumss 15073* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremsumss2 15074* Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
(((𝐴𝐵 ∧ ∀𝑘𝐴 𝐶 ∈ ℂ) ∧ (𝐵 ⊆ (ℤ𝑀) ∨ 𝐵 ∈ Fin)) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 if(𝑘𝐴, 𝐶, 0))

Theoremfsumcvg2 15075* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumsers 15076* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 21-Apr-2014.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → Σ𝑘𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumcvg3 15077* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Theoremfsumser 15078* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 15093 and fsump1i 15115, which should make our notation clear and from which, along with closure fsumcl 15081, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumcl2lem 15079* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → Σ𝑘𝐴 𝐵𝑆)

Theoremfsumcllem 15080* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑 → 0 ∈ 𝑆)       (𝜑 → Σ𝑘𝐴 𝐵𝑆)

Theoremfsumcl 15081* Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℂ)

Theoremfsumrecl 15082* Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℝ)

Theoremfsumzcl 15083* Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℤ)

Theoremfsumnn0cl 15084* Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ0)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℕ0)

Theoremfsumrpcl 15085* Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ+)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℝ+)

Theoremfsumzcl2 15086* A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘𝐴 𝐵 ∈ ℤ)

Theoremfsumadd 15087* The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))

Theoremfsumsplit 15088* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝑈 𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremfsumsplitf 15089* Split a sum into two parts. A version of fsumsplit 15088 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝑈 𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremsumsnf 15090* A sum of a singleton is the term. A version of sumsn 15092 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝐵    &   (𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremfsumsplitsn 15091* Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   𝑘𝐷    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝑉)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝑘 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘𝐴 𝐶 + 𝐷))

Theoremsumsn 15092* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremfsum1 15093* The finite sum of 𝐴(𝑘) from 𝑘 = 𝑀 to 𝑀 (i.e. a sum with only one term) is 𝐵 i.e. 𝐴(𝑀). (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)𝐴 = 𝐵)

Theoremsumpr 15094* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(𝑘 = 𝐴𝐶 = 𝐷)    &   (𝑘 = 𝐵𝐶 = 𝐸)    &   (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   (𝜑 → (𝐴𝑉𝐵𝑊))    &   (𝜑𝐴𝐵)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))

Theoremsumtp 15095* A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.)
(𝑘 = 𝐴𝐷 = 𝐸)    &   (𝑘 = 𝐵𝐷 = 𝐹)    &   (𝑘 = 𝐶𝐷 = 𝐺)    &   (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))    &   (𝜑 → (𝐴𝑉𝐵𝑊𝐶𝑋))    &   (𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺))

Theoremsumsns 15096* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
((𝑀𝑉𝑀 / 𝑘𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)

Theoremfsumm1 15097* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑁𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Theoremfzosump1 15098* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑁𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵))

Theoremfsum1p 15099* Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑀𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))

Theoremfsummsnunz 15100* A finite sum all of whose summands are integers is itself an integer (case where the summation set is the union of a finite set and a singleton). (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
((𝐴 ∈ Fin ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ)

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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 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