Definition df-bj-finsum 38298

Description: Finite summation in commutative monoids. This finite summation function can be extended to pairs ⟨𝑦, 𝑧⟩ where 𝑦 is a left-unital magma and 𝑧 is defined on a totally ordered set (choosing left-associative composition), or dropping unitality and requiring nonempty families, or on any monoids for families of permutable elements, etc. We use the term "summation", even though the definition stands for any unital, commutative and associative composition law. (Contributed by BJ, 9-Jun-2019.)

Assertion

Ref	Expression
df-bj-finsum	⊢ FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))

Distinct variable group:   𝑥,𝑦,𝑧,𝑡,𝑠,𝑓,𝑚,𝑛

Detailed syntax breakdown of Definition df-bj-finsum

Step	Hyp	Ref	Expression
1		cfinsum 38297	. 2 class FinSum
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1568	. . . . . 6 class 𝑦
5		ccmn 19999	. . . . . 6 class CMnd
6	4, 5	wcel 2144	. . . . 5 wff 𝑦 ∈ CMnd
7		vt	. . . . . . . 8 setvar 𝑡
8	7	cv 1568	. . . . . . 7 class 𝑡
9		cbs 17392	. . . . . . . 8 class Base
10	4, 9	cfv 6538	. . . . . . 7 class (Base‘𝑦)
11		vz	. . . . . . . 8 setvar 𝑧
12	11	cv 1568	. . . . . . 7 class 𝑧
13	8, 10, 12	wf 6534	. . . . . 6 wff 𝑧:𝑡⟶(Base‘𝑦)
14		cfn 8976	. . . . . 6 class Fin
15	13, 7, 14	wrex 3086	. . . . 5 wff ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)
16	6, 15	wa 401	. . . 4 wff (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))
17	16, 3, 11	copab 5166	. . 3 class {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))}
18		c1 11204	. . . . . . . . 9 class 1
19		vm	. . . . . . . . . 10 setvar 𝑚
20	19	cv 1568	. . . . . . . . 9 class 𝑚
21		cfz 13642	. . . . . . . . 9 class ...
22	18, 20, 21	co 7420	. . . . . . . 8 class (1...𝑚)
23	2	cv 1568	. . . . . . . . . 10 class 𝑥
24		c2nd 8000	. . . . . . . . . 10 class 2nd
25	23, 24	cfv 6538	. . . . . . . . 9 class (2nd ‘𝑥)
26	25	cdm 5650	. . . . . . . 8 class dom (2nd ‘𝑥)
27		vf	. . . . . . . . 9 setvar 𝑓
28	27	cv 1568	. . . . . . . 8 class 𝑓
29	22, 26, 28	wf1o 6537	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥)
30		vs	. . . . . . . . 9 setvar 𝑠
31	30	cv 1568	. . . . . . . 8 class 𝑠
32		c1st 7999	. . . . . . . . . . . 12 class 1st
33	23, 32	cfv 6538	. . . . . . . . . . 11 class (1st ‘𝑥)
34		cplusg 17433	. . . . . . . . . . 11 class +g
35	33, 34	cfv 6538	. . . . . . . . . 10 class (+g‘(1st ‘𝑥))
36		vn	. . . . . . . . . . 11 setvar 𝑛
37		cn 12338	. . . . . . . . . . 11 class ℕ
38	36	cv 1568	. . . . . . . . . . . . 13 class 𝑛
39	38, 28	cfv 6538	. . . . . . . . . . . 12 class (𝑓‘𝑛)
40	39, 25	cfv 6538	. . . . . . . . . . 11 class ((2nd ‘𝑥)‘(𝑓‘𝑛))
41	36, 37, 40	cmpt 5185	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛)))
42	35, 41, 18	cseq 14147	. . . . . . . . 9 class seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))
43	20, 42	cfv 6538	. . . . . . . 8 class (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)
44	31, 43	wceq 1569	. . . . . . 7 wff 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)
45	29, 44	wa 401	. . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))
46	45, 27	wex 1811	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))
47		cn0 12609	. . . . 5 class ℕ0
48	46, 19, 47	wrex 3086	. . . 4 wff ∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))
49	48, 30	cio 6492	. . 3 class (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)))
50	2, 17, 49	cmpt 5185	. 2 class (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))
51	1, 50	wceq 1569	1 wff FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))

This definition is used by:  bj-finsumval0  38299