Definition df-bj-finsum 37906

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 37905	. 2 class FinSum
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1569	. . . . . 6 class 𝑦
5		ccmn 19851	. . . . . 6 class CMnd
6	4, 5	wcel 2143	. . . . 5 wff 𝑦 ∈ CMnd
7		vt	. . . . . . . 8 setvar 𝑡
8	7	cv 1569	. . . . . . 7 class 𝑡
9		cbs 17270	. . . . . . . 8 class Base
10	4, 9	cfv 6538	. . . . . . 7 class (Base‘𝑦)
11		vz	. . . . . . . 8 setvar 𝑧
12	11	cv 1569	. . . . . . 7 class 𝑧
13	8, 10, 12	wf 6534	. . . . . 6 wff 𝑧:𝑡⟶(Base‘𝑦)
14		cfn 8944	. . . . . 6 class Fin
15	13, 7, 14	wrex 3089	. . . . 5 wff ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)
16	6, 15	wa 400	. . . 4 wff (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))
17	16, 3, 11	copab 5174	. . 3 class {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))}
18		c1 11102	. . . . . . . . 9 class 1
19		vm	. . . . . . . . . 10 setvar 𝑚
20	19	cv 1569	. . . . . . . . 9 class 𝑚
21		cfz 13536	. . . . . . . . 9 class ...
22	18, 20, 21	co 7412	. . . . . . . 8 class (1...𝑚)
23	2	cv 1569	. . . . . . . . . 10 class 𝑥
24		c2nd 7986	. . . . . . . . . 10 class 2nd
25	23, 24	cfv 6538	. . . . . . . . 9 class (2nd ‘𝑥)
26	25	cdm 5663	. . . . . . . 8 class dom (2nd ‘𝑥)
27		vf	. . . . . . . . 9 setvar 𝑓
28	27	cv 1569	. . . . . . . 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 1569	. . . . . . . 8 class 𝑠
32		c1st 7985	. . . . . . . . . . . 12 class 1st
33	23, 32	cfv 6538	. . . . . . . . . . 11 class (1st ‘𝑥)
34		cplusg 17311	. . . . . . . . . . 11 class +g
35	33, 34	cfv 6538	. . . . . . . . . 10 class (+g‘(1st ‘𝑥))
36		vn	. . . . . . . . . . 11 setvar 𝑛
37		cn 12234	. . . . . . . . . . 11 class ℕ
38	36	cv 1569	. . . . . . . . . . . . 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 5193	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛)))
42	35, 41, 18	cseq 14039	. . . . . . . . 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 1570	. . . . . . 7 wff 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)
45	29, 44	wa 400	. . . . . 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 1809	. . . . 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 12505	. . . . 5 class ℕ0
48	46, 19, 47	wrex 3089	. . . 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 5193	. 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 1570	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  37907