Definition df-sum 15612

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 15642. 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 15807). (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 15611	. 2 class Σ𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1539	. . . . . . . 8 class 𝑚
7		cuz 12753	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 6486	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3905	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		caddc 11031	. . . . . . . 8 class +
11		vn	. . . . . . . . 9 setvar 𝑛
12		cz 12489	. . . . . . . . 9 class ℤ
13	11	cv 1539	. . . . . . . . . . 11 class 𝑛
14	13, 1	wcel 2109	. . . . . . . . . 10 wff 𝑛 ∈ 𝐴
15	3, 13, 2	csb 3853	. . . . . . . . . 10 class ⦋𝑛 / 𝑘⦌𝐵
16		cc0 11028	. . . . . . . . . 10 class 0
17	14, 15, 16	cif 4478	. . . . . . . . 9 class if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
18	11, 12, 17	cmpt 5176	. . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
19	10, 18, 6	cseq 13926	. . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
20		vx	. . . . . . . 8 setvar 𝑥
21	20	cv 1539	. . . . . . 7 class 𝑥
22		cli 15409	. . . . . . 7 class ⇝
23	19, 21, 22	wbr 5095	. . . . . 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 3053	. . . 4 wff ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
26		c1 11029	. . . . . . . . 9 class 1
27		cfz 13428	. . . . . . . . 9 class ...
28	26, 6, 27	co 7353	. . . . . . . 8 class (1...𝑚)
29		vf	. . . . . . . . 9 setvar 𝑓
30	29	cv 1539	. . . . . . . 8 class 𝑓
31	28, 1, 30	wf1o 6485	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
32		cn 12146	. . . . . . . . . . 11 class ℕ
33	13, 30	cfv 6486	. . . . . . . . . . . 12 class (𝑓‘𝑛)
34	3, 33, 2	csb 3853	. . . . . . . . . . 11 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
35	11, 32, 34	cmpt 5176	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
36	10, 35, 26	cseq 13926	. . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
37	6, 36	cfv 6486	. . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
38	21, 37	wceq 1540	. . . . . . 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 1779	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
41	40, 5, 32	wrex 3053	. . . 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 847	. . 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 6440	. 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 1540	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  15613  sumeq1  15614  nfsum1  15615  nfsum  15616  sumeq2w  15617  sumeq2ii  15618  cbvsum  15620  cbvsumv  15621  sumeq2sdv  15628  zsum  15643  fsum  15645  sumeq2si  36175  cbvsumdavw  36252  cbvsumdavw2  36268