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Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclim2ser2 11301* 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 11302* 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 11303* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐶 · (𝐹𝑘)))       (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴))
 
Theoremclimlec2 11304* 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 11305* 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 11306* 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 11307* 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 11308* 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)
 
Theoremiser3shft 11309* Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.)
(𝜑𝐹𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴))
 
Theoremclimcau 11310* A converging sequence of complex numbers is a Cauchy sequence. The converse would require excluded middle or a different definition of Cauchy sequence (for example, fixing a rate of convergence as in climcvg1n 11313). 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)
 
Theoremclimrecvg1n 11311* A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)(abs‘((𝐹𝑘) − (𝐹𝑛))) < (𝐶 / 𝑛))       (𝜑𝐹 ∈ dom ⇝ )
 
Theoremclimcvg1nlem 11312* Lemma for climcvg1n 11313. We construct sequences of the real and imaginary parts of each term of 𝐹, show those converge, and use that to show that 𝐹 converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
(𝜑𝐹:ℕ⟶ℂ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)(abs‘((𝐹𝑘) − (𝐹𝑛))) < (𝐶 / 𝑛))    &   𝐺 = (𝑥 ∈ ℕ ↦ (ℜ‘(𝐹𝑥)))    &   𝐻 = (𝑥 ∈ ℕ ↦ (ℑ‘(𝐹𝑥)))    &   𝐽 = (𝑥 ∈ ℕ ↦ (i · (𝐻𝑥)))       (𝜑𝐹 ∈ dom ⇝ )
 
Theoremclimcvg1n 11313* A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
(𝜑𝐹:ℕ⟶ℂ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)(abs‘((𝐹𝑘) − (𝐹𝑛))) < (𝐶 / 𝑛))       (𝜑𝐹 ∈ dom ⇝ )
 
Theoremclimcaucn 11314* A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 11310 but adds the part that (𝐹𝑘) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))
 
Theoremserf0 11315* 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)
 
4.8.2  Finite and infinite sums
 
Syntaxcsu 11316 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-sumdc 11317* Define the sum of a series with an index set of integers 𝐴. The variable 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. In both cases we have an if expression so that we only need 𝐵 to be defined where 𝑘𝐴. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers). 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 11485). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
 
Theoremsumeq1 11318 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 11319 Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
𝑘𝐴       𝑘Σ𝑘𝐴 𝐵
 
Theoremnfsum 11320 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
 
Theoremsumdc 11321* Decidability of a subset of upper integers. (Contributed by Jim Kingdon, 1-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑 → ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)    &   (𝜑𝑁 ∈ ℤ)       (𝜑DECID 𝑁𝐴)
 
Theoremsumeq2 11322* 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 11323 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 11324* 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 11325* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
𝑘𝐵    &   𝑗𝐶    &   (𝑗 = 𝑘𝐵 = 𝐶)       Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
 
Theoremsumeq1i 11326* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
𝐴 = 𝐵       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶
 
Theoremsumeq2i 11327* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
(𝑘𝐴𝐵 = 𝐶)       Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶
 
Theoremsumeq12i 11328* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
𝐴 = 𝐵    &   (𝑘𝐴𝐶 = 𝐷)       Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷
 
Theoremsumeq1d 11329* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
 
Theoremsumeq2d 11330* Equality deduction for sum. Note that unlike sumeq2dv 11331, 𝑘 may occur in 𝜑. (Contributed by NM, 1-Nov-2005.)
(𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
 
Theoremsumeq2dv 11331* 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)
 
Theoremsumeq2ad 11332* Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
 
Theoremsumeq2sdv 11333* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
 
Theorem2sumeq2dv 11334* 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 11335* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)
 
Theoremsumeq12rdv 11336* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)
 
Theoremsumfct 11337* 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 Jim Kingdon, 18-Sep-2022.)
(∀𝑘𝐴 𝐵 ∈ ℂ → Σ𝑗𝐴 ((𝑘𝐴𝐵)‘𝑗) = Σ𝑘𝐴 𝐵)
 
Theoremfz1f1o 11338* 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)))
 
Theoremnnf1o 11339 Lemma for sum and product theorems. (Contributed by Jim Kingdon, 15-Aug-2022.)
(𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   (𝜑𝐺:(1...𝑁)–1-1-onto𝐴)       (𝜑𝑁 = 𝑀)
 
Theoremsumrbdclem 11340* Lemma for sumrbdc 11342. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))       ((𝜑𝐴 ⊆ (ℤ𝑁)) → (seq𝑀( + , 𝐹) ↾ (ℤ𝑁)) = seq𝑁( + , 𝐹))
 
Theoremfsum3cvg 11341* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremsumrbdc 11342* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (ℤ𝑁))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → DECID 𝑘𝐴)       (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹) ⇝ 𝐶))
 
Theoremsummodclem3 11343* Lemma for summodc 11346. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)    &   (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))    &   𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))       (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
 
Theoremsummodclem2a 11344* Lemma for summodc 11346. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))    &   𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)    &   (𝜑𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))
 
Theoremsummodclem2 11345* Lemma for summodc 11346. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))       ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
 
Theoremsummodc 11346* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))    &   𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))       (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))
 
Theoremzsumdc 11347* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑 → ∀𝑥𝑍 DECID 𝑥𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
 
Theoremisum 11348* 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)))
 
Theoremfsumgcl 11349* Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
 
Theoremfsum3 11350* The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 0)))‘𝑀))
 
Theoremsum0 11351 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
 
Theoremisumz 11352* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
(((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∨ 𝐴 ∈ Fin) → Σ𝑘𝐴 0 = 0)
 
Theoremfsumf1o 11353* 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)
 
Theoremisumss 11354* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐵 ⊆ (ℤ𝑀))    &   (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐵)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
 
Theoremfisumss 11355* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
 
Theoremisumss2 11356* Change the index set of a sum by adding zeroes. The nonzero elements are in the contained set 𝐴 and the added zeroes compose the rest of the containing set 𝐵 which needs to be summable. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Jim Kingdon, 24-Sep-2022.)
(𝜑𝐴𝐵)    &   (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)    &   (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)    &   (𝜑 → ((𝑀 ∈ ℤ ∧ 𝐵 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐵) ∨ 𝐵 ∈ Fin))       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 if(𝑘𝐴, 𝐶, 0))
 
Theoremfsum3cvg2 11357* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremfsumsersdc 11358* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → Σ𝑘𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremfsum3cvg3 11359* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
 
Theoremfsum3ser 11360* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11375 and fsump1 11383, which should make our notation clear and from which, along with closure fsumcl 11363, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremfsumcl2lem 11361* - 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 11362* - 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 11363* 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 11364* 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 11365* 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 11366* 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 11367* 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 11368* 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 11369* 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 11370* 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 11371* Split a sum into two parts. A version of fsumsplit 11370 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 11372* A sum of a singleton is the term. A version of sumsn 11374 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 11373* 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 11374* 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 11375* 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 11376* 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 11377* 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 11378* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
((𝑀𝑉𝑀 / 𝑘𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
 
Theoremfsumm1 11379* 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 11380* 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 11381* 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))
 
Theoremfsumsplitsnun 11382* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
((𝐴 ∈ Fin ∧ (𝑍𝑉𝑍𝐴) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 = (Σ𝑘𝐴 𝐵 + 𝑍 / 𝑘𝐵))
 
Theoremfsump1 11383* The addition of the next term in a finite sum of 𝐴(𝑘) is the current term plus 𝐵 i.e. 𝐴(𝑁 + 1). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝑁)𝐴 + 𝐵))
 
Theoremisumclim 11384* An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐵)       (𝜑 → Σ𝑘𝑍 𝐴 = 𝐵)
 
Theoremisumclim2 11385* A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘𝑍 𝐴)
 
Theoremisumclim3 11386* The sequence of partial finite sums of a converging infinite series converges to the infinite sum of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)       (𝜑𝐹 ⇝ Σ𝑘𝑍 𝐴)
 
Theoremsumnul 11387* The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 = ∅)
 
Theoremisumcl 11388* The sum of a converging infinite series is a complex number. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ∈ ℂ)
 
Theoremisummulc2 11389* An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐵 · Σ𝑘𝑍 𝐴) = Σ𝑘𝑍 (𝐵 · 𝐴))
 
Theoremisummulc1 11390* An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (Σ𝑘𝑍 𝐴 · 𝐵) = Σ𝑘𝑍 (𝐴 · 𝐵))
 
Theoremisumdivapc 11391* An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (Σ𝑘𝑍 𝐴 / 𝐵) = Σ𝑘𝑍 (𝐴 / 𝐵))
 
Theoremisumrecl 11392* The sum of a converging infinite real series is a real number. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ∈ ℝ)
 
Theoremisumge0 11393* An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐴)       (𝜑 → 0 ≤ Σ𝑘𝑍 𝐴)
 
Theoremisumadd 11394* Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 (𝐴 + 𝐵) = (Σ𝑘𝑍 𝐴 + Σ𝑘𝑍 𝐵))
 
Theoremsumsplitdc 11395* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝐴𝐵) ⊆ 𝑍)    &   ((𝜑𝑘𝑍) → DECID 𝑘𝐴)    &   ((𝜑𝑘𝑍) → DECID 𝑘𝐵)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐶, 0))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = if(𝑘𝐵, 𝐶, 0))    &   ((𝜑𝑘 ∈ (𝐴𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘 ∈ (𝐴𝐵)𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))
 
Theoremfsump1i 11396* Optimized version of fsump1 11383 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑁 = (𝐾 + 1)    &   (𝑘 = 𝑁𝐴 = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝐾𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆))    &   (𝜑 → (𝑆 + 𝐵) = 𝑇)       (𝜑 → (𝑁𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇))
 
Theoremfsum2dlemstep 11397* Lemma for fsum2d 11398- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.) (Revised by Jim Kingdon, 8-Oct-2022.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → ¬ 𝑦𝑥)    &   (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)    &   (𝜑𝑥 ∈ Fin)    &   (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)       ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
 
Theoremfsum2d 11398* Write a double sum as a sum over a two-dimensional region. Note that 𝐵(𝑗) is a function of 𝑗. (Contributed by Mario Carneiro, 27-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
 
Theoremfsumxp 11399* Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 ∈ (𝐴 × 𝐵)𝐷)
 
Theoremfsumcnv 11400* Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)    &   (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → Rel 𝐴)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴𝐶)
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