Definition df-prod 16085

Description: Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sum 15866 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 16084	. 2 class ∏𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1568	. . . . . . . 8 class 𝑚
7		cuz 12975	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 6546	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3899	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		vy	. . . . . . . . . . 11 setvar 𝑦
11	10	cv 1568	. . . . . . . . . 10 class 𝑦
12		cc0 11210	. . . . . . . . . 10 class 0
13	11, 12	wne 2958	. . . . . . . . 9 wff 𝑦 ≠ 0
14		cmul 11215	. . . . . . . . . . 11 class ·
15		cz 12703	. . . . . . . . . . . 12 class ℤ
16	3	cv 1568	. . . . . . . . . . . . . 14 class 𝑘
17	16, 1	wcel 2146	. . . . . . . . . . . . 13 wff 𝑘 ∈ 𝐴
18		c1 11211	. . . . . . . . . . . . 13 class 1
19	17, 2, 18	cif 4484	. . . . . . . . . . . 12 class if(𝑘 ∈ 𝐴, 𝐵, 1)
20	3, 15, 19	cmpt 5188	. . . . . . . . . . 11 class (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
21		vn	. . . . . . . . . . . 12 setvar 𝑛
22	21	cv 1568	. . . . . . . . . . 11 class 𝑛
23	14, 20, 22	cseq 14154	. . . . . . . . . 10 class seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
24		cli 15663	. . . . . . . . . 10 class ⇝
25	23, 11, 24	wbr 5105	. . . . . . . . 9 wff seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
26	13, 25	wa 401	. . . . . . . 8 wff (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
27	26, 10	wex 1811	. . . . . . 7 wff ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
28	27, 21, 8	wrex 3089	. . . . . 6 wff ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
29	14, 20, 6	cseq 14154	. . . . . . 7 class seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
30		vx	. . . . . . . 8 setvar 𝑥
31	30	cv 1568	. . . . . . 7 class 𝑥
32	29, 31, 24	wbr 5105	. . . . . 6 wff seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥
33	9, 28, 32	w3a 1103	. . . . 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 3089	. . . 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 13649	. . . . . . . . 9 class ...
36	18, 6, 35	co 7427	. . . . . . . 8 class (1...𝑚)
37		vf	. . . . . . . . 9 setvar 𝑓
38	37	cv 1568	. . . . . . . 8 class 𝑓
39	36, 1, 38	wf1o 6545	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
40		cn 12345	. . . . . . . . . . 11 class ℕ
41	22, 38	cfv 6546	. . . . . . . . . . . 12 class (𝑓‘𝑛)
42	3, 41, 2	csb 3850	. . . . . . . . . . 11 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
43	21, 40, 42	cmpt 5188	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
44	14, 43, 18	cseq 14154	. . . . . . . . 9 class seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
45	6, 44	cfv 6546	. . . . . . . 8 class (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
46	31, 45	wceq 1569	. . . . . . 7 wff 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
47	39, 46	wa 401	. . . . . 6 wff (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
48	47, 37	wex 1811	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
49	48, 5, 40	wrex 3089	. . . 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 861	. . 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 6498	. 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 1569	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 used by:  prodex  16086  prodeq1f  16087  prodeq1  16088  nfcprod1  16089  nfcprod  16090  prodeq2w  16091  prodeq2ii  16092  cbvprod  16094  cbvprodv  16095  prodeq1i  16097  prodeq2sdv  16103  zprod  16116  fprod  16120  prodeq2si  37120  prodeq12sdv  37134  cbvprodvw2  37163  cbvproddavw  37196  cbvproddavw2  37212