Definition df-isum 10564

Description: Define the sum of a series with an index set of integers 𝐴. 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. (Contributed by Jim Kingdon, 15-Feb-2022.)

Assertion

Ref	Expression
df-isum	⊢ 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)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘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-isum

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		vk	. . 3 setvar 𝑘
4	1, 2, 3	csu 10563	. 2 class Σ𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1284	. . . . . . . 8 class 𝑚
7		cuz 8913	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 4968	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 2984	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		vj	. . . . . . . . . 10 setvar 𝑗
11	10	cv 1284	. . . . . . . . 9 class 𝑗
12	11, 1	wcel 1434	. . . . . . . 8 wff 𝑗 ∈ 𝐴
13	12	wdc 776	. . . . . . 7 wff DECID 𝑗 ∈ 𝐴
14	13, 10, 8	wral 2353	. . . . . 6 wff ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
15		caddc 7255	. . . . . . . 8 class +
16		cc 7250	. . . . . . . 8 class ℂ
17		vn	. . . . . . . . 9 setvar 𝑛
18		cz 8645	. . . . . . . . 9 class ℤ
19	17	cv 1284	. . . . . . . . . . 11 class 𝑛
20	19, 1	wcel 1434	. . . . . . . . . 10 wff 𝑛 ∈ 𝐴
21	3, 19, 2	csb 2919	. . . . . . . . . 10 class ⦋𝑛 / 𝑘⦌𝐵
22		cc0 7252	. . . . . . . . . 10 class 0
23	20, 21, 22	cif 3373	. . . . . . . . 9 class if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
24	17, 18, 23	cmpt 3865	. . . . . . . 8 class (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
25	15, 16, 24, 6	cseq 9739	. . . . . . 7 class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ)
26		vx	. . . . . . . 8 setvar 𝑥
27	26	cv 1284	. . . . . . 7 class 𝑥
28		cli 10490	. . . . . . 7 class ⇝
29	25, 27, 28	wbr 3811	. . . . . 6 wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑥
30	9, 14, 29	w3a 920	. . . . 5 wff (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑥)
31	30, 5, 18	wrex 2354	. . . 4 wff ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑥)
32		c1 7253	. . . . . . . . 9 class 1
33		cfz 9318	. . . . . . . . 9 class ...
34	32, 6, 33	co 5590	. . . . . . . 8 class (1...𝑚)
35		vf	. . . . . . . . 9 setvar 𝑓
36	35	cv 1284	. . . . . . . 8 class 𝑓
37	34, 1, 36	wf1o 4967	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
38		cn 8315	. . . . . . . . . . 11 class ℕ
39		cle 7425	. . . . . . . . . . . . 13 class ≤
40	19, 6, 39	wbr 3811	. . . . . . . . . . . 12 wff 𝑛 ≤ 𝑚
41	19, 36	cfv 4968	. . . . . . . . . . . . 13 class (𝑓‘𝑛)
42	3, 41, 2	csb 2919	. . . . . . . . . . . 12 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
43	40, 42, 22	cif 3373	. . . . . . . . . . 11 class if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)
44	17, 38, 43	cmpt 3865	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
45	15, 16, 44, 32	cseq 9739	. . . . . . . . 9 class seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)
46	6, 45	cfv 4968	. . . . . . . 8 class (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚)
47	27, 46	wceq 1285	. . . . . . 7 wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚)
48	37, 47	wa 102	. . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚))
49	48, 35	wex 1422	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚))
50	49, 5, 38	wrex 2354	. . . 4 wff ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚))
51	31, 50	wo 662	. . 3 wff (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚)))
52	51, 26	cio 4931	. 2 class (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚))))
53	4, 52	wceq 1285	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)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)), ℂ)‘𝑚))))

This definition is referenced by:  sumeq1  10565  nfsum1  10566  nfsum  10567  sumeq2d  10569  sumeq2  10570