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Theorem fprodabs 14629
 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, 2syl6eleq 2708 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 6612 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 14576 . . . . . 6 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴)
65fveq2d 6152 . . . . 5 (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴))
74prodeq1d 14576 . . . . 5 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
86, 7eqeq12d 2636 . . . 4 (𝑎 = 𝑀 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
98imbi2d 330 . . 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 6612 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 14576 . . . . . 6 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴)
1211fveq2d 6152 . . . . 5 (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴))
1310prodeq1d 14576 . . . . 5 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
1412, 13eqeq12d 2636 . . . 4 (𝑎 = 𝑛 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)))
1514imbi2d 330 . . 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 6612 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 14576 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)
1817fveq2d 6152 . . . . 5 (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴))
1916prodeq1d 14576 . . . . 5 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
2018, 19eqeq12d 2636 . . . 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 330 . . 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 6612 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 14576 . . . . . 6 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴)
2423fveq2d 6152 . . . . 5 (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴))
2522prodeq1d 14576 . . . . 5 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
2624, 25eqeq12d 2636 . . . 4 (𝑎 = 𝑁 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
2726imbi2d 330 . . 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 6189 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
2928adantl 482 . . . . 5 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
30 fzsn 12325 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3130adantl 482 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3231prodeq1d 14576 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴))
33 simpr 477 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
34 uzid 11646 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3534, 2syl6eleqr 2709 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀𝑍)
36 fprodabs.3 . . . . . . . . . . . . 13 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
3736ralrimiva 2960 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
38 nfcsb1v 3530 . . . . . . . . . . . . . 14 𝑘𝑀 / 𝑘𝐴
3938nfel1 2775 . . . . . . . . . . . . 13 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
40 csbeq1a 3523 . . . . . . . . . . . . . 14 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
4140eleq1d 2683 . . . . . . . . . . . . 13 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
4239, 41rspc 3289 . . . . . . . . . . . 12 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
4337, 42mpan9 486 . . . . . . . . . . 11 ((𝜑𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4435, 43sylan2 491 . . . . . . . . . 10 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4544abscld 14109 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℝ)
4645recnd 10012 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℂ)
4729, 46eqeltrd 2698 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ)
48 prodsns 14627 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
4933, 47, 48syl2anc 692 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5032, 49eqtrd 2655 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5130prodeq1d 14576 . . . . . . . 8 (𝑀 ∈ ℤ → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
5251adantl 482 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
53 prodsns 14627 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5433, 44, 53syl2anc 692 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5552, 54eqtrd 2655 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = 𝑀 / 𝑘𝐴)
5655fveq2d 6152 . . . . 5 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = (abs‘𝑀 / 𝑘𝐴))
5729, 50, 563eqtr4rd 2666 . . . 4 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
5857expcom 451 . . 3 (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
59 simp3 1061 . . . . . . . 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 6632 . . . . . . . . . . 11 (𝑛 + 1) ∈ V
61 csbfv2g 6189 . . . . . . . . . . 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 2630 . . . . . . . . 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 6622 . . . . . . 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 477 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
67 elfzuz 12280 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ𝑀))
6867, 2syl6eleqr 2709 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘𝑍)
6968, 36sylan2 491 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7069adantlr 750 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7166, 70fprodp1s 14626 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴))
7271fveq2d 6152 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)))
73 fzfid 12712 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑀...𝑛) ∈ Fin)
74 elfzuz 12280 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7574, 2syl6eleqr 2709 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
7675, 36sylan2 491 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
7776adantlr 750 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
7873, 77fprodcl 14607 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ)
79 peano2uz 11685 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
8079, 2syl6eleqr 2709 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
81 nfcsb1v 3530 . . . . . . . . . . . . . 14 𝑘(𝑛 + 1) / 𝑘𝐴
8281nfel1 2775 . . . . . . . . . . . . 13 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
83 csbeq1a 3523 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8483eleq1d 2683 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8582, 84rspc 3289 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8637, 85mpan9 486 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
8780, 86sylan2 491 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
8878, 87absmuld 14127 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
8972, 88eqtrd 2655 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
90893adant3 1079 . . . . . . 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 14109 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ)
9291recnd 10012 . . . . . . . . 9 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ)
9366, 92fprodp1s 14626 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · (𝑛 + 1) / 𝑘(abs‘𝐴)))
94933adant3 1079 . . . . . . 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 2665 . . . . . 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 1261 . . . . 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 11690 . 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186  ⦋csb 3514  {csn 4148  ‘cfv 5847  (class class class)co 6604  ℂcc 9878  1c1 9881   + caddc 9883   · cmul 9885  ℤcz 11321  ℤ≥cuz 11631  ...cfz 12268  abscabs 13908  ∏cprod 14560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-prod 14561 This theorem is referenced by:  etransclem23  39778
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