Definition df-sumdc 11497

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 11665). (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 11496	. 2 class Σ𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1363	. . . . . . . 8 class 𝑚
7		cuz 9592	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 5254	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3153	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		vj	. . . . . . . . . 10 setvar 𝑗
11	10	cv 1363	. . . . . . . . 9 class 𝑗
12	11, 1	wcel 2164	. . . . . . . 8 wff 𝑗 ∈ 𝐴
13	12	wdc 835	. . . . . . 7 wff DECID 𝑗 ∈ 𝐴
14	13, 10, 8	wral 2472	. . . . . 6 wff ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
15		caddc 7875	. . . . . . . 8 class +
16		vn	. . . . . . . . 9 setvar 𝑛
17		cz 9317	. . . . . . . . 9 class ℤ
18	16	cv 1363	. . . . . . . . . . 11 class 𝑛
19	18, 1	wcel 2164	. . . . . . . . . 10 wff 𝑛 ∈ 𝐴
20	3, 18, 2	csb 3080	. . . . . . . . . 10 class ⦋𝑛 / 𝑘⦌𝐵
21		cc0 7872	. . . . . . . . . 10 class 0
22	19, 20, 21	cif 3557	. . . . . . . . 9 class if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
23	16, 17, 22	cmpt 4090	. . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
24	15, 23, 6	cseq 10518	. . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
25		vx	. . . . . . . 8 setvar 𝑥
26	25	cv 1363	. . . . . . 7 class 𝑥
27		cli 11421	. . . . . . 7 class ⇝
28	24, 26, 27	wbr 4029	. . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
29	9, 14, 28	w3a 980	. . . . 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 2473	. . . 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 7873	. . . . . . . . 9 class 1
32		cfz 10074	. . . . . . . . 9 class ...
33	31, 6, 32	co 5918	. . . . . . . 8 class (1...𝑚)
34		vf	. . . . . . . . 9 setvar 𝑓
35	34	cv 1363	. . . . . . . 8 class 𝑓
36	33, 1, 35	wf1o 5253	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
37		cn 8982	. . . . . . . . . . 11 class ℕ
38		cle 8055	. . . . . . . . . . . . 13 class ≤
39	18, 6, 38	wbr 4029	. . . . . . . . . . . 12 wff 𝑛 ≤ 𝑚
40	18, 35	cfv 5254	. . . . . . . . . . . . 13 class (𝑓‘𝑛)
41	3, 40, 2	csb 3080	. . . . . . . . . . . 12 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
42	39, 41, 21	cif 3557	. . . . . . . . . . 11 class if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)
43	16, 37, 42	cmpt 4090	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
44	15, 43, 31	cseq 10518	. . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))
45	6, 44	cfv 5254	. . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
46	26, 45	wceq 1364	. . . . . . 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 1503	. . . . 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 2473	. . . 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 709	. . 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 5213	. 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 1364	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  11498  nfsum1  11499  nfsum  11500  sumeq2  11502  cbvsum  11503  zsumdc  11527  fsum3  11530