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Definition df-bj-finsum 37227
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))} ↦ (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑡,𝑠,𝑓,𝑚,𝑛

Detailed syntax breakdown of Definition df-bj-finsum
StepHypRef Expression
1 cfinsum 37226 . 2 class FinSum
2 vx . . 3 setvar 𝑥
3 vy . . . . . . 7 setvar 𝑦
43cv 1534 . . . . . 6 class 𝑦
5 ccmn 19799 . . . . . 6 class CMnd
64, 5wcel 2104 . . . . 5 wff 𝑦 ∈ CMnd
7 vt . . . . . . . 8 setvar 𝑡
87cv 1534 . . . . . . 7 class 𝑡
9 cbs 17235 . . . . . . . 8 class Base
104, 9cfv 6559 . . . . . . 7 class (Base‘𝑦)
11 vz . . . . . . . 8 setvar 𝑧
1211cv 1534 . . . . . . 7 class 𝑧
138, 10, 12wf 6555 . . . . . 6 wff 𝑧:𝑡⟶(Base‘𝑦)
14 cfn 8979 . . . . . 6 class Fin
1513, 7, 14wrex 3066 . . . . 5 wff 𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)
166, 15wa 395 . . . 4 wff (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))
1716, 3, 11copab 5212 . . 3 class {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))}
18 c1 11148 . . . . . . . . 9 class 1
19 vm . . . . . . . . . 10 setvar 𝑚
2019cv 1534 . . . . . . . . 9 class 𝑚
21 cfz 13538 . . . . . . . . 9 class ...
2218, 20, 21co 7426 . . . . . . . 8 class (1...𝑚)
232cv 1534 . . . . . . . . . 10 class 𝑥
24 c2nd 8007 . . . . . . . . . 10 class 2nd
2523, 24cfv 6559 . . . . . . . . 9 class (2nd𝑥)
2625cdm 5684 . . . . . . . 8 class dom (2nd𝑥)
27 vf . . . . . . . . 9 setvar 𝑓
2827cv 1534 . . . . . . . 8 class 𝑓
2922, 26, 28wf1o 6558 . . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)
30 vs . . . . . . . . 9 setvar 𝑠
3130cv 1534 . . . . . . . 8 class 𝑠
32 c1st 8006 . . . . . . . . . . . 12 class 1st
3323, 32cfv 6559 . . . . . . . . . . 11 class (1st𝑥)
34 cplusg 17288 . . . . . . . . . . 11 class +g
3533, 34cfv 6559 . . . . . . . . . 10 class (+g‘(1st𝑥))
36 vn . . . . . . . . . . 11 setvar 𝑛
37 cn 12258 . . . . . . . . . . 11 class
3836cv 1534 . . . . . . . . . . . . 13 class 𝑛
3938, 28cfv 6559 . . . . . . . . . . . 12 class (𝑓𝑛)
4039, 25cfv 6559 . . . . . . . . . . 11 class ((2nd𝑥)‘(𝑓𝑛))
4136, 37, 40cmpt 5233 . . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛)))
4235, 41, 18cseq 14029 . . . . . . . . 9 class seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))
4320, 42cfv 6559 . . . . . . . 8 class (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)
4431, 43wceq 1535 . . . . . . 7 wff 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)
4529, 44wa 395 . . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))
4645, 27wex 1774 . . . . 5 wff 𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))
47 cn0 12518 . . . . 5 class 0
4846, 19, 47wrex 3066 . . . 4 wff 𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))
4948, 30cio 6509 . . 3 class (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)))
502, 17, 49cmpt 5233 . 2 class (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))))
511, 50wceq 1535 1 wff FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))))
Colors of variables: wff setvar class
This definition is referenced by:  bj-finsumval0  37228
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