Definition df-sum 15880

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 15909. 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 16077). (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)⟶1-1onto𝐴 ∧ 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 15879	. 2 class Σ𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1569	. . . . . . . 8 class 𝑚
7		cuz 12989	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 6555	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3900	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		caddc 11227	. . . . . . . 8 class +
11		vn	. . . . . . . . 9 setvar 𝑛
12		cz 12717	. . . . . . . . 9 class ℤ
13	11	cv 1569	. . . . . . . . . . 11 class 𝑛
14	13, 1	wcel 2147	. . . . . . . . . 10 wff 𝑛 ∈ 𝐴
15	3, 13, 2	csb 3851	. . . . . . . . . 10 class ⦋𝑛 / 𝑘⦌𝐵
16		cc0 11224	. . . . . . . . . 10 class 0
17	14, 15, 16	cif 4485	. . . . . . . . 9 class if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
18	11, 12, 17	cmpt 5189	. . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
19	10, 18, 6	cseq 14168	. . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
20		vx	. . . . . . . 8 setvar 𝑥
21	20	cv 1569	. . . . . . 7 class 𝑥
22		cli 15677	. . . . . . 7 class ⇝
23	19, 21, 22	wbr 5106	. . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
24	9, 23	wa 401	. . . . 5 wff (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
25	24, 5, 12	wrex 3090	. . . 4 wff ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
26		c1 11225	. . . . . . . . 9 class 1
27		cfz 13663	. . . . . . . . 9 class ...
28	26, 6, 27	co 7440	. . . . . . . 8 class (1...𝑚)
29		vf	. . . . . . . . 9 setvar 𝑓
30	29	cv 1569	. . . . . . . 8 class 𝑓
31	28, 1, 30	wf1o 6554	. . . . . . 7 wff 𝑓:(1...𝑚)⟶1-1onto𝐴
32		cn 12359	. . . . . . . . . . 11 class ℕ
33	13, 30	cfv 6555	. . . . . . . . . . . 12 class (𝑓‘𝑛)
34	3, 33, 2	csb 3851	. . . . . . . . . . 11 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
35	11, 32, 34	cmpt 5189	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
36	10, 35, 26	cseq 14168	. . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
37	6, 36	cfv 6555	. . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
38	21, 37	wceq 1570	. . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
39	31, 38	wa 401	. . . . . 6 wff (𝑓:(1...𝑚)⟶1-1onto𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
40	39, 29	wex 1812	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)⟶1-1onto𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
41	40, 5, 32	wrex 3090	. . . 4 wff ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)⟶1-1onto𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
42	25, 41	wo 861	. . 3 wff (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)⟶1-1onto𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
43	42, 20	cio 6507	. 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)⟶1-1onto𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
44	4, 43	wceq 1570	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)⟶1-1onto𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))

This definition is used by:  sumex  15881  sumeq1  15882  nfsum1  15883  nfsum  15884  sumeq2w  15885  sumeq2ii  15886  cbvsum  15888  cbvsumv  15889  sumeq2sdv  15896  zsum  15910  fsum  15912  sumeq2si  37168  cbvsumdavw  37245  cbvsumdavw2  37261