Definition df-bj-finsum 37740

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 37739	. 2 class FinSum
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1558	. . . . . 6 class 𝑦
5		ccmn 19803	. . . . . 6 class CMnd
6	4, 5	wcel 2141	. . . . 5 wff 𝑦 ∈ CMnd
7		vt	. . . . . . . 8 setvar 𝑡
8	7	cv 1558	. . . . . . 7 class 𝑡
9		cbs 17228	. . . . . . . 8 class Base
10	4, 9	cfv 6517	. . . . . . 7 class (Base‘𝑦)
11		vz	. . . . . . . 8 setvar 𝑧
12	11	cv 1558	. . . . . . 7 class 𝑧
13	8, 10, 12	wf 6513	. . . . . 6 wff 𝑧:𝑡⟶(Base‘𝑦)
14		cfn 8923	. . . . . 6 class Fin
15	13, 7, 14	wrex 3085	. . . . 5 wff ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)
16	6, 15	wa 399	. . . 4 wff (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))
17	16, 3, 11	copab 5161	. . 3 class {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))}
18		c1 11071	. . . . . . . . 9 class 1
19		vm	. . . . . . . . . 10 setvar 𝑚
20	19	cv 1558	. . . . . . . . 9 class 𝑚
21		cfz 13509	. . . . . . . . 9 class ...
22	18, 20, 21	co 7392	. . . . . . . 8 class (1...𝑚)
23	2	cv 1558	. . . . . . . . . 10 class 𝑥
24		c2nd 7965	. . . . . . . . . 10 class 2nd
25	23, 24	cfv 6517	. . . . . . . . 9 class (2nd ‘𝑥)
26	25	cdm 5645	. . . . . . . 8 class dom (2nd ‘𝑥)
27		vf	. . . . . . . . 9 setvar 𝑓
28	27	cv 1558	. . . . . . . 8 class 𝑓
29	22, 26, 28	wf1o 6516	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥)
30		vs	. . . . . . . . 9 setvar 𝑠
31	30	cv 1558	. . . . . . . 8 class 𝑠
32		c1st 7964	. . . . . . . . . . . 12 class 1st
33	23, 32	cfv 6517	. . . . . . . . . . 11 class (1st ‘𝑥)
34		cplusg 17269	. . . . . . . . . . 11 class +g
35	33, 34	cfv 6517	. . . . . . . . . 10 class (+g‘(1st ‘𝑥))
36		vn	. . . . . . . . . . 11 setvar 𝑛
37		cn 12207	. . . . . . . . . . 11 class ℕ
38	36	cv 1558	. . . . . . . . . . . . 13 class 𝑛
39	38, 28	cfv 6517	. . . . . . . . . . . 12 class (𝑓‘𝑛)
40	39, 25	cfv 6517	. . . . . . . . . . 11 class ((2nd ‘𝑥)‘(𝑓‘𝑛))
41	36, 37, 40	cmpt 5180	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛)))
42	35, 41, 18	cseq 14011	. . . . . . . . 9 class seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))
43	20, 42	cfv 6517	. . . . . . . 8 class (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)
44	31, 43	wceq 1559	. . . . . . 7 wff 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)
45	29, 44	wa 399	. . . . . 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 1798	. . . . 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 12478	. . . . 5 class ℕ0
48	46, 19, 47	wrex 3085	. . . 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 6471	. . 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 5180	. 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 1559	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  37741