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Definition df-sumdc 11135
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. 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 11303). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
Assertion
Ref Expression
df-sumdc 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))))
Distinct variable groups:   𝑓,𝑘,𝑚,𝑛,𝑥,𝑗   𝐴,𝑓,𝑚,𝑛,𝑥,𝑗   𝐵,𝑓,𝑚,𝑛,𝑥,𝑗
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)

Detailed syntax breakdown of Definition df-sumdc
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
3 vk . . 3 setvar 𝑘
41, 2, 3csu 11134 . 2 class Σ𝑘𝐴 𝐵
5 vm . . . . . . . . 9 setvar 𝑚
65cv 1330 . . . . . . . 8 class 𝑚
7 cuz 9338 . . . . . . . 8 class
86, 7cfv 5123 . . . . . . 7 class (ℤ𝑚)
91, 8wss 3071 . . . . . 6 wff 𝐴 ⊆ (ℤ𝑚)
10 vj . . . . . . . . . 10 setvar 𝑗
1110cv 1330 . . . . . . . . 9 class 𝑗
1211, 1wcel 1480 . . . . . . . 8 wff 𝑗𝐴
1312wdc 819 . . . . . . 7 wff DECID 𝑗𝐴
1413, 10, 8wral 2416 . . . . . 6 wff 𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴
15 caddc 7635 . . . . . . . 8 class +
16 vn . . . . . . . . 9 setvar 𝑛
17 cz 9066 . . . . . . . . 9 class
1816cv 1330 . . . . . . . . . . 11 class 𝑛
1918, 1wcel 1480 . . . . . . . . . 10 wff 𝑛𝐴
203, 18, 2csb 3003 . . . . . . . . . 10 class 𝑛 / 𝑘𝐵
21 cc0 7632 . . . . . . . . . 10 class 0
2219, 20, 21cif 3474 . . . . . . . . 9 class if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
2316, 17, 22cmpt 3989 . . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
2415, 23, 6cseq 10230 . . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
25 vx . . . . . . . 8 setvar 𝑥
2625cv 1330 . . . . . . 7 class 𝑥
27 cli 11059 . . . . . . 7 class
2824, 26, 27wbr 3929 . . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
299, 14, 28w3a 962 . . . . 5 wff (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
3029, 5, 17wrex 2417 . . . 4 wff 𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
31 c1 7633 . . . . . . . . 9 class 1
32 cfz 9802 . . . . . . . . 9 class ...
3331, 6, 32co 5774 . . . . . . . 8 class (1...𝑚)
34 vf . . . . . . . . 9 setvar 𝑓
3534cv 1330 . . . . . . . 8 class 𝑓
3633, 1, 35wf1o 5122 . . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto𝐴
37 cn 8732 . . . . . . . . . . 11 class
38 cle 7813 . . . . . . . . . . . . 13 class
3918, 6, 38wbr 3929 . . . . . . . . . . . 12 wff 𝑛𝑚
4018, 35cfv 5123 . . . . . . . . . . . . 13 class (𝑓𝑛)
413, 40, 2csb 3003 . . . . . . . . . . . 12 class (𝑓𝑛) / 𝑘𝐵
4239, 41, 21cif 3474 . . . . . . . . . . 11 class if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)
4316, 37, 42cmpt 3989 . . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
4415, 43, 31cseq 10230 . . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))
456, 44cfv 5123 . . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
4626, 45wceq 1331 . . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
4736, 46wa 103 . . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
4847, 34wex 1468 . . . . 5 wff 𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
4948, 5, 37wrex 2417 . . . 4 wff 𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
5030, 49wo 697 . . 3 wff (∃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)))
5150, 25cio 5086 . 2 class (℩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))))
524, 51wceq 1331 1 wff 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))))
Colors of variables: wff set class
This definition is referenced by:  sumeq1  11136  nfsum1  11137  nfsum  11138  sumeq2  11140  cbvsum  11141  zsumdc  11165  fsum3  11168
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