Theorem fprodabs 15322

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 2923	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		oveq2 7158	. . . . . . 7 ⊢ (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
5	4	prodeq1d 15269	. . . . . 6 ⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴)
6	5	fveq2d 6668	. . . . 5 ⊢ (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴))
7	4	prodeq1d 15269	. . . . 5 ⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
8	6, 7	eqeq12d 2837	. . . 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 343	. . 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 7158	. . . . . . 7 ⊢ (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
11	10	prodeq1d 15269	. . . . . 6 ⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴)
12	11	fveq2d 6668	. . . . 5 ⊢ (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴))
13	10	prodeq1d 15269	. . . . 5 ⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
14	12, 13	eqeq12d 2837	. . . 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 343	. . 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 7158	. . . . . . 7 ⊢ (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
17	16	prodeq1d 15269	. . . . . 6 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)
18	17	fveq2d 6668	. . . . 5 ⊢ (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴))
19	16	prodeq1d 15269	. . . . 5 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
20	18, 19	eqeq12d 2837	. . . 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 343	. . 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 7158	. . . . . . 7 ⊢ (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
23	22	prodeq1d 15269	. . . . . 6 ⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴)
24	23	fveq2d 6668	. . . . 5 ⊢ (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴))
25	22	prodeq1d 15269	. . . . 5 ⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
26	24, 25	eqeq12d 2837	. . . 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 343	. . 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 6708	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ⦋𝑀 / 𝑘⦌(abs‘𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
29	28	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
30		fzsn 12943	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
31	30	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
32	31	prodeq1d 15269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴))
33		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
34		uzid 12252	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
35	34, 2	eleqtrrdi 2924	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
36		fprodabs.3	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
37	36	ralrimiva 3182	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
38		nfcsb1v 3906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴
39	38	nfel1 2994	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ
40		csbeq1a 3896	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑀 → 𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
41	40	eleq1d 2897	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
42	39, 41	rspc 3610	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
43	37, 42	mpan9 509	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
44	35, 43	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
45	44	abscld 14790	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘⦋𝑀 / 𝑘⦌𝐴) ∈ ℝ)
46	45	recnd 10663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘⦋𝑀 / 𝑘⦌𝐴) ∈ ℂ)
47	29, 46	eqeltrd 2913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ)
48		prodsns 15320	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
49	33, 47, 48	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
50	32, 49	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴))
51	30	prodeq1d 15269	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
52	51	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
53		prodsns 15320	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
54	33, 44, 53	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
55	52, 54	eqtrd 2856	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
56	55	fveq2d 6668	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴))
57	29, 50, 56	3eqtr4rd 2867	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
58	57	expcom 416	. . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
59		simp3 1134	. . . . . . . 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 7183	. . . . . . . . . . 11 ⊢ (𝑛 + 1) ∈ V
61		csbfv2g 6708	. . . . . . . . . . 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 2830	. . . . . . . . 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 7168	. . . . . . 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 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
67		elfzuz 12898	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀))
68	67, 2	eleqtrrdi 2924	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ 𝑍)
69	68, 36	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
70	69	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
71	66, 70	fprodp1s 15319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴))
72	71	fveq2d 6668	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴)))
73		fzfid 13335	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑛) ∈ Fin)
74		elfzuz 12898	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
75	74, 2	eleqtrrdi 2924	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
76	75, 36	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
77	76	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
78	73, 77	fprodcl 15300	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ)
79		peano2uz 12295	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
80	79, 2	eleqtrrdi 2924	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
81		nfcsb1v 3906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴
82	81	nfel1 2994	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ
83		csbeq1a 3896	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → 𝐴 = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
84	83	eleq1d 2897	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
85	82, 84	rspc 3610	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
86	37, 85	mpan9 509	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
87	80, 86	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
88	78, 87	absmuld 14808	. . . . . . . . 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 2856	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)))
90	89	3adant3 1128	. . . . . . 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 14790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ)
92	91	recnd 10663	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ)
93	66, 92	fprodp1s 15319	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)))
94	93	3adant3 1128	. . . . . . 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 2866	. . . . . 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 1115	. . . . 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 12300	. 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 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  ⦋csb 3882  {csn 4560  ‘cfv 6349  (class class class)co 7150  ℂcc 10529  1c1 10532   + caddc 10534   · cmul 10536  ℤcz 11975  ℤ_≥cuz 12237  ...cfz 12886  abscabs 14587  ∏cprod 15253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-fz 12887  df-fzo 13028  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-prod 15254

This theorem is referenced by:  etransclem23  42536