Definition df-prod 16073

Description: Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sum 15854 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 16072	. 2 class ∏𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1569	. . . . . . . 8 class 𝑚
7		cuz 12965	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 6538	. . . . . . 7 class (ℤ≥‘𝑚)
9	1, 8	wss 3899	. . . . . 6 wff 𝐴 ⊆ (ℤ≥‘𝑚)
10		vy	. . . . . . . . . . 11 setvar 𝑦
11	10	cv 1569	. . . . . . . . . 10 class 𝑦
12		cc0 11200	. . . . . . . . . 10 class 0
13	11, 12	wne 2956	. . . . . . . . 9 wff 𝑦 ≠ 0
14		cmul 11205	. . . . . . . . . . 11 class ·
15		cz 12693	. . . . . . . . . . . 12 class ℤ
16	3	cv 1569	. . . . . . . . . . . . . 14 class 𝑘
17	16, 1	wcel 2145	. . . . . . . . . . . . 13 wff 𝑘 ∈ 𝐴
18		c1 11201	. . . . . . . . . . . . 13 class 1
19	17, 2, 18	cif 4482	. . . . . . . . . . . 12 class if(𝑘 ∈ 𝐴, 𝐵, 1)
20	3, 15, 19	cmpt 5186	. . . . . . . . . . 11 class (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
21		vn	. . . . . . . . . . . 12 setvar 𝑛
22	21	cv 1569	. . . . . . . . . . 11 class 𝑛
23	14, 20, 22	cseq 14144	. . . . . . . . . 10 class seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
24		cli 15651	. . . . . . . . . 10 class ⇝
25	23, 11, 24	wbr 5103	. . . . . . . . 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 1812	. . . . . . 7 wff ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
28	27, 21, 8	wrex 3087	. . . . . 6 wff ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
29	14, 20, 6	cseq 14144	. . . . . . 7 class seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
30		vx	. . . . . . . 8 setvar 𝑥
31	30	cv 1569	. . . . . . 7 class 𝑥
32	29, 31, 24	wbr 5103	. . . . . 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 3087	. . . 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 13639	. . . . . . . . 9 class ...
36	18, 6, 35	co 7420	. . . . . . . 8 class (1...𝑚)
37		vf	. . . . . . . . 9 setvar 𝑓
38	37	cv 1569	. . . . . . . 8 class 𝑓
39	36, 1, 38	wf1o 6537	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
40		cn 12335	. . . . . . . . . . 11 class ℕ
41	22, 38	cfv 6538	. . . . . . . . . . . 12 class (𝑓‘𝑛)
42	3, 41, 2	csb 3847	. . . . . . . . . . 11 class ⦋(𝑓‘𝑛) / 𝑘⦌𝐵
43	21, 40, 42	cmpt 5186	. . . . . . . . . 10 class (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
44	14, 43, 18	cseq 14144	. . . . . . . . 9 class seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
45	6, 44	cfv 6538	. . . . . . . 8 class (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
46	31, 45	wceq 1570	. . . . . . 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 1812	. . . . 5 wff ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
49	48, 5, 40	wrex 3087	. . . 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 6492	. 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 1570	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  16074  prodeq1f  16075  prodeq1  16076  nfcprod1  16077  nfcprod  16078  prodeq2w  16079  prodeq2ii  16080  cbvprod  16082  cbvprodv  16083  prodeq1i  16085  prodeq2sdv  16091  zprod  16104  fprod  16108  prodeq2si  36993  prodeq12sdv  37007  cbvprodvw2  37036  cbvproddavw  37069  cbvproddavw2  37085