Definition df-sum 15729

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 15759. 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 15924). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Assertion

Ref	Expression
df-sum	⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		vk	. . 3 setvar 𝑘
4	1, 2, 3	csu 15728	. 2 class Σ𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1536	. . . . . . . 8 class 𝑚
7		cuz 12897	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 6568	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3976	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		caddc 11181	. . . . . . . 8 class +
11		vn	. . . . . . . . 9 setvar 𝑛
12		cz 12633	. . . . . . . . 9 class ℤ
13	11	cv 1536	. . . . . . . . . . 11 class 𝑛
14	13, 1	wcel 2108	. . . . . . . . . 10 wff 𝑛 ∈ 𝐴
15	3, 13, 2	csb 3921	. . . . . . . . . 10 class ⦋𝑛 / 𝑘⦌𝐵
16		cc0 11178	. . . . . . . . . 10 class 0
17	14, 15, 16	cif 4548	. . . . . . . . 9 class if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
18	11, 12, 17	cmpt 5249	. . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
19	10, 18, 6	cseq 14046	. . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
20		vx	. . . . . . . 8 setvar 𝑥
21	20	cv 1536	. . . . . . 7 class 𝑥
22		cli 15524	. . . . . . 7 class ⇝
23	19, 21, 22	wbr 5166	. . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
24	9, 23	wa 395	. . . . 5 wff (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
25	24, 5, 12	wrex 3076	. . . 4 wff ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
26		c1 11179	. . . . . . . . 9 class 1
27		cfz 13561	. . . . . . . . 9 class ...
28	26, 6, 27	co 7443	. . . . . . . 8 class (1...𝑚)
29		vf	. . . . . . . . 9 setvar 𝑓
30	29	cv 1536	. . . . . . . 8 class 𝑓
31	28, 1, 30	wf1o 6567	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
32		cn 12287	. . . . . . . . . . 11 class ℕ
33	13, 30	cfv 6568	. . . . . . . . . . . 12 class (𝑓‘𝑛)
34	3, 33, 2	csb 3921	. . . . . . . . . . 11 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
35	11, 32, 34	cmpt 5249	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
36	10, 35, 26	cseq 14046	. . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
37	6, 36	cfv 6568	. . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
38	21, 37	wceq 1537	. . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
39	31, 38	wa 395	. . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
40	39, 29	wex 1777	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
41	40, 5, 32	wrex 3076	. . . 4 wff ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
42	25, 41	wo 846	. . 3 wff (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
43	42, 20	cio 6518	. 2 class (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
44	4, 43	wceq 1537	1 wff Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))

This definition is referenced by:  sumex  15730  sumeq1  15731  nfsum1  15732  nfsum  15733  sumeq2w  15734  sumeq2ii  15735  cbvsum  15737  cbvsumv  15738  sumeq2sdv  15745  zsum  15760  fsum  15762  sumeq2si  36158  cbvsumdavw  36237  cbvsumdavw2  36253