Definition df-sumdc 11831

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. In both cases we have an if expression so that we only need 𝐵 to be defined where 𝑘 ∈ 𝐴. 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 11999). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)

Assertion

Ref	Expression
df-sumdc	⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		vk	. . 3 setvar 𝑘
4	1, 2, 3	csu 11830	. 2 class Σ𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1374	. . . . . . . 8 class 𝑚
7		cuz 9690	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 5294	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3177	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		vj	. . . . . . . . . 10 setvar 𝑗
11	10	cv 1374	. . . . . . . . 9 class 𝑗
12	11, 1	wcel 2180	. . . . . . . 8 wff 𝑗 ∈ 𝐴
13	12	wdc 838	. . . . . . 7 wff DECID 𝑗 ∈ 𝐴
14	13, 10, 8	wral 2488	. . . . . 6 wff ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
15		caddc 7970	. . . . . . . 8 class +
16		vn	. . . . . . . . 9 setvar 𝑛
17		cz 9414	. . . . . . . . 9 class ℤ
18	16	cv 1374	. . . . . . . . . . 11 class 𝑛
19	18, 1	wcel 2180	. . . . . . . . . 10 wff 𝑛 ∈ 𝐴
20	3, 18, 2	csb 3104	. . . . . . . . . 10 class ⦋𝑛 / 𝑘⦌𝐵
21		cc0 7967	. . . . . . . . . 10 class 0
22	19, 20, 21	cif 3582	. . . . . . . . 9 class if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
23	16, 17, 22	cmpt 4124	. . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
24	15, 23, 6	cseq 10636	. . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
25		vx	. . . . . . . 8 setvar 𝑥
26	25	cv 1374	. . . . . . 7 class 𝑥
27		cli 11755	. . . . . . 7 class ⇝
28	24, 26, 27	wbr 4062	. . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
29	9, 14, 28	w3a 983	. . . . 5 wff (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
30	29, 5, 17	wrex 2489	. . . 4 wff ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
31		c1 7968	. . . . . . . . 9 class 1
32		cfz 10172	. . . . . . . . 9 class ...
33	31, 6, 32	co 5974	. . . . . . . 8 class (1...𝑚)
34		vf	. . . . . . . . 9 setvar 𝑓
35	34	cv 1374	. . . . . . . 8 class 𝑓
36	33, 1, 35	wf1o 5293	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
37		cn 9078	. . . . . . . . . . 11 class ℕ
38		cle 8150	. . . . . . . . . . . . 13 class ≤
39	18, 6, 38	wbr 4062	. . . . . . . . . . . 12 wff 𝑛 ≤ 𝑚
40	18, 35	cfv 5294	. . . . . . . . . . . . 13 class (𝑓‘𝑛)
41	3, 40, 2	csb 3104	. . . . . . . . . . . 12 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
42	39, 41, 21	cif 3582	. . . . . . . . . . 11 class if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)
43	16, 37, 42	cmpt 4124	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
44	15, 43, 31	cseq 10636	. . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))
45	6, 44	cfv 5294	. . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
46	26, 45	wceq 1375	. . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
47	36, 46	wa 104	. . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
48	47, 34	wex 1518	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
49	48, 5, 37	wrex 2489	. . . 4 wff ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
50	30, 49	wo 712	. . 3 wff (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)))
51	50, 25	cio 5252	. 2 class (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))
52	4, 51	wceq 1375	1 wff Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))

This definition is referenced by:  sumeq1  11832  nfsum1  11833  nfsum  11834  sumeq2  11836  cbvsum  11837  zsumdc  11861  fsum3  11864