Theorem fprodabs 11619

Description: The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)

Hypotheses

Ref	Expression
fprodabs.1	⊢ 𝑍 = (ℤ≥‘𝑀)
fprodabs.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
fprodabs.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)

Assertion

Ref	Expression
fprodabs	⊢ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))

Distinct variable groups:   𝑘,𝑀   𝑘,𝑁   𝑘,𝑍   𝜑,𝑘

Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem fprodabs

Dummy variables 𝑎 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodabs.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		fprodabs.1	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
3	1, 2	eleqtrdi 2270	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		oveq2 5882	. . . . . . 7 ⊢ (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
5	4	prodeq1d 11567	. . . . . 6 ⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴)
6	5	fveq2d 5519	. . . . 5 ⊢ (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴))
7	4	prodeq1d 11567	. . . . 5 ⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
8	6, 7	eqeq12d 2192	. . . 4 ⊢ (𝑎 = 𝑀 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
9	8	imbi2d 230	. . 3 ⊢ (𝑎 = 𝑀 → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))))
10		oveq2 5882	. . . . . . 7 ⊢ (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
11	10	prodeq1d 11567	. . . . . 6 ⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴)
12	11	fveq2d 5519	. . . . 5 ⊢ (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴))
13	10	prodeq1d 11567	. . . . 5 ⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
14	12, 13	eqeq12d 2192	. . . 4 ⊢ (𝑎 = 𝑛 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)))
15	14	imbi2d 230	. . 3 ⊢ (𝑎 = 𝑛 → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))))
16		oveq2 5882	. . . . . . 7 ⊢ (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
17	16	prodeq1d 11567	. . . . . 6 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)
18	17	fveq2d 5519	. . . . 5 ⊢ (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴))
19	16	prodeq1d 11567	. . . . 5 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
20	18, 19	eqeq12d 2192	. . . 4 ⊢ (𝑎 = (𝑛 + 1) → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴)))
21	20	imbi2d 230	. . 3 ⊢ (𝑎 = (𝑛 + 1) → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))))
22		oveq2 5882	. . . . . . 7 ⊢ (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
23	22	prodeq1d 11567	. . . . . 6 ⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴)
24	23	fveq2d 5519	. . . . 5 ⊢ (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴))
25	22	prodeq1d 11567	. . . . 5 ⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
26	24, 25	eqeq12d 2192	. . . 4 ⊢ (𝑎 = 𝑁 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
27	26	imbi2d 230	. . 3 ⊢ (𝑎 = 𝑁 → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))))
28		csbfv2g 5552	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ⦋𝑀 / 𝑘⦌(abs‘𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
29	28	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
30		fzsn 10063	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
31	30	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
32	31	prodeq1d 11567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴))
33		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
34		uzid 9540	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
35	34, 2	eleqtrrdi 2271	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
36		fprodabs.3	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
37	36	ralrimiva 2550	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
38		nfcsb1v 3090	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴
39	38	nfel1 2330	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ
40		csbeq1a 3066	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑀 → 𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
41	40	eleq1d 2246	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
42	39, 41	rspc 2835	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
43	37, 42	mpan9 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
44	35, 43	sylan2 286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
45	44	abscld 11185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘⦋𝑀 / 𝑘⦌𝐴) ∈ ℝ)
46	45	recnd 7984	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘⦋𝑀 / 𝑘⦌𝐴) ∈ ℂ)
47	29, 46	eqeltrd 2254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ)
48		prodsns 11606	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
49	33, 47, 48	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
50	32, 49	eqtrd 2210	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
51	30	prodeq1d 11567	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
52	51	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
53		prodsns 11606	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
54	33, 44, 53	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
55	52, 54	eqtrd 2210	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
56	55	fveq2d 5519	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
57	29, 50, 56	3eqtr4rd 2221	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
58	57	expcom 116	. . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
59		simp3 999	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
60		peano2uz 9581	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
61		csbfv2g 5552	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘𝑀) → ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴) = (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
62	60, 61	syl 14	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴) = (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
63	62	eqcomd 2183	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴) = ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴))
64	63	3ad2ant2 1019	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴) = ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴))
65	59, 64	oveq12d 5892	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)))
66		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
67		elfzuz 10018	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀))
68	67, 2	eleqtrrdi 2271	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ 𝑍)
69	68, 36	sylan2 286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
70	69	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
71	66, 70	fprodp1s 11605	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴))
72	71	fveq2d 5519	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴)))
73		eluzel2 9531	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
74	73	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ)
75		eluzelz 9535	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
76	75	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ ℤ)
77	74, 76	fzfigd 10428	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑛) ∈ Fin)
78		elfzuz 10018	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
79	78, 2	eleqtrrdi 2271	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
80	79, 36	sylan2 286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
81	80	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
82	77, 81	fprodcl 11610	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ)
83	60, 2	eleqtrrdi 2271	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
84		nfcsb1v 3090	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴
85	84	nfel1 2330	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ
86		csbeq1a 3066	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → 𝐴 = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
87	86	eleq1d 2246	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
88	85, 87	rspc 2835	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
89	37, 88	mpan9 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
90	83, 89	sylan2 286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
91	82, 90	absmuld 11198	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴)) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)))
92	72, 91	eqtrd 2210	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)))
93	92	3adant3 1017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)))
94	70	abscld 11185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ)
95	94	recnd 7984	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ)
96	66, 95	fprodp1s 11605	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)))
97	96	3adant3 1017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)))
98	65, 93, 97	3eqtr4d 2220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
99	98	3exp 1202	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑀) → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))))
100	99	com12 30	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝜑 → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))))
101	100	a2d 26	. . 3 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))))
102	9, 15, 21, 27, 58, 101	uzind4 9586	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
103	3, 102	mpcom 36	1 ⊢ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ⦋csb 3057  {csn 3592  ‘cfv 5216  (class class class)co 5874  ℂcc 7808  1c1 7811   + caddc 7813   · cmul 7815  ℤcz 9251  ℤ_≥cuz 9526  ...cfz 10006  abscabs 11001  ∏cprod 11553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-proddc 11554

This theorem is referenced by: (None)