Definition df-bj-finsum 37969

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 37968	. 2 class FinSum
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1569	. . . . . 6 class 𝑦
5		ccmn 19881	. . . . . 6 class CMnd
6	4, 5	wcel 2146	. . . . 5 wff 𝑦 ∈ CMnd
7		vt	. . . . . . . 8 setvar 𝑡
8	7	cv 1569	. . . . . . 7 class 𝑡
9		cbs 17294	. . . . . . . 8 class Base
10	4, 9	cfv 6543	. . . . . . 7 class (Base‘𝑦)
11		vz	. . . . . . . 8 setvar 𝑧
12	11	cv 1569	. . . . . . 7 class 𝑧
13	8, 10, 12	wf 6539	. . . . . 6 wff 𝑧:𝑡⟶(Base‘𝑦)
14		cfn 8952	. . . . . 6 class Fin
15	13, 7, 14	wrex 3092	. . . . 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 5178	. . 3 class {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))}
18		c1 11119	. . . . . . . . 9 class 1
19		vm	. . . . . . . . . 10 setvar 𝑚
20	19	cv 1569	. . . . . . . . 9 class 𝑚
21		cfz 13553	. . . . . . . . 9 class ...
22	18, 20, 21	co 7423	. . . . . . . 8 class (1...𝑚)
23	2	cv 1569	. . . . . . . . . 10 class 𝑥
24		c2nd 7994	. . . . . . . . . 10 class 2nd
25	23, 24	cfv 6543	. . . . . . . . 9 class (2nd ‘𝑥)
26	25	cdm 5666	. . . . . . . 8 class dom (2nd ‘𝑥)
27		vf	. . . . . . . . 9 setvar 𝑓
28	27	cv 1569	. . . . . . . 8 class 𝑓
29	22, 26, 28	wf1o 6542	. . . . . . 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 7993	. . . . . . . . . . . 12 class 1st
33	23, 32	cfv 6543	. . . . . . . . . . 11 class (1st ‘𝑥)
34		cplusg 17335	. . . . . . . . . . 11 class +g
35	33, 34	cfv 6543	. . . . . . . . . 10 class (+g‘(1st ‘𝑥))
36		vn	. . . . . . . . . . 11 setvar 𝑛
37		cn 12251	. . . . . . . . . . 11 class ℕ
38	36	cv 1569	. . . . . . . . . . . . 13 class 𝑛
39	38, 28	cfv 6543	. . . . . . . . . . . 12 class (𝑓‘𝑛)
40	39, 25	cfv 6543	. . . . . . . . . . 11 class ((2nd ‘𝑥)‘(𝑓‘𝑛))
41	36, 37, 40	cmpt 5197	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛)))
42	35, 41, 18	cseq 14057	. . . . . . . . 9 class seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))
43	20, 42	cfv 6543	. . . . . . . 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 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 1812	. . . . 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 12522	. . . . 5 class ℕ0
48	46, 19, 47	wrex 3092	. . . 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 6497	. . 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 5197	. 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 used by:  bj-finsumval0  37970