Definition df-prod 15952

Description: Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sum 15735 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.)

Assertion

Ref	Expression
df-prod	⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))

Distinct variable groups:   𝑓,𝑘,𝑚,𝑛,𝑥,𝑦   𝐴,𝑓,𝑚,𝑛,𝑥,𝑦   𝐵,𝑓,𝑚,𝑛,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)

Detailed syntax breakdown of Definition df-prod

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		vk	. . 3 setvar 𝑘
4	1, 2, 3	cprod 15951	. 2 class ∏𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1536	. . . . . . . 8 class 𝑚
7		cuz 12903	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 6573	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3976	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		vy	. . . . . . . . . . 11 setvar 𝑦
11	10	cv 1536	. . . . . . . . . 10 class 𝑦
12		cc0 11184	. . . . . . . . . 10 class 0
13	11, 12	wne 2946	. . . . . . . . 9 wff 𝑦 ≠ 0
14		cmul 11189	. . . . . . . . . . 11 class ·
15		cz 12639	. . . . . . . . . . . 12 class ℤ
16	3	cv 1536	. . . . . . . . . . . . . 14 class 𝑘
17	16, 1	wcel 2108	. . . . . . . . . . . . 13 wff 𝑘 ∈ 𝐴
18		c1 11185	. . . . . . . . . . . . 13 class 1
19	17, 2, 18	cif 4548	. . . . . . . . . . . 12 class if(𝑘 ∈ 𝐴, 𝐵, 1)
20	3, 15, 19	cmpt 5249	. . . . . . . . . . 11 class (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
21		vn	. . . . . . . . . . . 12 setvar 𝑛
22	21	cv 1536	. . . . . . . . . . 11 class 𝑛
23	14, 20, 22	cseq 14052	. . . . . . . . . 10 class seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
24		cli 15530	. . . . . . . . . 10 class ⇝
25	23, 11, 24	wbr 5166	. . . . . . . . 9 wff seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
26	13, 25	wa 395	. . . . . . . 8 wff (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
27	26, 10	wex 1777	. . . . . . 7 wff ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
28	27, 21, 8	wrex 3076	. . . . . 6 wff ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
29	14, 20, 6	cseq 14052	. . . . . . 7 class seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
30		vx	. . . . . . . 8 setvar 𝑥
31	30	cv 1536	. . . . . . 7 class 𝑥
32	29, 31, 24	wbr 5166	. . . . . 6 wff seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥
33	9, 28, 32	w3a 1087	. . . . 5 wff (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)
34	33, 5, 15	wrex 3076	. . . 4 wff ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)
35		cfz 13567	. . . . . . . . 9 class ...
36	18, 6, 35	co 7448	. . . . . . . 8 class (1...𝑚)
37		vf	. . . . . . . . 9 setvar 𝑓
38	37	cv 1536	. . . . . . . 8 class 𝑓
39	36, 1, 38	wf1o 6572	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
40		cn 12293	. . . . . . . . . . 11 class ℕ
41	22, 38	cfv 6573	. . . . . . . . . . . 12 class (𝑓‘𝑛)
42	3, 41, 2	csb 3921	. . . . . . . . . . 11 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
43	21, 40, 42	cmpt 5249	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
44	14, 43, 18	cseq 14052	. . . . . . . . 9 class seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
45	6, 44	cfv 6573	. . . . . . . 8 class (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
46	31, 45	wceq 1537	. . . . . . 7 wff 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
47	39, 46	wa 395	. . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
48	47, 37	wex 1777	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
49	48, 5, 40	wrex 3076	. . . 4 wff ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
50	34, 49	wo 846	. . 3 wff (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
51	50, 30	cio 6523	. 2 class (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
52	4, 51	wceq 1537	1 wff ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))

This definition is referenced by:  prodex  15953  prodeq1f  15954  prodeq1  15955  nfcprod1  15956  nfcprod  15957  prodeq2w  15958  prodeq2ii  15959  cbvprod  15961  cbvprodv  15962  prodeq1i  15964  prodeq2sdv  15971  zprod  15985  fprod  15989  prodeq2si  36168  prodeq12sdv  36184  cbvprodvw2  36213  cbvproddavw  36246  cbvproddavw2  36262