Definition df-bj-finsum 36991

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 36990	. 2 class FinSum
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1533	. . . . . 6 class 𝑦
5		ccmn 19778	. . . . . 6 class CMnd
6	4, 5	wcel 2099	. . . . 5 wff 𝑦 ∈ CMnd
7		vt	. . . . . . . 8 setvar 𝑡
8	7	cv 1533	. . . . . . 7 class 𝑡
9		cbs 17213	. . . . . . . 8 class Base
10	4, 9	cfv 6554	. . . . . . 7 class (Base‘𝑦)
11		vz	. . . . . . . 8 setvar 𝑧
12	11	cv 1533	. . . . . . 7 class 𝑧
13	8, 10, 12	wf 6550	. . . . . 6 wff 𝑧:𝑡⟶(Base‘𝑦)
14		cfn 8974	. . . . . 6 class Fin
15	13, 7, 14	wrex 3060	. . . . 5 wff ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)
16	6, 15	wa 394	. . . 4 wff (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))
17	16, 3, 11	copab 5215	. . 3 class {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))}
18		c1 11159	. . . . . . . . 9 class 1
19		vm	. . . . . . . . . 10 setvar 𝑚
20	19	cv 1533	. . . . . . . . 9 class 𝑚
21		cfz 13538	. . . . . . . . 9 class ...
22	18, 20, 21	co 7424	. . . . . . . 8 class (1...𝑚)
23	2	cv 1533	. . . . . . . . . 10 class 𝑥
24		c2nd 8002	. . . . . . . . . 10 class 2nd
25	23, 24	cfv 6554	. . . . . . . . 9 class (2nd ‘𝑥)
26	25	cdm 5682	. . . . . . . 8 class dom (2nd ‘𝑥)
27		vf	. . . . . . . . 9 setvar 𝑓
28	27	cv 1533	. . . . . . . 8 class 𝑓
29	22, 26, 28	wf1o 6553	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥)
30		vs	. . . . . . . . 9 setvar 𝑠
31	30	cv 1533	. . . . . . . 8 class 𝑠
32		c1st 8001	. . . . . . . . . . . 12 class 1st
33	23, 32	cfv 6554	. . . . . . . . . . 11 class (1st ‘𝑥)
34		cplusg 17266	. . . . . . . . . . 11 class +g
35	33, 34	cfv 6554	. . . . . . . . . 10 class (+g‘(1st ‘𝑥))
36		vn	. . . . . . . . . . 11 setvar 𝑛
37		cn 12264	. . . . . . . . . . 11 class ℕ
38	36	cv 1533	. . . . . . . . . . . . 13 class 𝑛
39	38, 28	cfv 6554	. . . . . . . . . . . 12 class (𝑓‘𝑛)
40	39, 25	cfv 6554	. . . . . . . . . . 11 class ((2nd ‘𝑥)‘(𝑓‘𝑛))
41	36, 37, 40	cmpt 5236	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛)))
42	35, 41, 18	cseq 14021	. . . . . . . . 9 class seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))
43	20, 42	cfv 6554	. . . . . . . 8 class (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)
44	31, 43	wceq 1534	. . . . . . 7 wff 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)
45	29, 44	wa 394	. . . . . 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 1774	. . . . 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 12524	. . . . 5 class ℕ0
48	46, 19, 47	wrex 3060	. . . 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 6504	. . 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 5236	. 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 1534	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 referenced by:  bj-finsumval0  36992