Definition df-prod 15994

Description: Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sum 15775 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 15993	. 2 class ∏𝑘 ∈ 𝐴 𝐵
5		vm	. . . . . . . . 9 setvar 𝑚
6	5	cv 1569	. . . . . . . 8 class 𝑚
7		cuz 12888	. . . . . . . 8 class ℤ≥
8	6, 7	cfv 6533	. . . . . . 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 11125	. . . . . . . . . 10 class 0
13	11, 12	wne 2955	. . . . . . . . 9 wff 𝑦 ≠ 0
14		cmul 11130	. . . . . . . . . . 11 class ·
15		cz 12616	. . . . . . . . . . . 12 class ℤ
16	3	cv 1569	. . . . . . . . . . . . . 14 class 𝑘
17	16, 1	wcel 2145	. . . . . . . . . . . . 13 wff 𝑘 ∈ 𝐴
18		c1 11126	. . . . . . . . . . . . 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 14066	. . . . . . . . . 10 class seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
24		cli 15572	. . . . . . . . . 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 3086	. . . . . 6 wff ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
29	14, 20, 6	cseq 14066	. . . . . . 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 3086	. . . 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 13562	. . . . . . . . 9 class ...
36	18, 6, 35	co 7414	. . . . . . . 8 class (1...𝑚)
37		vf	. . . . . . . . 9 setvar 𝑓
38	37	cv 1569	. . . . . . . 8 class 𝑓
39	36, 1, 38	wf1o 6532	. . . . . . 7 wff 𝑓:(1...𝑚)–1-1-onto→𝐴
40		cn 12258	. . . . . . . . . . 11 class ℕ
41	22, 38	cfv 6533	. . . . . . . . . . . 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 14066	. . . . . . . . 9 class seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
45	6, 44	cfv 6533	. . . . . . . 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 3086	. . . 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 6487	. 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  15995  prodeq1f  15996  prodeq1  15997  nfcprod1  15998  nfcprod  15999  prodeq2w  16000  prodeq2ii  16001  cbvprod  16003  cbvprodv  16004  prodeq1i  16006  prodeq2sdv  16012  zprod  16025  fprod  16029  prodeq2si  36825  prodeq12sdv  36839  cbvprodvw2  36868  cbvproddavw  36901  cbvproddavw2  36917