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Theorem fprodabs 16010
Description: The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)
Hypotheses
Ref Expression
fprodabs.1 𝑍 = (ℤ𝑀)
fprodabs.2 (𝜑𝑁𝑍)
fprodabs.3 ((𝜑𝑘𝑍) → 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.
StepHypRef Expression
1 fprodabs.2 . . 3 (𝜑𝑁𝑍)
2 fprodabs.1 . . 3 𝑍 = (ℤ𝑀)
31, 2eleqtrdi 2851 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 7439 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 15956 . . . . . 6 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴)
65fveq2d 6910 . . . . 5 (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴))
74prodeq1d 15956 . . . . 5 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
86, 7eqeq12d 2753 . . . 4 (𝑎 = 𝑀 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
98imbi2d 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 7439 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 15956 . . . . . 6 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴)
1211fveq2d 6910 . . . . 5 (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴))
1310prodeq1d 15956 . . . . 5 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
1412, 13eqeq12d 2753 . . . 4 (𝑎 = 𝑛 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)))
1514imbi2d 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 7439 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 15956 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)
1817fveq2d 6910 . . . . 5 (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴))
1916prodeq1d 15956 . . . . 5 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
2018, 19eqeq12d 2753 . . . 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)))
2120imbi2d 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 7439 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 15956 . . . . . 6 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴)
2423fveq2d 6910 . . . . 5 (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴))
2522prodeq1d 15956 . . . . 5 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
2624, 25eqeq12d 2753 . . . 4 (𝑎 = 𝑁 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
2726imbi2d 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 6955 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
2928adantl 481 . . . . 5 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
30 fzsn 13606 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3130adantl 481 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3231prodeq1d 15956 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴))
33 simpr 484 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
34 uzid 12893 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3534, 2eleqtrrdi 2852 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀𝑍)
36 fprodabs.3 . . . . . . . . . . . . 13 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
3736ralrimiva 3146 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
38 nfcsb1v 3923 . . . . . . . . . . . . . 14 𝑘𝑀 / 𝑘𝐴
3938nfel1 2922 . . . . . . . . . . . . 13 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
40 csbeq1a 3913 . . . . . . . . . . . . . 14 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
4140eleq1d 2826 . . . . . . . . . . . . 13 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
4239, 41rspc 3610 . . . . . . . . . . . 12 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
4337, 42mpan9 506 . . . . . . . . . . 11 ((𝜑𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4435, 43sylan2 593 . . . . . . . . . 10 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4544abscld 15475 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℝ)
4645recnd 11289 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℂ)
4729, 46eqeltrd 2841 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ)
48 prodsns 16008 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
4933, 47, 48syl2anc 584 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5032, 49eqtrd 2777 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5130prodeq1d 15956 . . . . . . . 8 (𝑀 ∈ ℤ → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
5251adantl 481 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
53 prodsns 16008 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5433, 44, 53syl2anc 584 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5552, 54eqtrd 2777 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = 𝑀 / 𝑘𝐴)
5655fveq2d 6910 . . . . 5 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = (abs‘𝑀 / 𝑘𝐴))
5729, 50, 563eqtr4rd 2788 . . . 4 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
5857expcom 413 . . 3 (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
59 simp3 1139 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
60 ovex 7464 . . . . . . . . . . 11 (𝑛 + 1) ∈ V
61 csbfv2g 6955 . . . . . . . . . . 11 ((𝑛 + 1) ∈ V → (𝑛 + 1) / 𝑘(abs‘𝐴) = (abs‘(𝑛 + 1) / 𝑘𝐴))
6260, 61ax-mp 5 . . . . . . . . . 10 (𝑛 + 1) / 𝑘(abs‘𝐴) = (abs‘(𝑛 + 1) / 𝑘𝐴)
6362eqcomi 2746 . . . . . . . . 9 (abs‘(𝑛 + 1) / 𝑘𝐴) = (𝑛 + 1) / 𝑘(abs‘𝐴)
6463a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘(𝑛 + 1) / 𝑘𝐴) = (𝑛 + 1) / 𝑘(abs‘𝐴))
6559, 64oveq12d 7449 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · (𝑛 + 1) / 𝑘(abs‘𝐴)))
66 simpr 484 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
67 elfzuz 13560 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ𝑀))
6867, 2eleqtrrdi 2852 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘𝑍)
6968, 36sylan2 593 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7069adantlr 715 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7166, 70fprodp1s 16007 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴))
7271fveq2d 6910 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)))
73 fzfid 14014 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑀...𝑛) ∈ Fin)
74 elfzuz 13560 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7574, 2eleqtrrdi 2852 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
7675, 36sylan2 593 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
7776adantlr 715 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
7873, 77fprodcl 15988 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ)
79 peano2uz 12943 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
8079, 2eleqtrrdi 2852 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
81 nfcsb1v 3923 . . . . . . . . . . . . . 14 𝑘(𝑛 + 1) / 𝑘𝐴
8281nfel1 2922 . . . . . . . . . . . . 13 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
83 csbeq1a 3913 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8483eleq1d 2826 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8582, 84rspc 3610 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8637, 85mpan9 506 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
8780, 86sylan2 593 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
8878, 87absmuld 15493 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
8972, 88eqtrd 2777 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
90893adant3 1133 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
9170abscld 15475 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ)
9291recnd 11289 . . . . . . . . 9 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ)
9366, 92fprodp1s 16007 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · (𝑛 + 1) / 𝑘(abs‘𝐴)))
94933adant3 1133 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (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)))
9565, 90, 943eqtr4d 2787 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
96953exp 1120 . . . . 5 (𝜑 → (𝑛 ∈ (ℤ𝑀) → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))))
9796com12 32 . . . 4 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))))
9897a2d 29 . . 3 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))))
999, 15, 21, 27, 58, 98uzind4 12948 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
1003, 99mpcom 38 1 (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  csb 3899  {csn 4626  cfv 6561  (class class class)co 7431  cc 11153  1c1 11156   + caddc 11158   · cmul 11160  cz 12613  cuz 12878  ...cfz 13547  abscabs 15273  cprod 15939
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-prod 15940
This theorem is referenced by:  etransclem23  46272
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