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Definition df-sum 14265
 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) 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.)
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 14264 . 2 class Σ𝑘𝐴 𝐵
5 vm . . . . . . . . 9 setvar 𝑚
65cv 1474 . . . . . . . 8 class 𝑚
7 cuz 11563 . . . . . . . 8 class
86, 7cfv 5804 . . . . . . 7 class (ℤ𝑚)
91, 8wss 3540 . . . . . 6 wff 𝐴 ⊆ (ℤ𝑚)
10 caddc 9818 . . . . . . . 8 class +
11 vn . . . . . . . . 9 setvar 𝑛
12 cz 11254 . . . . . . . . 9 class
1311cv 1474 . . . . . . . . . . 11 class 𝑛
1413, 1wcel 1977 . . . . . . . . . 10 wff 𝑛𝐴
153, 13, 2csb 3499 . . . . . . . . . 10 class 𝑛 / 𝑘𝐵
16 cc0 9815 . . . . . . . . . 10 class 0
1714, 15, 16cif 4036 . . . . . . . . 9 class if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
1811, 12, 17cmpt 4643 . . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
1910, 18, 6cseq 12663 . . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
20 vx . . . . . . . 8 setvar 𝑥
2120cv 1474 . . . . . . 7 class 𝑥
22 cli 14063 . . . . . . 7 class
2319, 21, 22wbr 4583 . . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
249, 23wa 383 . . . . 5 wff (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
2524, 5, 12wrex 2897 . . . 4 wff 𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
26 c1 9816 . . . . . . . . 9 class 1
27 cfz 12197 . . . . . . . . 9 class ...
2826, 6, 27co 6549 . . . . . . . 8 class (1...𝑚)
29 vf . . . . . . . . 9 setvar 𝑓
3029cv 1474 . . . . . . . 8 class 𝑓
3128, 1, 30wf1o 5803 . . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto𝐴
32 cn 10897 . . . . . . . . . . 11 class
3313, 30cfv 5804 . . . . . . . . . . . 12 class (𝑓𝑛)
343, 33, 2csb 3499 . . . . . . . . . . 11 class (𝑓𝑛) / 𝑘𝐵
3511, 32, 34cmpt 4643 . . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3610, 35, 26cseq 12663 . . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
376, 36cfv 5804 . . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3821, 37wceq 1475 . . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3931, 38wa 383 . . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4039, 29wex 1695 . . . . 5 wff 𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4140, 5, 32wrex 2897 . . . 4 wff 𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4225, 41wo 382 . . 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 5766 . 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 1475 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  14266  sumeq1  14267  nfsum1  14268  nfsum  14269  sumeq2w  14270  sumeq2ii  14271  cbvsum  14273  zsum  14296  fsum  14298
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