Theorem fprodabs 15899

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 2838	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		oveq2 7361	. . . . . . 7 ⊢ (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
5	4	prodeq1d 15845	. . . . . 6 ⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴)
6	5	fveq2d 6830	. . . . 5 ⊢ (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴))
7	4	prodeq1d 15845	. . . . 5 ⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
8	6, 7	eqeq12d 2745	. . . 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 340	. . 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 7361	. . . . . . 7 ⊢ (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
11	10	prodeq1d 15845	. . . . . 6 ⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴)
12	11	fveq2d 6830	. . . . 5 ⊢ (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴))
13	10	prodeq1d 15845	. . . . 5 ⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
14	12, 13	eqeq12d 2745	. . . 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 340	. . 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 7361	. . . . . . 7 ⊢ (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
17	16	prodeq1d 15845	. . . . . 6 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)
18	17	fveq2d 6830	. . . . 5 ⊢ (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴))
19	16	prodeq1d 15845	. . . . 5 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
20	18, 19	eqeq12d 2745	. . . 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 340	. . 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 7361	. . . . . . 7 ⊢ (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
23	22	prodeq1d 15845	. . . . . 6 ⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴)
24	23	fveq2d 6830	. . . . 5 ⊢ (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴))
25	22	prodeq1d 15845	. . . . 5 ⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
26	24, 25	eqeq12d 2745	. . . 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 340	. . 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 6873	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ⦋𝑀 / 𝑘⦌(abs‘𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
29	28	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
30		fzsn 13487	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
31	30	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
32	31	prodeq1d 15845	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴))
33		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
34		uzid 12768	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
35	34, 2	eleqtrrdi 2839	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
36		fprodabs.3	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
37	36	ralrimiva 3121	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
38		nfcsb1v 3877	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴
39	38	nfel1 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ
40		csbeq1a 3867	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑀 → 𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
41	40	eleq1d 2813	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
42	39, 41	rspc 3567	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
43	37, 42	mpan9 506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
44	35, 43	sylan2 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
45	44	abscld 15364	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘⦋𝑀 / 𝑘⦌𝐴) ∈ ℝ)
46	45	recnd 11162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘⦋𝑀 / 𝑘⦌𝐴) ∈ ℂ)
47	29, 46	eqeltrd 2828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ)
48		prodsns 15897	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
49	33, 47, 48	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
50	32, 49	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
51	30	prodeq1d 15845	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
52	51	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
53		prodsns 15897	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
54	33, 44, 53	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
55	52, 54	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
56	55	fveq2d 6830	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
57	29, 50, 56	3eqtr4rd 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
58	57	expcom 413	. . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
59		simp3 1138	. . . . . . . 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		ovex 7386	. . . . . . . . . . 11 ⊢ (𝑛 + 1) ∈ V
61		csbfv2g 6873	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ V → ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴) = (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
62	60, 61	ax-mp 5	. . . . . . . . . 10 ⊢ ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴) = (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)
63	62	eqcomi 2738	. . . . . . . . 9 ⊢ (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴) = ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)
64	63	a1i 11	. . . . . . . 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 7371	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
67		elfzuz 13441	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀))
68	67, 2	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ 𝑍)
69	68, 36	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
70	69	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
71	66, 70	fprodp1s 15896	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴))
72	71	fveq2d 6830	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴)))
73		fzfid 13898	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑛) ∈ Fin)
74		elfzuz 13441	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
75	74, 2	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
76	75, 36	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
77	76	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
78	73, 77	fprodcl 15877	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ)
79		peano2uz 12820	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
80	79, 2	eleqtrrdi 2839	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
81		nfcsb1v 3877	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴
82	81	nfel1 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ
83		csbeq1a 3867	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → 𝐴 = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
84	83	eleq1d 2813	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
85	82, 84	rspc 3567	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
86	37, 85	mpan9 506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
87	80, 86	sylan2 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
88	78, 87	absmuld 15382	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴)) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)))
89	72, 88	eqtrd 2764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)))
90	89	3adant3 1132	. . . . . . 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)))
91	70	abscld 15364	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ)
92	91	recnd 11162	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ)
93	66, 92	fprodp1s 15896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)))
94	93	3adant3 1132	. . . . . . 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)))
95	65, 90, 94	3eqtr4d 2774	. . . . . 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))
96	95	3exp 1119	. . . . 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))))
97	96	com12 32	. . . 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))))
98	97	a2d 29	. . 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))))
99	9, 15, 21, 27, 58, 98	uzind4 12825	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
100	3, 99	mpcom 38	1 ⊢ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438  ⦋csb 3853  {csn 4579  ‘cfv 6486  (class class class)co 7353  ℂcc 11026  1c1 11029   + caddc 11031   · cmul 11033  ℤcz 12489  ℤ_≥cuz 12753  ...cfz 13428  abscabs 15159  ∏cprod 15828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-fz 13429  df-fzo 13576  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-prod 15829

This theorem is referenced by:  etransclem23  46239