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Definition df-sum 15673
Description: 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. These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers) by summo 15703. 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 15868). (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))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘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 15672 . 2 class Σ𝑘𝐴 𝐵
5 vm . . . . . . . . 9 setvar 𝑚
65cv 1532 . . . . . . . 8 class 𝑚
7 cuz 12860 . . . . . . . 8 class
86, 7cfv 6553 . . . . . . 7 class (ℤ𝑚)
91, 8wss 3949 . . . . . 6 wff 𝐴 ⊆ (ℤ𝑚)
10 caddc 11149 . . . . . . . 8 class +
11 vn . . . . . . . . 9 setvar 𝑛
12 cz 12596 . . . . . . . . 9 class
1311cv 1532 . . . . . . . . . . 11 class 𝑛
1413, 1wcel 2098 . . . . . . . . . 10 wff 𝑛𝐴
153, 13, 2csb 3894 . . . . . . . . . 10 class 𝑛 / 𝑘𝐵
16 cc0 11146 . . . . . . . . . 10 class 0
1714, 15, 16cif 4532 . . . . . . . . 9 class if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
1811, 12, 17cmpt 5235 . . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
1910, 18, 6cseq 14006 . . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
20 vx . . . . . . . 8 setvar 𝑥
2120cv 1532 . . . . . . 7 class 𝑥
22 cli 15468 . . . . . . 7 class
2319, 21, 22wbr 5152 . . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
249, 23wa 394 . . . . 5 wff (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
2524, 5, 12wrex 3067 . . . 4 wff 𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
26 c1 11147 . . . . . . . . 9 class 1
27 cfz 13524 . . . . . . . . 9 class ...
2826, 6, 27co 7426 . . . . . . . 8 class (1...𝑚)
29 vf . . . . . . . . 9 setvar 𝑓
3029cv 1532 . . . . . . . 8 class 𝑓
3128, 1, 30wf1o 6552 . . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto𝐴
32 cn 12250 . . . . . . . . . . 11 class
3313, 30cfv 6553 . . . . . . . . . . . 12 class (𝑓𝑛)
343, 33, 2csb 3894 . . . . . . . . . . 11 class (𝑓𝑛) / 𝑘𝐵
3511, 32, 34cmpt 5235 . . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3610, 35, 26cseq 14006 . . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
376, 36cfv 6553 . . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3821, 37wceq 1533 . . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3931, 38wa 394 . . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4039, 29wex 1773 . . . . 5 wff 𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4140, 5, 32wrex 3067 . . . 4 wff 𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4225, 41wo 845 . . 3 wff (∃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)))
4342, 20cio 6503 . 2 class (℩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))))
444, 43wceq 1533 1 wff 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))))
Colors of variables: wff setvar class
This definition is referenced by:  sumex  15674  sumeq1  15675  nfsum1  15676  nfsum  15677  sumeq2w  15678  sumeq2ii  15679  cbvsum  15681  zsum  15704  fsum  15706
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