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Definition df-sum 10104
Description: 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). 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. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Assertion
Ref Expression
df-sum Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))))
Distinct variable groups:   𝑓,𝑘,𝑚,𝑛,𝑥   𝐴,𝑓,𝑚,𝑛,𝑥   𝐵,𝑓,𝑚,𝑛,𝑥
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)

Detailed syntax breakdown of Definition df-sum
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
3 vk . . 3 setvar 𝑘
41, 2, 3csu 10103 . 2 class Σ𝑘𝐴 𝐵
5 vm . . . . . . . . 9 setvar 𝑚
65cv 1258 . . . . . . . 8 class 𝑚
7 cuz 8569 . . . . . . . 8 class
86, 7cfv 4930 . . . . . . 7 class (ℤ𝑚)
91, 8wss 2945 . . . . . 6 wff 𝐴 ⊆ (ℤ𝑚)
10 caddc 6950 . . . . . . . 8 class +
11 cc 6945 . . . . . . . 8 class
12 vn . . . . . . . . 9 setvar 𝑛
13 cz 8302 . . . . . . . . 9 class
1412cv 1258 . . . . . . . . . . 11 class 𝑛
1514, 1wcel 1409 . . . . . . . . . 10 wff 𝑛𝐴
163, 14, 2csb 2880 . . . . . . . . . 10 class 𝑛 / 𝑘𝐵
17 cc0 6947 . . . . . . . . . 10 class 0
1815, 16, 17cif 3359 . . . . . . . . 9 class if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
1912, 13, 18cmpt 3846 . . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
2010, 11, 19, 6cseq 9375 . . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ)
21 vx . . . . . . . 8 setvar 𝑥
2221cv 1258 . . . . . . 7 class 𝑥
23 cli 10030 . . . . . . 7 class
2420, 22, 23wbr 3792 . . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥
259, 24wa 101 . . . . 5 wff (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)
2625, 5, 13wrex 2324 . . . 4 wff 𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)
27 c1 6948 . . . . . . . . 9 class 1
28 cfz 8976 . . . . . . . . 9 class ...
2927, 6, 28co 5540 . . . . . . . 8 class (1...𝑚)
30 vf . . . . . . . . 9 setvar 𝑓
3130cv 1258 . . . . . . . 8 class 𝑓
3229, 1, 31wf1o 4929 . . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto𝐴
33 cn 7990 . . . . . . . . . . 11 class
3414, 31cfv 4930 . . . . . . . . . . . 12 class (𝑓𝑛)
353, 34, 2csb 2880 . . . . . . . . . . 11 class (𝑓𝑛) / 𝑘𝐵
3612, 33, 35cmpt 3846 . . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3710, 11, 36, 27cseq 9375 . . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)
386, 37cfv 4930 . . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
3922, 38wceq 1259 . . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
4032, 39wa 101 . . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
4140, 30wex 1397 . . . . 5 wff 𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
4241, 5, 33wrex 2324 . . . 4 wff 𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
4326, 42wo 639 . . 3 wff (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)))
4443, 21cio 4893 . 2 class (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))))
454, 44wceq 1259 1 wff Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))))
Colors of variables: wff set class
This definition is referenced by:  sumeq1  10105  nfsum1  10106  nfsum  10107
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